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Critique of Pure Reason

Lecture Notes: Analytic and Synthetic Judgments

G. J. Mattey

Thus far, we have been concerned with presentations of objects. Presentations of specific objects are called ‘intuitions,’ while presentations of a number of objects through some common characteristics they share are called ‘concepts.’ We have noted that the fundamental problem in metaphysics is to discover how pure or a priori concepts generated by the understanding without recourse to sensuous presentation apply to intuitions which are, for human beings, sensuous presentations.

It was mentioned in the discussion of the letter to Herz that a related problem is to show how principles governing pure concepts apply to objects of sensuous intuition. In the human mind, principles are manifest in “judgments,” which combine concept and intuition, concept and concept, or judgment and judgment. So I might judge, respectively, that the golf ball in my hand is round, that what is round is extended, or that if I drop the ball straight down, then it will bounce straight up. Here, we shall be concerned primarily with a distinction between two kinds of judgment, the analytic judgment and the synthetic judgment.


To understand the account of judgments given in the Introduction to the Critique, it is necessary to introduce a few more definitions, which unfortunately are not given until the middle of the book, at A320/B376-7. We begin with presentation, which is the most generic sort of mental state. Presentation with consciousness is called “perception.” (We are not told in the cited passage what presentation without consciousness might be.) Perceptions, in turn, are divided between the subjective and the objective. Subjective perceptions are “the modification of the subject’s state,” and they are “sensations.” They are called “subjective” because they refer to the subject of presentation rather than the objects of presentation.

Objective perceptions, perceptions which refer to something other than the state of the subject, are called “cognitions.” The two kinds of cognition are intuitions and concepts.

An intuition refers directly to the object and is singular; a concept refers to the object indirectly, by means of a characteristic that may be common to several things. (A320/B376-7)
Concepts are then divided into empirical concepts and pure concepts, i.e., those which arise from abstraction from sensuous intuitions and those which arise from the understanding independently of sensuous intuition.


Kant begins the Introduction by making a distinction between two ways in which cognition is related to experience. In the order of time, experience precedes cognition. “For what else might rouse our cognitive power to its operation if objects stirring our senses did not do so?” (B1). (In the A edition, Kant states that “Experience is, without doubt, the first product to which our understanding gives rise” and “is our first instruction” (A1).) As we have stated, Kant held firmly to the view that the only source of intuition in human beings is sensuous. The immediate result of this “rousing” is “presentations,” which Kant seems to be identifying here with sensations. (The A edition states that the understanding gives rise to experience “by working on the raw material of sense impressions” (A1).)

But these presentations are not yet experience, which Kant viewed as a connected whole. So the “stirring” of our sense by objects brings about a second result: “they set in motion our understanding’s activity, by which it compares these presentations, connects or separates them, and thus processes the raw material of sense impressions into a cognition of objects which is called experience” (B1). Here, Kant seems to mean by “experience” not any single intuition or concept, but rather a system of intuitions and concepts.

In the next paragraph, Kant makes a suggestion that will turn out to be his main contention. “It might well be that even our experiential cognition is composite, consisting of what we receive through impressions and what our own cognitive power supplies from itself (sense impressions merely prompting it to do so).” (B1). Given that aspects of experiential cognition can have different origins, it can then be the case that “our own cognitive power” acting by itself can supply “a priori cognitions” which are one ingredient in experience. Then the task of the philosopher is to separate these cognitions from those which are based in the sense impressions we receive from objects.

A priori cognitions are contrasted with empirical or a posteriori cognitions, i.e., cognitions which arise from experience. Kant recognized that we make judgments that are in a sense “a priori” when we reason from a universal principle without recourse to some particular experience, as when we judge that a house that has been undermined will cave in. He was not interested in such judgments, but only in those which do not at all rely on experience (which in this case informs us that bodies have weight). He notes that even a judgment which uses an a priori concept, such as causality, may still involve an empirical concept, such as change. The judgment “Every change has a cause” cannot be derived from any experience, and so it is in that sense a priori, but it is not “pure” because it contains an empirical cognition as a component. The empirical cognition is of change, which Kant will later claim is a feature only of objects of experience.

A Priori Judgments

Kant had already stated in the letter to Herz that the human mind supplies us with a priori concepts. But how do we know which concepts (or intuitions, for that matter) have their origin entirely in the mind? Kant thought that there are two criteria that “are safe indicators of a priori cognition” (B4). These are necessity and strict universality.

