nLab Critique of Pure Reason

  • Kant

    Kritik der Reinen Vernunft (Critique of Pure Reason), abbreviated KrV.

    • first edition 1781. Usually and in this article denoted by “A”, e.g. “A111” refers to page 111 of the original first edition. But the Akademieausgabe uses A¹.

    • second edition 1787. Usually and in this article denoted by “B”. Akademieausgabe uses A².

    The second edition was “improved here and there”; some passages have been replaced entirely, like in the Transcendental Induction.


Introduction and Prefaces

I. Transcendental Doctrine of Elements. 1st Part. Transcendental Aesthetics

The science of all the principles of sensibility a priori, I call transcendental aesthetic.* There must, then, be such a science forming the first part of the transcendental doctrine of elements, in contradistinction to that part which contains the principles of pure thought, and which is called transcendental logic.

First Secton. On Space.

For Kant space is not a concept but an intuition, more precisely an intuition, “which serves for the foundation of all external intuitions”. Kant develops this view in the following steps (A23-24/B38-40)

  1. Space is not a conception which has been derived from outward experiences. For, in order that certain sensations may relate to something without me (that is, to something which occupies a different part of space from that in which I am); in like manner, in order that I may represent them not merely as without, of, and near to each other, but also in separate places, the representation of space must already exist as a foundation. …

  2. Space then is a necessary representation a priori, which serves for the foundation of all external intuitions. We never can imagine or make a representation to ourselves of the non-existence of space, though we may easily enough think that no objects are found in it. …

  3. Space is no discursive, or as we say, general conception of the relations of things, but a pure intuition. For, in the first place, we can only represent to ourselves one space, and, when we talk of divers spaces, we mean only parts of one and the same space. Moreover, these parts cannot antecede this one all-embracing space, as the component parts from which the aggregate can be made up, but can be cogitated only as existing in it. Space is essentially one, and multiplicity in it, consequently the general notion of spaces, of this or that space, depends solely upon limitations. …

The intuition (in the ordinary sense) Kant has concerning space is similar to a number theorist who considers “the” natural numbers as the object of her studies, though she certainly has limitations in specifying what she means. Indeed, common way to do so have limitation: for instance, one can not uniquely determine the natural numbers by a list of axioms, as Gödel's fist incompleteness theorem demonstrates, nor is it satisfying to describe the natural numbers by a concrete model, because there are several ways to do so.

Due to his intuition that there is “one” space, Kant discards the first two out of the three possibilities he discusses. Since Kant this intuition has been dropped mostly by geometers in favor of the first introducing pioneering concepts like Riemannian manifolds in 1854. In his very early work Bertrand Russell tried to adapt Kant’s work to this new view (see An Essay on the Foundations of Geometry). Later Kant’s idea appeared again in the framework of noncommutative geometry (Petitot 2009).

Actually, all three possibilities discussed by Kant appear in modern mathematics:

  1. the “discursive, or …, general conception” is just the most common way to describe a space by a list of axioms, e.g. a topological space or a metric space.

  2. a space that “antecede this one all-embracing space, as the component parts from which the aggregate can be made up” (German: “vor dem einigen allbefassenden Raume gleichsam als dessen Bestandtheile (daraus seine Zusammensetzung möglich sei) vorhergehen”). A modern concept that matches this description quite well is the concept of a classifying space, that does not contain each member of a certain class of spaces as a subspace, but from which “antecede” each such space by pullback.

  3. “Space is essentially one, and multiplicity in it, consequently the general notion of spaces, of this or that space, depends solely upon limitations.” (German: “Er [Der Raum] ist wesentlich einig, das Mannigfaltige in ihm, mithin auch der allgemeine Begriff von Räumen überhaupt beruht lediglich auf Einschränkungen.”) A situation like this is found if any space in a geometrical theory can be considered as a subspace of one fix space. Good examples of such situations are given, as mentioned, by noncommutative geometry (Petitot 2009), which is concerned with subspaces of the space of bounded operators on a separable Hilbert space, or by descriptive set theory that can be understood as the study of subsets of the real line.

