<ai1> The Transcendental Doctrine of Judgment

The System of all Axioms of Pure Understanding

[B187] In the previous chapter, I considered only the general preconditions for the transcendental faculty of judgment to have the right to use the pure concepts of understanding for making synthetic judgments. My task is now to specify the judgments which the understanding actually makes apriori, within the limits set by the critical philosophy. I need to lay them out systematically, in accordance with their interrelations; and my table of categories must undoubtedly provide the natural and infallible guidance in this. For all pure apriori knowledge of the understanding must consist in the application of just these categories to possible experience. So it is the relation of the categories to sensibility in general [B188] which will reveal all the transcendental axioms of the use of the understanding, completely and in a system.

<ai2> Apriori axioms, or ‘foundational propositions’, are called ‘foundational’, not merely because they contain within themselves the foundation for other judgments, but also because they themselves are not founded upon any superior or more general knowledge. However, this special characteristic does not exempt them from all proof. Certainly a proof could not be carried further on the objective side, since an axiom is the foundation for all knowledge of its object. But this does not mean that it is impossible to construct a proof from the subjective sources of the possibility of knowledge of an object in general. Indeed, it is necessary to do so, since otherwise the axiom would attract the strongest suspicion of being merely an ungrounded assertion.

<ai3> Secondly, I shall confine myself to the axioms which relate to the categories. So my restricted area of inquiry excludes the principles of the Transcendental Aesthetic, according to which space and time are the preconditions for the possibility of all things as appearances; and it also excludes their negative side, namely that they cannot apply to things in themselves.

<ai4> Similarly, mathematical axioms are not a part of this system, because they derive only from intuition, and not from the pure concepts of the understanding. [B189] However, the possibility of mathematical axioms necessarily has a place here, because they are synthetic apriori judgments. But it is not to prove their soundness and self-evident certainty (since this needs no proof at all), but only to make comprehensible and to deduce the possibility of evident apriori knowledge of this nature.

<ai5> However, I must also say something about the axiom of analytic judgments, and contrast it with the synthetic axioms which I am essentially concerned with. It is precisely this contrast which frees the theory of synthetic axioms from all misunderstanding, and lays their special nature clearly before the eyes.

<ai6> The System of the Axioms of Pure Understanding

On the Supreme Axiom of All Analytic Judgments

Whatever the content of our knowledge, and however it relates to its object, the universal (though negative) precondition of absolutely all our judgments is that they are not self-contradictory. Any self-contradictory judgment is intrinsically nothing, even without reference to its object. However, [B190] even if a judgment contains no contradiction, yet it can combine concepts in a way which is not found in the object, or without there being any ground, whether apriori or aposteriori, which would justify such a judgment. So a judgment can be either false or groundless, despite being free of any internal contradiction.

<ai7> The proposition that ‘Nothing has a predicate which contradicts it’ is called the Principle of Contradiction, and it is a universal (though merely negative) criterion of all truth. However, it belongs merely to logic, because it is valid of any item of knowledge, simply as knowledge in general, and irrespective of its content. It says that the contradiction entirely annihilates and cancels the knowledge.

<ai8> However, it can also be used positively — that is, not simply to eliminate falsehood and error (in so far as it arises from a contradiction), but also to know truth. For if a judgment is analytic (whether negative or affirmative), it must always be possible for its truth to be sufficiently known on the basis of the principle of contradiction. For it is always correct to deny the opposite of what already exists and is thought in the knowledge of the object as its concept; whereas its concept itself must necessarily be asserted of it, [B191] because its opposite would contradict the object.

<ai9> Consequently, we must accept the principle of contradiction as the universal and completely sufficient principle of all analytic knowledge. However, its authority and usefulness do not extend as far as its being a sufficient criterion of truth. For the fact that no knowledge can go against it without annihilating itself certainly makes it a necessary condition of our knowledge, but not the basis for determining its truth. Since I am now essentially concerned with the synthetic part of our knowledge, I shall of course always be careful not to go against this inviolable axiom, but I can never expect from it any enlightenment as to the truth of synthetic knowledge.

