<s1>[B169] The Transcendental Analytic

The structure of general logic corresponds exactly to the division between the higher cognitive faculties. These are understanding, judgment, and reason. So, in its analytic part, the discipline of logic deals with concepts, judgments, and inferences, exactly in accordance with the functions and order of these cognitive faculties — though we usually use the term ‘understanding’ in a wider sense, to cover all three.

<s2> [B170] This purely formal logic ignores the content of knowledge (whether pure or empirical), and is concerned only with the form of thought (i.e. inferential knowledge) in general. Consequently, in its analytic part, it can also cover the basic set of rules which reason must obey. This is because the form of reason has its own fixed prescription, which we can have insight into apriori, simply by analysing rational functions into their elements, and without considering the particular nature of the knowledge involved.

<s3> Transcendental logic cannot imitate general logic in such an analysis, since it is restricted to a specific content, namely pure apriori knowledge. For it emerges that the transcendental use of reason is not objectively valid at all, and hence does not belong to a logic of truth, i.e. to the Analytic. Instead, as a logic of illusion, it requires a separate part of the curriculum, called the Transcendental Dialectic.

<s4> So understanding and judgment do belong to the analytic part of transcendental logic, because it contains the set of rules for their objectively valid, and hence true use. Only reason is utterly dialectical, in its ambition to discover something about objects apriori, and to extend knowledge beyond the limits of possible experience. [B171] Its illusory assertions are utterly incompatible with the sort of rules the Analytic must contain.

<s5> So the Analytic of Axioms will be a set of rules only for the faculty of judgment. They tell it how the concepts of understanding, which contain the precondition for rules apriori, are to be applied to appearances. Consequently, while I am discussing what are properly the axioms of understanding, I shall indicate my purpose more accurately by calling it the doctrine of judgment.

<s6> Introduction

If the understanding in general is defined as the faculty of rules, then judgment is the faculty of subsuming under rules — that is, of deciding whether or not something falls under a given rule (or is an instance to which a given law applies). General logic does not and cannot include any prescriptions for judgment. For since it takes no account of any content of knowledge, the only thing left for it to do is to dissect analytically [B172] the mere form of knowledge in concepts, judgments, and inferences, and thus to establish formal rules for all use of the understanding.

<s7> Now if logic aimed to show, quite universally, how one is to subsume things under these rules (i.e. to decide whether something comes under them or not), then this could be done only by means of yet another rule. But precisely because it is a rule, the faculty of judgment would need a completely new instruction as to how to apply it. So it emerges that, while the understanding is capable of being taught and educated through rules, judgment is a special innate gift, which cannot be taught at all, but only exercised.

<s8> Hence judgment is also the essential ingredient of so-called ‘mother wit’, which no teaching can replace the lack of. Teaching can provide people who have limited understandings with abundant rules drawn from the insights of others, and, as it were, graft these rules into their understandings. However, the ability to take advantage of these rules must lie within the learners themselves. If they lack the natural gift of judgment, they will always be liable to misapply any rule they may be prescribed for how to take advantage of rules.*

<s10> So a doctor, a [B173] lawyer, or a political scientist can have many fine medical, judicial, or political rules in their heads, to the extent that they themselves can teach their disciplines from first principles; yet they can easily go wrong in the application of these rules. One way they can go wrong is because they lack innate judgment (though not understanding); and although they have insight into the general in abstract, they cannot decide whether a particular case comes under the general. The other way they can go wrong is because they are not sufficiently trained to make such judgments through examples and case studies.

<s11> The one great benefit of examples is that they sharpen the faculty of judgment. But as far as the correctness and precision of the insights of the understanding are concerned, examples are usually harmful, since it is rare that they completely fulfil the conditions specified in a rule (as particular cases falling within the terms of the rule). In addition, examples often make the understanding less inclined to strive for insight into its rules in general, despite the fact that its rules are self-sufficient, and do not depend on the particular circumstances of experience. The consequence is that we tend to use the rules of understanding as a check-list rather than as a set of fundamental axioms. So examples are the [B174] baby-walker of judgment; but no-one can do without them if they lack judgment as an innate gift.