Making such comparisons maybe a word of caution is in order to note that what Kant meant by the intuition of e.g. “space” is certainly not exactly any of the objects described above. But Kant’s “formal intuition” (“formale Anschauung”), which he mentions later in the Transcendental Deduction as a footnote in B §26, comes probably quite close.

Second Section. On Time.

“Time is nothing other than the form of inner sense, i.e., of the intuition of our self and our inner state.” (A33/B50)

I. Transcendental Doctrine of Elements. 2nd Part. Transcendental Logic

Introduction. The Idea of a Transcendental Logic.

In the development of Kant’s distinction between general logic and transcendental logic his Attempt to Introduce the Concept of Negative Quantities into Philosophy from 1763 constitutes a crucial step.

I. On logic in general. (A50-5/B74-9)

In this paragraph, the scope of logic as a science is clarified, distinguishing it from aesthetics and structuring its subdisciplines, to then be able to show the place of transcendental logic among them in the following paragraph. To do so, Kant firstly provides a plethora of terminology (“receptivity of impressions”, “spontaneity of concepts”, “empirical”, “sensibility”, “receptivity”), to then distinguish between the science of aesthetics and logic, the former concerning the rules of sensibility and the latter the rules of thinking. (A50-2/B74-6).

Secondly (A52/B76), Kant divides the science of logic into general logic (synonymous with elementary logic) which contains the necessary rules of thinking, regardless of its object and particular logic which contains the necessary rules of thinking about certain kinds of objects. Such a set of rules is called “organon of this or that science”. In modern language this is the methodology? of a scientific discipline. Finally (A53-5/B77-9), general logic is then again subdivided into pure logic—which “abstract[s] from all empirical conditions under which our understanding is exercised”—and applied logic—which deals with the conditions of thinking “under the subjective empirical conditions that psychology teaches us”, which are only given a posteriori, and which can all be given only empirically.

II. On transcendental logic. (A55-7/B79-82)

In this paragraph, the concept of transcendental logic as a particular kind of applied logic is established, contrasting it to pure logic. Pure logic abstracts from all contents of cognition [Erkenntnis] and is only concerned with the form of thinking in general. Transcendental logic contains the “rules of pure thinking of an object” (A55/B80). Kant then notes the difference between cognition a priori and transcendental cognition. The former is merely any cognition obtained without recourse to intuition. The latter is cognition concerning the possibility or use of cognition a priori. (A56-7/B80-1) Hence, transcendental aesthetics would describe the pure forms of intuition, which can then be drawn on to justify e.g. the possibility of geometric cognition a priori. Analogously there would need to be a transcendental logic, which would be the science of the possibilities and rules of thinking about objects a priori. (A56-7/B81-2)

III. On the division of general logic into analytic and dialectic. (A57-62/B82-6)

In this paragraph, general logic is established as a negative criterion of truth, while arguing against its capability to serve as a positive (sufficient) criterion. To show, that general pure logic is only capable of providing a negative criterion of truth, Kant firstly supposes a definition of truth which is: Truth “is the agreement of cognition with its object” (A58/B82). Now, using this supposition as a premise, he attempts to prove the self-contradictory nature of a “certain and general criterion of truth of any cognition” (A58/B82) according to the following schema:

Truth “is the agreement of cognition with its object” (A58/B82), hence “a cognition is false if it does not agree with the object to which it is related even if it contains something that could well be valid of other objects”
“[A] general criterion of truth [found in general logic] would be that which was valid of all cognitions without any distinction among their objects” (A58/B83) since general logic abstracts from all objects and only concerns the pure form of thinking.
A general criterion of truth found in general logic is self-contradictory. (A59/B83)