<ai10>However, despite the fact that this famous axiom is devoid of content and merely formal, there is another way of formulating it, involving a synthesis which is introduced out of carelessness, and quite unnecessarily. This formulation goes: ‘It is impossible for something to be and not to be at the same time.’ One objection to it is the redundant addition of its self-evident certainty (through the word ‘impossible’), when this should be self-evident from the proposition. More seriously, the proposition is infected by the restriction to time, so that it says something like: ‘A [B192] thing A which is B cannot at the same time be not-B; but it can easily be both (B or not-B) successively.’ For example, a person who is young cannot be old at the same time; but one and the same person can easily be young at one time, and not-young (i.e. old) at another time. But since the principle of contradiction is a merely logical proposition, it must not limit its claims to time relationships; and hence a formulation such as the above goes completely against its purpose.

<ai11> The source of the misunderstanding is simply this. First, the predicate of a thing is separated from the concept of the thing; and then the opposite of this predicate is connected with the predicate. But this never gives rise to a contradiction with the subject, but only with its predicate which is connected with the subject synthetically. And indeed, a contradiction arises only when the first and second predicates are affirmed at the same time. If I say: ‘An unlearned person is not learned’, the qualification ‘at the same time’ must be added, since someone who is unlearned at one time can perfectly well be learned at another time. But if I say: ‘No unlearned person is learned’, then the proposition is analytic. This is because the property (of unlearnedness) is now one of the components of the concept of the subject, and thus the negative proposition is immediately obvious from the principle of contradiction, without the qualification ‘at the same time’ needing to be added. This is also the reason why I changed the formulation [B193] of the principle above, so that it clearly expresses the nature of an analytic proposition.

<ai12> The System of the Axioms of Pure Understanding

On the Supreme Axiom of All Synthetic Judgments

Explaining the possibility of synthetic judgments is a task which general logic has nothing at all to do with — it doesn’t even need to know its name. But in a transcendental logic, it is the most important business of all; and even the only one, if we are considering the possibility of synthetic apriori judgments, and also the preconditions and scope of their validity. For on completing this task, transcendental logic can fully satisfy its goal of determining the scope and limits of pure reason.

<ai13> In an analytic judgment, I stay with the given concept, in order to discern something about it. If it is an affirmative judgment, I attribute to this concept only what was already thought in it; and if it is a negative judgment, I exclude only the opposite of the concept. But in synthetic judgments, I must go beyond the given concept, to consider its relation to something which is quite different from what was thought in the concept. [B194] So this relation is never one of identity or of contradiction, and hence not one such that the truth or falsehood of the judgment can be established by considering the judgment as it is in itself.

<ai14> Granted that one must go beyond a given concept in order to compare it synthetically with another concept, there must be a third thing, in which alone the synthesis of two concepts can take place. But what is this third thing which mediates all synthetic judgments? There is only one totality which includes all our representations, namely inner sense, and its apriori form, which is time. The synthesis of representations depends on the imagination; but their synthetic unity (which is required for judgment) depends on the unity of apperception. Thus the possibility of synthetic judgments must be sought in these three items, and the possibility of pure synthetic judgments in particular, since all of them contain the sources of apriori representations. Indeed, for these very reasons, they are necessary for there to be any knowledge of objects, since it depends entirely on the synthesis of representations.

<ai15> For knowledge to have objective reality is for it to relate to an object, which gives it sense and meaning. So if knowledge is to have such reality, it must be possible for an object to be given in some way or other. Without an object, a concept is empty; and although we have indeed thought something through the concept, [B195] we have not actually known anything through this thought, and we have merely been playing with representations. When I say that an object is ‘given’, I mean that it is presented directly in intuition, and not merely through the mediation of something else. So for an object to be given is nothing other than for its representation to relate to experience, whether actual or merely possible.