<s12> Now although general logic cannot give any prescriptions to judgment, the situation is quite different with transcendental logic — so much so, that the essential function of transcendental logic seems to be that of correcting or confirming the faculty of judgment in its use of pure understanding, through specific rules. For if the role of philosophy is to extend the scope of understanding in the realm of pure apriori knowledge, and hence to be a discipline with positive content, then it seems that philosophy is quite unnecessary — or rather that it is wrong to bring it in at all. This is because, despite all the efforts made up till now, little or no territory has been gained by using philosophy. But if philosophy is a critique, then its sharp-wittedness and technique of scrutiny will put it in demand for preventing errors of judgment in the use of the few pure concepts of understanding that we have (even though its usefulness is then only negative).

<s13> However, transcendental philosophy has a unique feature: not only can it point out the rule which is given in the relevant pure concept of understanding (or rather the universal precondition for rules), but it can also point out apriori the instance to which the rules must be applied. [B175] This makes transcendental philosophy superior in this respect to all other disciplines (except maths). The source of its superiority lies precisely in the fact that it deals with concepts which must relate to their objects apriori. Hence its objective validity cannot be established aposteriori, since any attempt to establish its validity in this way would completely fail to do justice to its exalted status. Instead, transcendental philosophy must specify not only the concepts, but also the universal but sufficient defining characteristics of the preconditions under which objects can be given in conformity with these concepts. For otherwise the concepts would lack any content, and hence would be mere logical forms rather than pure concepts of understanding.

<s14> Now this transcendental doctrine of judgment will contain two chapters. The first is the Schematism of the Pure Understanding, which deals with the sensory preconditions under which alone pure concepts of understanding can be used. The second is the Axioms of the Pure Understanding, which deals with the synthetic judgments which flow apriori from pure concepts of understanding under these preconditions, and which are the basis for all other apriori knowledge.

<s15>[B176]The Transcendental Doctrine of Judgment

Whenever an object is brought under a concept, the representation of the object must have something in common with the representation of the concept. In other words, the concept must include whatever is represented as coming under it in the object. For this is precisely what is meant by the maxim that an object is included under a concept. For example, the empirical concept of a plate has something in common with the pure geometrical concept of a circle, in that the circularity which is thought in the plate can be intuited in the circle.

<s16> Now pure concepts of understanding can never be found in any intuition, since they have nothing at all in common with empirical intuitions (or indeed with sensory intuitions in general). So how is it possible for empirical intuitions to come under pure concepts, and hence for a category to be applied to appearances? For no-one will say that a particular category, for example that of causality, could also be discerned by the senses, [B177] and be included in appearance. And this question, which is so natural and yet so profound, is the essential reason which makes a transcendental doctrine of judgment necessary — namely so that it will be shown, quite generally, how it is possible for pure concepts of understanding to be applied to appearances. In no other discipline is it necessary to give a detailed explanation of how general concepts apply to its subject matter, because the concepts through which the subject matter is thought in general are not so heterogeneous and distinct from those which represent it as it is given in concrete detail.

<s17> Now it is obvious that there must be some third thing, which has to have something in common both with the category and with the appearance, and which makes it possible to apply the former to the latter. This mediating representation will have to be pure (free from anything empirical), and yet both intellectual and sensory. Such a thing is the transcendental schema.

<s18> A concept of the understanding contains pure synthetic unity of a multiplicity in general. Time contains an apriori multiplicity in pure intuition, since it is the formal precondition of the multiplicity of inner sense, and hence of the connecting together of all representations. Now a transcendental determination of time has something in common with a category (which brings about its unity), in so far as it is universal and is grounded in an apriori rule. [B178] On the other hand, it has something in common with an appearance, in so far as time is involved in every empirical representation of the multiplicity. Hence a category can be applied to appearances through the mediation of a transcendental determination of time, since, as the schema of the concepts of understanding, it mediates the bringing of appearances under the category.

<s19> After what has been proved in the Deduction of the Categories, I hope that no-one will be in any doubt over the answer to the following question: namely whether these pure concepts of the understanding are merely of empirical use, or whether they are also of transcendental use. To put it another way: whether they relate to appearances apriori only as preconditions of a possible experience, or whether, as preconditions of the possibility of things in general, they can be extended to objects in themselves, without any restriction to what we perceive through our senses.

<s20> For we saw that concepts cannot exist, or have any significance whatever, where no object is given, either for the concepts themselves, or at least for the elements of which they are composed. Hence concepts certainly cannot extend to things in themselves, without reference to whether or how these things might be given to us. We also saw that the only way in which objects are given to us is through modulation of our senses. Finally, we saw that pure apriori concepts must contain apriori, not only the [B179] function of the understanding instantiated in the category, but also formal preconditions of sensibility, namely those of inner sense. These preconditions include the universal and necessary precondition for a category to be applicable to any object.