Nevertheless, since “[g]eneral logic analyses the entire formal business of the understanding and reason into its elements, and presents these as principles of all logical assessment of our cognition. This part of logic can therefore be called an analytic [our emphasis], and is on that very account at least the negative touchstone of truth.” (A60/B84) On the other hand, “general logic, which is merely a canon for judging, has been used as if it were an organon for the actual production of at least the semblance of objective assertions, and thus in fact it has thereby been misused. Now general logic, as a putative organon, is called dialectic.” (A61/B85)

IV. On the division of transcendental logic into the transcendental analytic and dialectic. (A62-4/B87-8)

The purpose of this paragraph is to narrow down the scope of transcendental logic, applying the division established in the last paragraph. From A62-3/B87:

The use of this pure cognition [described in transcendental logic], however, depends on this as its condition: that objects are given to us in intuition, to which it can be applied. For without intuition all of our cognition would lack objects and therefore remain completely empty. The part of transcendental logic, therefore, that expounds the elements of the pure cognition of the understanding and the principles without which no object can be thought at all, is the transcendental analytic and at the same time a logic of truth [our emphasis] For no cognition can contradict it without at the same time losing all content, i.e., all relation to any object …

One may summarize, that the transcendental logic, in its analytic part, provides the necessary criterion of truth, not only regarding its logical form, which is already accomplished by general pure logic, but also regarding the necessary form which truth has to have as it necessarily relates to objects. Kant continues (A63/B88):

[Transcendental logic] should properly be only a canon for the assessment of empirical use [of pure cognitions of the understanding], it is misused if one […] dares to synthetically judge, assert, and decide about objects in general with the pure understanding alone. The use of the pure understanding would in this case therefore be dialectical. The second part of the transcendental logic must therefore be a critique of this dialectical illusion [our emphasis].

First Division. The Transcendental Analytic

First Book. The Analytik of Concepts

First Chapter. On the Transcendental Clue [Leitfaden] to the Discovery of all Pure Concepts of the Understanding
Second Section. On the logical function of the understanding in judgments.
Table of Judgements
\,1. Quantity of judgments
2. Quality
\,3. Relation
4. Modality

Note that at least in (Jäsche Logik, 1800) the negative and infinite judgements collapse in pure logic:

Nach dem Principium der Ausschließung jedes Dritten (exclusi tertii) ist die Sphäre eines Begriffs relativ auf eine andre entweder ausschließend oder einschließend. - Da nun die Logik bloß mit der Form des Urteils, nicht mit den Begriffen ihrem Inhalte nach, es zu tun hat: so ist die Unterscheidung der unendlichen von den negativen Urteilen nicht zu dieser Wissenschaft gehörig.

Third Section. On the pure concepts of the understanding or categories.

“… there arise exactly as many pure concepts of the understanding … as there were logical functions of all possible judgments in the previous table:” (A79/B105 f.)

Table of Categories
\,1. Of Quantity
2. Of Quality
\,3. Of Relation
Of Inherence and Subsistence
\qquad (substantia et accidens)
Of Causality and Dependence
\qquad (cause and effect)
Of Community (reciprocity
\qquad between agent and patient)
4. Of Modality
Existence - Non-existence
Necessity - Contingency
Second Chapter. On the Deduction of the Pure Concepts of the Understanding [Transcendental Deduction]

Kant explains the transcendental deduction as follows:

I therefore call the explanation of the way in which concepts can relate to objects a priori their transcenden­tal deduction. (B117)

Kant begins by describing the way in which the manifold of intuition is formed:

Yet the combination (conjunctio) of a manifold in general can never come to us through the senses, and therefore cannot already be contained in the pure form of sensible intuition; for it is an act of the spontaneity of the power of representation. (B129)

He justifies the transcendental unity in the following sentence:

I am therefore conscious of the identi­cal self in regard to the manifold of the representations that are given to me in an intuition because I call them all together my representa­tions, which constitute one. (B135)