We arrive at these criteria by noting the limitations of experience. First, “experience does indeed teach us that something is thus or thus, but not that it cannot be otherwise” (B3, cf. A1). So if there are any judgments which we must think of as necessarily true, they would be a priori judgments. Second, experience can only teach us that things have always been a certain way, but not that all things without exception are a certain way. So any judgment which we must think of as applying to all things without exception would be an a priori judgment. Kant thought the two criteria are inseparably bound together, and indeed most commentators have found it difficult to find any difference in meaning between them.

Kant was quite convinced that there are a priori judgments. All judgments of mathematics are a priori, as is the metaphysical judgment “all change must have a cause” (B5). In the latter case, the fact that the a priori concept of a cause appears in the predicate of the judgment insures the a priori character of the judgment itself. If it applies to objects at all, concept of a cause carries with it necessity and strict universality. For this reason, Kant claimed that Hume’s treatment of the concept, as the product of a habit based on repeated association, is entirely inadequate. The concept which Hume called “cause” has only “subjective necessity” and is an empirical concept (Prolegomena, Preface, Ak 4:257-258).

Kant and Hume

Kant seems to have been familiar with Hume mostly from the Enquiry Concerning Human Understanding. In that book, Hume gave a definition of a cause as “an object followed by another, and whose appearance always conveys the thought to that other” (Section 7, “The Idea of Necessary Connection”). Kant was correct that the necessity in this definition is purely subjective. Hume gave this definition only after all his attempts to discover an objective necessary connection, i.e., a connection that necessarily holds between objects, had failed.

All events seem entirely loose and separate. One event follows another; but we can never observe any tie between them. They seem conjoined, but never connected. And as we can have no idea of any thing, which never appeared to our outward sense or inward sentiment, the necessary conclusion seems to be, that we have no idea of connection or power at all, and that these words are absolutely without meaning, when employed in philosophical reasonings, or common life. (Section 7, “The Idea of Necessary Connection”)

Kant claims against Hume that the fact that we can never observe a tie between objects does not mean that we have no concept of such a tie. To be sure, we have no empirical concept of a cause; Hume was right in this. The dispute hinges on whether we actually do have an a priori concept of a cause. This possibility is precluded by Hume’s fundamental principle that all our ideas are copies of antecedent impressions, which in Kant’s terms means that all cognitions arise from experience—a principle whose denial was fundamental for Kant.

More generally, Kant denied the empiricist standpoint of Locke that had been embraced by Hume.

Whence has [the mind] all the materials of reason and knowledge? To this I answer, in one word, EXPERIENCE. In that all our knowledge is founded; and from that it ultimately derives itself. Our observation employed either, about external sensible objects, or about the internal operations of our own mind perceived and reflected upon by ourselves, is that which supplies our understanding with all the materials of thinking. These two are the fountains of knolwedge, from whence all the ideas we have, or can naturally have, do spring. (An Essay concerning Human Understanding, Book II, Chapter 1, Section 2)
A priori concepts are just the sort of “ideas” (as Locke and Hume would call them) whose possibility an empiricist denies.

In the Prolegomena, Kant pursued the matter further. He acknowledged that Hume had done a great service to philosophy by questioning the credentials of reason regarding its use of allegedly a priori concepts. Indeed, he even attributed his own rejection of dogmatism to his recollection of Hume. He conceded that Hume was right regarding a key point.

He challenged reason, which pretends to have given birth to this concept of herself, to answer him by what right she thinks anything could be so constituted that if that thing be posited, something else also must necessarily be posited; for this is the meaning of the concept of cause. He demonstrated irrefutably that it was entirely impossible for reason to think a priori and by means of concepts such a combination as involves necessity. We cannot at all see why, in consequence of the existence of one thing, another must necessarily exist, or how the concept of such a combination can arise a priori. (Preamble, Ak 4:257)
As a result, Hume concluded that cause is not an a priori cognition at all. Kant thought that if this were true, it would be the death of metaphysics, since metaphysics has as its primary subject-matter a priori cognitions. It is this conclusion, not its premises, that Kant called “hasty and mistaken” (Ak 4:258).

What was the flaw in Hume’s reasoning, then? The answer is that he looked for the source of the necessary connection in experience. As Kant put it, after he, Kant, had discovered a number of other such a priori concepts, “I proceeded to the deduction of these concepts, which I was now certain were not derived from experience, as Hume had tried, but sprang from the pure understanding” (Ak 4:260). So at this point, “solving Hume’s problem” requires a proof that the concept of cause (and all the other a priori concepts central to metaphysics) has its origin in the pure understanding.