<s21> I shall use the expression ‘the schema of the concept of understanding’ for this formal and pure precondition of sensibility, to which the application of the concept of the understanding is restricted. And I shall use the expression ‘the schematism of pure understanding’ for what the understanding does with these schemata.

<s22> In itself, a schema is never anything other than a product of the imagination. However, a schema must not be confused with an image, since the only goal of the synthesis performed by the imagination is not a single intuition, but unity in the process of reducing sensibility in general to something specific. So if I place five points in a row, thus: · · · · · — then this is an image of the number five. On the other hand, if I think only of a number in general (which could be five or a hundred), then this thought is the representation of a method for representing a multitude (e.g. a thousand) by means of an image which corresponds to a particular concept, rather than the image itself. If it were the image of a thousand, I would hardly be able to review it, and compare it with the concept. So what I call the ‘schema’ of a concept is this representation of a general process by which the imagination generates an image corresponding to a concept.

<s23> [B180] In fact, our pure sensory concepts are based on schemata, and not on images of objects. No image of a triangle in general could ever correspond adequately to our concept of it. It could never do justice to the generality of the concept, because the concept is valid for all triangles, whether right-angled, acute, etc., whereas an image would always be limited to one of this range of possibilities. The schema of a triangle can exist only in thought, and it amounts to a rule determining how the imagination is to synthesise shapes in space.

<s24> It is even less the case that a particular object of experience (or an image of it) ever fully encapsulates its empirical concept. Rather, the concept always relates immediately to the schema of the imagination, as a rule for generating a particular intuition in accordance with a specific general concept. The concept of a dog indicates a rule, in accordance with which my imagination has the general ability to draw the figure of such a four-footed animal, without being restricted to any one particular figure which experience presents me with, or even to any possible particular image which I can form.

<s25> This schematism of our understanding, in respect of appearances and their form alone, is an art hidden in the depths of the human soul. It is only with difficulty that we can ever divine its actual [B181] manner of working by natural means, and lay it bare before our eyes. All we can say is that an image is a product of the empirical faculty of the productive imagination, whereas a schema of sensory concepts (such as of shapes in space) is a product of pure imagination apriori, and is a sort of outline sketch. It is only through and in accordance with the schema that images first become possible. However, the images can never be connected with the concept other than through the mediation of the schema which they indicate; and in themselves they are not fully congruent to the schema.

<s26> By contrast, the schema of a pure concept of understanding is something which cannot be brought into relation with any image at all. The schema is only the pure synthesis, in conformity with a rule of unity for concepts in general, which the category expresses. It is a transcendental product of the imagination, which deals with the determination of inner sense in general, in accordance with the preconditions imposed by its form (namely time) on all representations. This is because representations must be connected together apriori under a single concept, in order to conform to the unity of apperception.

<s27> I shall not waste time on a tedious and boring analysis of what is required for transcendental schemata of pure concepts of understanding in general. Instead, I shall set them out in the order of the categories, and in connection with them.

<s28> [B182] The pure image of all quanta which are objects of outer sense is space; and that of all objects of the senses whatever is time. The pure schema of quantity, as a concept of the understanding, is number, which is a representation which treats as a whole the successive addition of one unit to another (of the same kind). Thus number is nothing other than the unity of the synthesis of the multiplicity of a uniform intuition in general, because I create time itself in the apprehension of intuition.

<s29> In the pure concept of the understanding, reality is that which corresponds to a sensation in general, and hence is that whose concept in itself indicates a being (in time). Negation is that whose concept represents an absence of being (in time). Thus the opposition between the two arises from the contrast between one and the same time being a filled or an empty time. But time is only the form of intuition, and hence of objects as appearances. Consequently, that in appearances which corresponds to sensation is the transcendental matter of all objects as things in themselves (i.e. thingliness, or reality).

<s30> Now every sensation has a quantity or degree to which it fills one and the same time, i.e. inner sense in respect of one and the same representation of an object. This can be greater or less, until it disappears into nothing (= 0 = negation). Hence there is a relation and connection between reality and negation, or rather [B183] a transition from the one to the other, which makes it possible to represent every reality as a quantum. And the schema of a reality, as the quantity of something in so far is it fills time, is just this continuous and uniform production of the reality in time. One can either start with a sensation which has a certain degree of reality, and then diminishes over time to a point when it disappears; or one can start with negation, and then it gradually increases to its full quantity.