<ai16> The concepts of space and time are completely pure of anything empirical, and it is absolutely certain that they are represented completely apriori in the mind. Nevertheless, they would lack any objective validity or sense and meaning, unless it were established how they necessarily apply to objects of experience. Indeed, their representation is merely a schema, which is always related to the reproductive imagination. And it is the reproductive imagination which summons up the objects of experience, without which the concepts of space and time would have no meaning. And the same is true of every kind of concept.

<ai17> So it is the possibility of experience that gives objective reality to all our apriori knowledge. Now experience depends on the synthetic unity of appearances, that is, on a synthesis, in accordance with concepts, of the object of appearances in general. Without this synthesis, there would never be any knowledge, but only a confused jumble of perceptions. These would not fit together into any unitary network in accordance with rules of a thoroughly connected (possible) consciousness, and hence they would not fit into the transcendental and necessary unity of apperception.

<ai18> [B196] So experience has principles of its form underlying it apriori as its foundation, namely universal rules of unity in the synthesis of appearances. The objective reality of these rules, as necessary preconditions of experience, can always be shown in experience, and even in the possibility of experience. But without this relation to actual or possible experience, synthetic apriori propositions are completely impossible, because they have no third thing (namely a pure object) in which the synthetic unity of their concepts could demonstrate objective reality.

<ai19> We have so much a priori knowledge, expressed in synthetic judgments, about space in general, or about the figures which the productive imagination draws in it, that we actually need no experience at all for this knowledge. Nevertheless, it follows from the above that this knowledge would be nothing, and we would be dealing with a mere figment of the brain, if space were not to be considered as a precondition of the appearances which constitute the material for outer experience. Consequently, those synthetic judgments relate, if only indirectly, to possible experience — or rather to the very possibility of possible experience. And this is the sole foundation of the objective validity of their synthesis.

<ai20> Experience, as empirical synthesis, is the one way of knowing which gives reality to apriori synthesis; and it does so by virtue of its very possibility. Consequently, apriori knowledge also attains truth (correspondence [B197] with its object) only in so far as it includes nothing more than is necessary for the synthetic unity of experience in general.

<ai21> So the supreme principle of all synthetic judgments is: ‘Every object comes under the necessary preconditions of the synthetic unity of the multiplicity of intuition in a possible experience.’

<ai22> The way in which synthetic apriori judgments are possible is by relating the following to the possibility of experiential knowledge in general:

• the formal preconditions of apriori intuition;
• the synthesis of the imagination;
• the necessary unity of this synthesis in a transcendental apperception.

We can then say that the preconditions of the possibility of experience in general are at the same time preconditions of the possibility of objects of experience. This is why they have objective validity in a synthetic apriori judgment.

<ai23>The System of the Axioms of Pure Understanding

That there are any axioms at all in any area of knowledge, is due entirely to the pure understanding. It is not only the faculty of rules [B198] as to what happens, but it is also the very source of the axioms in accordance with which everything which can be presented to us only as an object necessarily comes under rules. For without rules, appearances could never supply knowledge of an object corresponding to themselves.

<ai24> Despite the fact that even the laws of nature are considered to be fundamental laws of the empirical use of understanding, nevertheless they bear the stamp of necessity. Hence there is at least the presumption that they are particular instances of foundations which are valid apriori and independently of any experience. But all laws of nature of every kind come under higher axioms of the understanding, in that they merely apply these axioms to particular instances of appearances. So it is only the axioms which supply the concept containing the general precondition and as it were the parameter for a rule. On the other hand, it is only experience which supplies the particular instance that comes under the rule.

<ai25> There is really no danger of confusing merely empirical axioms with axioms of the pure understanding, or the other way round. This confusion can easily be avoided, since it is easy to perceive the difference between the necessity arising from concepts (which is a mark of the pure understanding), and its absence from every empirical proposition, however general its validity. However, there are pure apriori axioms which I may not attribute exclusively to the pure understanding, because they are derived, not from pure concepts, but [B199] from pure intuitions (though by means of the understanding) — whereas the understanding is the faculty of concepts. Mathematics has the same sort of axioms, but their application to experience still always depends on the pure understanding — as also their objective validity, and even the possibility of such synthetic apriori knowledge (its deduction).