We have already seen that for Kant, the concept of a cause carries with it necessity and strict universality, and thus is a priori. Hume’s challenge is that the concept of necessity cannot be taken for granted, but is as obscure as any in metaphysics, so that we need “to fix, if possible, the precise meaning” of this term. (Enquiry, Section 7). And he thought it could be meaningful only if it is derived from some impression of sense or reflection. He found its meaning to be what Kant called “subjective,” in that it is based in uniformity of observation and habit. Even if we think of a cause as “necessary,” we have to have some basis for understanding this necessity as “objective.” And this Hume could not find.

Now one might adopt a Leibnizian definition of necessity is that whose opposite implies a contradiction. Hume held that no such definition could apply to the concept of cause, since it is always possible to imagine an event without the event which has always preceded it and which we call “cause.” What is needed is a notion of necessity which connects two objects of a certain kind together. This kind of necessity cannot be found in the concept of that object. This is most likely why Kant stated that Hume failed to find a connection “a priori and by means of concepts.” The real problem with Hume’s procedure, then, is that it looks to experience for the connection in the object as presented to the mind.

The solution will begin by shifting the source of investigation. Rather than look to the concept of any individual object or kind of object, we must look at what it is to be an object of experience. Kant will claim that it is pure concepts such as cause and effect which make objects of experience possible. This is the strategy that Hume had overlooked.

Mathematics and Metaphysics

Hume’s reaction to his failure to establish the objective necessity of cause and effect was to give up on most of what had traditionally passed as metaphysics and regard it as (in Kant’s description) “the delusion of a supposed rational insight” (B20). He notoriously ended the Enquiry with these words.

When we run over libraries, persuaded of these principles, what havoc must we make? If we take in hand any volume; of divinity or school metaphysics, for instance; let us ask, Does it contain any abstract reasoning concerning quantity and number? No. Does it contain any experimental reasoning concerning matters of fact? No. Commit it then to the flames: For it can contain nothing but sophistry and illusion. (Section 12, “Of the Academical or Sceptical Philosophy”)
Kant noted that Hume still allowed for an empirical metaphysics, but “overlooked the positive injury which results if reason be deprived of its most important prospects, which can alone supply to the will the highest aim for all its endeavors” (Prolegomena, Preface, Ak 4:258, note 2).

The interests of reason in metaphysics are in “God, freedom, and immortality” (A3/B7). Even if we could prove that there are a priori principles governing mathematics and the world of experienced objects (appearances), reason would be frustrated if it were cut off from its investigation of these three areas. Philosophers generally have pursued these investigations without first asking a preliminary question. It never occurred to them to inquire about the “range, validity, and value” of our a priori cognitions (A3/B7).

Why have metaphysicians neglected this fundamental task? One reason is that as long as one is careful and avoids contradiction, one can never be refuted in one’s metaphysical claims. Unfortunately, along with this safety come an inability to get what we might now call “traction” in experience, so that we have no way of knowing whether the result of our efforts is purely imaginary.

It may be thought that there is a way to get leverage for metaphysical conclusions—through the analysis of concepts. The greatest error made by philosophers in the past (Hume included) was with to assume that this is the way mathematics proceeds. Given this assumption, metaphysicians looked to the success of mathematics as a model for the success of metaphysics: proceeding by definitions, axioms, and demonstrations, as we saw in the last lecture. (See “The Discipline of Pure Reason in its Dogmatic Use,” beginning at A712/B740, for details.)

Analytic and Synthetic Judgments

The mistake of the metaphysicians was their view that mathematical judgments are “analytic.” Kant describes an analytic judgment as one which is merely “elucidatory,” that is, where what is implicit is made explicit. His examples use subject/predicate judgments, such as that a square has four sides. The content of the predicate is already thought in the subject, though perhaps “covertly.” Such judgments “do not through the predicate add anything to the concept of the subject; rather, they only dissect the concept, breaking it up into components which had already been thought in it (although thought confusedly)” (A7/B11).

The principle for determining the truth of analytic judgments is the principle of contradiction. The judgment that a square is five-sided, for example, is self-contradictory because the concept of having four sides is already thought in the concept of a square. It seems that analytic judgments are of little use in metaphyics, but as we saw last time, the philosophy of Wolff and Baumgarten rested entirely on definitions and demonstrations based on the principle of contradiction.