<s31> The schema of substance is the permanence of the real in time. In other words, it is the representation of substance as a substrate of the empirical determination of time in general, and hence as that which remains the same, while everything else changes. (Time does not pass; but the existence of changeable things passes away in time. So, since time is itself unchanging and enduring, what corresponds to it in appearance is that which is unchanging in its existence, namely substance. And it is only by reference to substance that it can be determined whether appearances are successive or co-existent in time.)

<s32> The schema of cause, and of the causality of a thing in general, is the real which, whenever it is supposed, something else always follows. So it consists in the successiveness of the multiplicity, in so far as it is subject to a rule.

<s33> The schema of community (mutual interaction), or the reciprocal causality of substances in respect of their accidents, is the simultaneity of the determinations [B184] of the one with those of the other, in accordance with a general rule.

<s34> The schema of possibility is the correspondence of the synthesis of different representations with the preconditions of time in general. For example, contradictories cannot exist in one and the same thing at the same time, but only one after another. So this schema is the determination of the representation of a thing to some time or other.

<s35> The schema of actuality is existence at a determinate time.

<s36> The schema of necessity is the existence of an object at every time.

<s37 It can be seen from all this, that the schema of each category contains and makes representable the following:

• in the case of the schema of quantity, the production (synthesis) of time itself, through the successive apprehension of an object;
• in the case of the schema of quality, the synthesis of sensation (perception) with the representation of time, or the filling of time;
• in the case of the schema of relation, the relationship of perceptions to each other at every time (that is, in accordance with a rule for the determination of time);
• finally, in the case of the schema of modality and its categories, time itself, as the correlate of determining whether and how an object belongs to time.

Hence the schemata are nothing other than apriori determinations of time in accordance with rules. These rules apply to all possible objects, and (following the order of the categories), they concern the series of time, the content of time, the order of time, [B185] and finally the totality of time.

<s38> Now from this it is clear that the schematism of the understanding through the transcendental synthesis of the imagination amounts to nothing other than the unity of every multiplicity of intuition in inner sense. Hence it amounts indirectly to the unity of apperception, as the function which corresponds to the inner sense (which is merely receptive). Thus the schemata of the pure concepts of understanding are the true and only preconditions for providing these concepts with a relationship to objects, and hence for giving them meaning.

<s39> So, ultimately, the categories have no application outside any possible experience. Their only role is to subject appearances to the universal rules of synthesis, grounded in a synthesis which is apriori necessary, because it is necessary for all consciousness to be unified in an originative apperception. In this way, the categories make appearances amenable to holistic connection in a single experience.

<s40> However, all our knowledge lies within the totality of all possible experience. Transcendental truth consists in this universal relationship to possible experience, and it precedes and makes possible all empirical truth.

<s41> Another observation is that, although it is the schemata of sensibility which first actualise the categories, [B186] yet they themselves also restrict them. That is, they limit them to requirements which lie outside the understanding — namely in sensibility. Hence the schema is essentially only the phenomenon, or the sensory concept of an object, in conformity with the category. (Number is the quantity of phenomena, sensation is the reality of phenomena, the constant and permanent in things is the substance of phenomena, — and eternity is the necessity of phenomena, etc.)

<s42> Now if we leave out a restricting requirement, it might seem that we are extending a previously curtailed concept. So the categories, in their pure meaning and without any requirements arising from sensibility, should be valid of things in general as they are (instead of their schemata representing things only as they appear). They should therefore have a meaning which was independent of any schemata, and had a much wider extension.

<s43> In fact, the pure concepts of understanding do indeed retain some meaning, even after the removal of any sensory requirement. However, it is the purely logical meaning that there is a unity of representations, and nothing else. But no object of the representations is given, and hence no meaning which could result in a concept of the object.

<s44> Take the concept of substance, for example. If the sensory determination of permanence is left out, it would mean nothing more than a something which can be thought as a subject, without being a predicate of something else. But I cannot do anything with this representation, since it tells me [B187] nothing at all about the determinations which a thing must have in order to count as a primary subject of this sort.

<s45> So, without schemata, the categories are only functions of the understanding for concepts, and they do not represent any object. But meaning consists in reference to an object, and the categories acquire this meaning from sensibility, which actualises the understanding, while at the same time restricting it.

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