<ai26> Therefore I shall not include among my axioms those of mathematics. I shall however include those axioms which are the apriori foundation of the possibility and objective validity of mathematical axioms. Hence they are to be considered as the principle of the mathematical axioms; and they proceed from concepts to intuition, and not from intuition to concepts.

<ai27> In the application of the pure concepts of understanding to possible experience, the way their synthesis is carried out is either mathematical or dynamical, depending on whether the synthesis involves merely intuition, or the existence of an appearance in general. But the apriori preconditions for intuition in relation to a possible experience are absolutely necessary, whereas those for the existence of the object of a possible empirical intuition are in themselves only contingent. Hence the mathematical axioms can be called necessary without qualification (i.e. self-evident), whereas, although the dynamical axioms do indeed bear the stamp of apriori necessity, they do so only under the precondition of empirical thought in an experience, and hence through something else and [B200] indirectly. It follows that, although the certainty of the dynamical axioms (which depends on their universal relation to experience) is unaffected, they do not have the immediate self-evidence which is exclusive to the mathematical axioms. However, it will be easier to come to a judgment about this after reading the System of Axioms.

<ai28> The table of categories gives us natural guidance for drawing up the table of axioms, because the axioms are nothing other than rules for the application of the categories to objects. A complete list of the axioms of pure understanding is as follows:

<ai29> I have chosen these names deliberately, in order to emphasise the extent to which these axioms are more or less self-evident, and the differences in the ways they are applied. As will soon be shown, the first two sets of axioms, derived from the categories of quantity and quality, are significantly different from the other two. This goes both for their degree of self-evidence, [B201] and for the way they determine appearances apriori (as long as only the form of the first two is considered). Although they are both completely certain, the first two are immediately self-evident, whereas the second two require a reasoned proof. So I shall call the first set mathematical axioms, and the second set dynamical axioms.*

<ai32> It should be carefully noted that here I am no more concerned with [B202] the axioms of mathematics in the one case, than with the axioms of general (physical) dynamics in the other, but only with the axioms of pure understanding in relation to inner sense, whatever the representations given in it. For it is only through the axioms of pure understanding that all the others become possible. So I have given the axioms their names more by virtue of how they are applied, than by virtue of their content. I shall now proceed to consider them in the same order as they are laid out in the table.

<ai33> 1.

The principle of the Axioms of Intuition is: All intuitions are extensive quantities.

As far as their form is concerned, all appearances include an intuition in space and time, which is the apriori foundation of all of them. So they cannot be apprehended (i.e. they cannot be brought into empirical consciousness) except through the synthesis of the multiplicity, by means of which the representations of a particular space or time are generated — that is, except through the unification of that which is homogeneous, and the consciousness of [B203] the synthetic unity of this homogeneous multiplicity. Now the consciousness of a multiple homogeneity in intuition in general, in so far as it first makes the representation of an object possible, is the concept of a quantum. Thus even the perception of an object as appearance is possible only through the same synthetic unity of the multiplicity of the given sensory intuition, as that through which the unity of the unification of a multiple homogeneity is thought in the concept of a quantity. In other words, appearances are all quantities, and in particular extensive quantities, because as intuitions in space or time, they must be represented through the same synthesis as that through which space and time in general are determined.