A second kind of judgment is “synthetic,” a judgment in which the predicate is not already thought in the subject. These judgments therefore are “expansive.” Kant allowed that all judgments of experience are synthetic. One could not, for example, find the concept of having weight in the concept of a body. Rather, one must look to the experience from which the concept of body was abstracted and find that heaviness is accompanies all the other characteristics that are found in that concept, such as “extension, impenetrability, shape, etc.” (A8/B12). Kant went on to assert that nearly all judgments in mathematics and metaphysics are synthetic. (Exceptions include analytic truths such as that A = A.)

In the Introduction, Kant tries to show that mathematical judgments are synthetic by showing they are not analytic. That is, Kant argues that, as in the case of the judgments that 7 + 5 = 12 and that a straight line is the shortest distance between two points, the predicate is not contained in the subject.

But negative claims are hard to establish. Kant asserted that in thinking the union between 7 and 5, “we are not thinking in any way at all what that single number is that unites the two” (B15). Leibniz had tried to show that such judgments are analytic, and perhaps someone could come up with a better proof at some point.

This deficiency can be overcome by giving an account of the nature of mathematical objects such as numbers and lines. Such an account is given, but it lies almost at the end of the book, in a section entitled “The Discipline of Pure Reason in its Dogmatic Use” (A712-738/B740-766). In this section, Kant argues that mathematical concepts (unlike those of philosophy) are constructed. His example throughout is the concept of a triangle. In the arithmetic case, the number twelve is constructed by the addition of five units to seven units. So we can see why Kant said that we need to use an aid like our fingers to determine the truth of the judgment: it is about a constructed concept.

In the Inaugural Dissertation, Kant gave a concrete counter-example to the claim that all mathematical judgments are analytic. This example turns up in the Prolegomena but not, curiously, in the Critique. There are in space “incongruent counterparts,” like a right-hand glove and a left-hand glove. Conceptually, the two are indistinguishable, so no analysis of concepts could reveal the fact a right hand cannot fit into a left-hand glove, and vice-versa. Intuition, the presentation of right- and left-hand gloves, is required to reveal the geometrical difference between the two.

Hume Revisited

We can now return to Kant’s remarks on Hume in the Prolegomena, which are summarized at B19-20 in the second edition of the Critique. Kant held that if Hume had recognized that mathematical judgments are synthetic, he would have seen that the problem with metaphysical judgments (which Hume recognized as being synthetic) would apply to mathematical judgments as well. We could form nothing but empirical mathematical concepts, and our judgments of mathematics would lack the character of necessity. Had Hume thought of this connection, “he would have been led into considerations which must needs be similar to those that now occupy us, but which would have gained inestimably from his inimitably elegant style” (Prolegomena, Preamble, Ak 4:273).

It is interesting to note that in the earlier Treatise of Human Nature, Part II, Hume was guilty of what Kant thought he was “far too acute to do,” namely, “subjecting the axioms of mathematics . . . to experience” (Prolegomena, Preamble, Ak 4:273). Hume wrote in that early work that geometry is “built on ideas, which are not exact, and maxims, which are not precisely true” because its ideas all are derived from the impressions of sense of finite size (A Treatise of Human Nature, Book I, Part II, Section 4).

When geometry decides any thing concerning the proportions of quantity, we ought not to look for the utmost precision and exactness. None of its proofs extend so far. It takes the dimensions and proportions of figures justly; but roughly, and with some liberty. Its errors are never considerable; nor would it err at all, did it not aspire to such an absolute perfection. (A Treatise of Human Nature, Book I, Part II, Section 4)
This view is not found in the Enquiry. But it does raise an interesting point. If geometrical objects are really constructed, as Kant claimed, what exactly are they constructed from? If they are constructed from sensible points, then he is subject to Hume’s objections to the precision of geometry.

The Plan of the Critique

The general issue in the Critique is how synthetic judgments a priori are possible. (“Now the proper problem of pure reason is contained in this question: How are synthetic judgments possible a priori” (B19).) Kant thought that synthetic judgments in different domains of inquiry must be treated differently. Thus the question of how mathematical judgments are possible is separated from the question of how judgments of “pure natural science” are possible. These latter judgments include the conservation of matter, the law of inertia, and the reciprocity of action and reaction (B21, footnote 243). Much of the rest of the Critique is devoted to exposing the errors of dogmatic metaphysics. At the end, there is a discussion of philosophical method.

The basic structure of the book is as follows.

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