Presupposing that the human understanding only operates with concepts rather than with intuitions, Kant defines “understanding” in the following way:

Understanding is, generally speaking, the faculty of cognitions. (B137)

To advance his argument, Kant then needs the synthetic unity, because:

The synthetic unity of con­sciousness is … an objective condition of all cognition, … (B138)

Now a question regarding the purpose of the categories arises since for Kant

[the Categories] are only rules for an understanding whose entire capacity consists in thinking, i.e., in the action of bringing the synthesis of the manifold that is given to it in intuition from elsewhere to the unity of apperception, which therefore cognizes nothing at all by itself, but only combines and orders the material for cognition, the in­tuition, which must be given to it through the object. (B145)

While “Space and time are valid, as conditions of the possibility of how objects can be given to us, no further than for objects of the senses, hence only for ex­perience.” (B148) the categories are rules regarding how understanding may occur.

The following quote enlightens the connection between natural laws, which Kant describes as derivable from nature, and the categories, which are conditions of possible experience:

Particular laws, because they concern em­pirically determined appearances, cannot be completely derived from the categories, although they all stand under them. (B165)

In the end Kant arrives at the result that “no a priori cognition is possible for us except solely of objects of possible experience.” (B166)

Second Section [A]. On the a priori grounds for the possibility of experience. [A deduction]

In the first edition Kant provides a more thorough description on how the formation of concepts in our mind works. This description was left out in the corresponding chapter of the second edition (“B deduction”). Concepts are provided by a threefold synthesis:

  1. The synthesis of apprehension in the intuition. All representations belong “as modifications of the mind … to inner sense, and as such all of our cognitions are in the end subjected to the formal condition of inner sense, namely time” (A99). Such representations may be though of as states of the mind. One may be inclined to construe them as objects in a category as “Every intuition [Anschauung] contains a manifold in itself” (A98/9)

  2. The synthesis of reproduction in the imagination. “It is … a merely empirical law in accordance with which rep­resentations that have often followed or accompanied one another are finally associated with each other and thereby placed in a connection in accordance with which, even without the presence of the object, one of these representations brings about a transition of the mind to the other in accordance with a constant rule [beständige Regel]. This law of reproduction, however, presupposes that the appearances themselves are actually subject to such a rule …” (A100)

  3. The synthesis of recognition in the concept. “Without consciousness that that which we think is the very same as what we thought a moment before, all reproduction in the series of representations would be in vain. … The word ”concept“ itself could already lead us to this remark. For it is this one consciousness that unifies the manifold that has been successively intuited, and then also reproduced, into one representation.” (A103) The crucial point is this unity of consciousness.

So a [constant] rule appears as a relationship between two states of consciousness while rules appear as a connection of a concept to such states. Kant speaks also of a unity of rule that “determines every manifold”. As for a modern formalization the former would could be interpreted as a morphism and the letter reminds one of a diagram that has the concept as its cone. When taking sets as object the rules should be some kind of relation. But it is not clear from Kant’s formulation if the class of morphisms is restricted in some way, e.g. to functions in one direction.

Kant continues to state that these three step process also happens for a priori concepts. This is to say for our cognition (Erkenntnis) thereof. But this unity of the consciousness means that for each cognition a concept is assigned to the intuition in some way, i.e. that there is a corresponding “rule of intuitions” (cognition(conceptRegelderAnschauungen)cognition \coloneqq (concept \to Regel\, der\, Anschauungen) ) (A106). An instance of this is e.g. “object in general” (“Objekt überhaupt”). Every instance comes with a necessity that there is such a concept and a rule of intuitions. But a necessity can only origin from some “transcendental condition”, that Kant also calls transcendental apperception. These are the pure concepts of the understanding we sought.