<ai34> By an extensive quantity, I mean one such that the representation of its parts makes possible the representation of the whole, and hence necessarily precedes it. I cannot represent a line, however short, without first pulling it out in thought — that is, starting from a point and generating all its parts one after another, and thus first drawing this intuition. The same is the case with any stretch of time, however small. The only thing I think in it is the successive progression from one moment to the next, in which a determinate quantity of time is ultimately generated through all the instants and their being added together. Since pure intuition in all appearances is either space or time, [B204] every appearance, as intuition, is an extensive quantity, in that it can become known only through successive synthesis (from part to part) in apprehension. Therefore all appearances are already intuited as aggregates (a heap) of previously given parts. This is not true of all kinds of quantities, but only of those which are represented to us and apprehended by us as extensive.

<ai35> The mathematics of extension (geometry), together with its axioms, is founded upon this successive synthesis performed by the productive imagination in generating figures. Its axioms give apriori expression to the preconditions of sensory intuition, which are the necessary prerequisites for the schema of a pure concept of outer appearance — for example, that there can be only one straight line between two points; that two straight lines do not enclose any space; and so on. These are axioms which essentially involve only quanta as such.

<ai36> But as for quantity (i.e. the answer to the question: how many something is), there are strictly speaking no axioms, even though there are all sorts of propositions which are synthetic and immediately certain (they need no proof). The propositions that the result will be the same if the same number is added to the same number, or that the result will be the same if the same number is subtracted from the same number, are analytic. This is because I am immediately conscious that the [B205] act of producing the quantity is identical in each case. But these propositions are not axioms, since an axiom must be synthetic apriori.

<ai37> On the other hand, self-evident propositions about the relations between numbers are certainly synthetic, but they are not universal like those of geometry. So on this account they cannot be called axioms, but only numerical formulae. 7+5=12 is not an analytic proposition, since I do not think the number 12, either in the representation of 7, or in the representation of 5, or in the representation of combining them together. It is irrelevant that I must think 12 in the sum of the two, since a proposition is analytic only if I actually think the predicate in the representation of the subject. But even though it is a synthetic proposition, it is only singular.

<ai38> Here, in so far as it is a question simply of the synthesis of things of the same kind (namely units), this synthesis can take place only in one way, even though the application of these numbers is subsequently universal. Suppose I say, ‘It is possible to draw a triangle by means of three lines, of which two taken together are longer than the third.’ Here I have nothing but the function of the productive imagination, which enables the lines to be drawn longer or shorter, and hence to be put together with any angles whatever. By contrast, the number 7 can be generated only in a single way; and the same is true of the number 12, which is generated by synthesising it with the number 5. Such propositions must not be called axioms, [B206] because there would be infinitely many of them. Instead, they must be called numerical formulae.

<ai39> This transcendental axiom of the mathematics of appearances greatly extends our apriori knowledge. For it is this axiom alone that makes the perfect accuracy of pure mathematics applicable to objects of experience. Without this axiom, the applicability of mathematics to experience would not be self-evident in this way — indeed, the lack of it has been the source of much controversy. Appearances are not things in themselves. Empirical intuition is possible only through pure intuition (of space and time); and so what geometry says of the pure intuition of space is, without dispute, also valid of empirical intuition. Nor can this conclusion be avoided by saying that objects of the senses cannot conform to the rules of construction in space, for example on the grounds that lines or angles are infinitely divisible. For this would be to deny the objective validity of space, and with it the objective validity of the whole of mathematics; and one would no longer know why or how far mathematics is applicable to appearances.

<ai40> The synthesis of spaces and times, as the essential form of all intuition, is what makes possible at the same time:

• the apprehension of appearance; hence
• every outer experience; and consequently also
• all knowledge of the objects of experience.

So, what pure mathematics proves of space or time is also necessarily valid of the objects of experience.

<ai41> Any objections to this are merely sophistries of a wrongly taught [B207] reason, which erroneously tries to free the objects of the senses from the formal precondition of our sensibility. So, despite the fact that they are mere appearances, objects are represented as objects in themselves, which are given to the understanding. But if this were the case, there could certainly be no synthetic apriori knowledge of them at all, and hence there could be no such knowledge of them through pure concepts of space. And the science which determines these concepts, namely geometry, would itself be impossible.

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