As example Kant provides: a triangle as an object by being conscious of the composition of three straight lines in accordance with a rule according to which such an intuition can always be exhibited. Moreover

cognitionofouterappearances=(bodytheunityofthemanifold thatisthoughtthrough[theconceptofbody]). cognition\, of\, outer\, appearances = \left(body \to \substack{the\, unity\, of\, the\, manifold\\ that\, is\, thought\, through\, [the\, concept\, of\, body]}\right).

Further one should remark that a priori concepts are distinguished by their numerical unity. Kant elaborates on this distinction in the next book.

Achourioti and van Lambalgen’s formalization

Starting from the description of the A deduction above Achourioti and van Lambalgen (2011/2012) suggested a formalization of Kant’s logic in terms of geometric logic. The idea is to assign each moment of time, i.e. impression, a model. Then a directed system of such models represents the syntheses of apprehension and reproduction. Taking the limit along this directed system (“synthesis of recognition”) gives the concept. Geometric implications come into play as they are exactly the formulas whose truth is stable under taking the limit.

Although the natural category for interpreting geometric logic is a geometric category (e.g. a topos), Achourioti and van Lambalgen work only in Set for technical convenience. From a philosophical point of view they remark that the underlying category should have additional structure. They elaborate on a specific finite set model of their interpretation, for which they can show a completeness theorem. We will state their main results with small reformulations.

Kantmathematical logic and model theory
timedirected index category
a manifold of representationobject in the underlying category
synthesis of apprehensionobjects of the index category of a directed system (inside the underlying category)
synthesis of reproductionmorphisms of the index category of a directed system (inside the underlying category)
synthesis of recognitionlimit along the directed system

Second Book. The Analytic of Principles

Appendix. On the amphiboly of the concepts of reflection through the confusion of the empirical use of the understanding with the transcendental.
  1. Identity and difference. If an object is presented to us several times, but always with the same inner determinations (_qualitas et quantitas_), then it is always exactly the same if it counts as an object of pure understanding, not many but only one thing (_numerica identitas_); but if it is appearance, then the issue is not the comparison of concepts, but rather, however identical everything may be in regard to that, the difference of the places of these appearances at the same time is still an adequate ground for the numerical difference of the object (of the senses) itself. (A263/B319)

II. Transcendental Doctrine of Methods.

On Aristotelian logic

Regarding Aristotelian logic:

Concepts [Begriffe] … serve as predicates of possible judgements. (A69/B94)

Compare to the identification of concepts with the types in the sense of type theory, see the references here.


Publications of Kant’s original work include:

Related primary literature from Kant comprises

  • G. B. Jäsche (ed.), Immanuel Kants Logik ein Handbuch zu Vorlesungen, pp. 419-582 in Kant, Schriften zur Metaphysik und Logik 2, Suhrkamp Frankfurt a. M. 1985[1800].

Secondary literature includes:

Achourioti and van Lambalgen’s approach appeared in

  • Achourioti and van Lambalgen, A formalization of Kant’s transcendental logic. (June 2011). Review of Symbolic Logic 4 (2):254-289.

  • Achourioti and van Lambalgen, The Completeness of Kant’s Table of Judgments & Consequences for Philosophy of Mathematics. Talk given at the Munich Center for Mathematical Philosophy (November 2011). Video.

Later the authors elaborated on the table of judgements in

  • Achourioti and van Lambalgen, Kant’s Logic Revisited (January 2016). In: Proceedings of PhML 2012, Saint Petersburg. Also published (2017) in: The IfCoLog Journal of Logics and their Applications 4(4), 845-865.

For criticism of Kant’s work on the categories, see

  • Gilbert Ryle, Categories, Proceedings of the Aristotelian Society, New Series, Vol. 38 (1937 - 1938), pp. 189-206.

Kant’s line of approach was, in principle, much more enlightened than Aristotle’s had been. Unfortunately his execution was hopelessly misguided.

category: reference

Last revised on August 6, 2020 at 15:40:42. See the history of this page for a list of all contributions to it.