Kant on the Synthetic A Priori

Immanuel Kant (1724–1804) was the most significant German philosopher of the eighteenth century, and was a key figure in the Enlightenment. He wrote most of his most famous philosophical works relatively late in his professional life, having only achieved a permanent position as professor in 1770, at the age of forty-six. From 1781 to 1798 Kant published a series of tremendously influential philosophical works, including the Critique of Pure Reason (1781/​7), the Groundwork of the Metaphysics of Morals (1785), the Metaphysical Foundations of Natural Science (1786), the Critique of Practical Reason (1788), and the Critique of the Power of Judgment (1790).

We’ll be primarily focusing on Kant’s project in the Critique of Pure Reason, as well as the moral philosophy of the Groundwork for the Metaphysics of Morals and the Critique of Practical Reason. There are certain aspects of Kant’s project in the Critique of Pure Reason that should be very familiar to those with knowledge of prominent European philosophers of the 17th and 18th centuries. For example, like Descartes, Locke, and Hume, Kant wishes to articulate the nature and extent of human knowledge, and to do so in a manner which proceeds from an analysis of the nature of human cognitive capacities. Kant believes that human reason requires a ‘critique’—a kind of process of self-examination or scrutiny—as to whether it is fit to provide us with knowledge. Kant ultimately argues that human reason is not fit to provide us with knowledge of a mind-independent reality transcending human experience. In this sense Kant is deeply sympathetic with ‘empiricist’ critiques by Locke, Berkeley, and Hume of ‘rationalist’ philosophers such as Descartes and Leibniz.

However, Kant also argues that we have much deeper and more extensive knowledge of the world we experience (or could possibly experience) than his empiricist predecessors would allow. For example, Kant argues that we have knowledge of necessary truths concerning aspects of the empirical world (such as that every event in the empirical world has a cause), as well as truths which are universal in scope or extent. Thus Kant articulates a view that is directly opposed to the kinds of skeptical arguments Hume discusses in his Treatise and first Enquiry.

Kant thus thinks that we have knowledge of the necessary and universal laws governing the empirical world—or more simple, “nature”—while he nevertheless argues that we are almost wholly ignorant of the fundamental reality which underlies or grounds the existence of the natural world. In this way Kant combines various aspects of both the traditional rationalist and empiricist positions. Like Locke and Hume, Kant thinks we must realize that the boundaries of human knowledge stop at experience, and thus that we must be extraordinarily circumspect concerning any claim made about what reality is like independent of all human experience. But, like Descartes and Leibniz, Kant thinks that central parts of human knowledge nevertheless exhibit characteristics of necessity and universality, and that, contrary to Hume’s skeptical arguments, we can have good reason to think that they do.

Kant thus critiques pure reason in order to show its nature and limits, and thereby curb the pretensions of various metaphysical systems articulated on the basis of a firm faith that reason alone allows us to scrutinize the very depths of reality. But Kant also argues that the legitimate domain of reason is more extensive and more substantive than previous empiricist critiques had allowed. In this way Kant salvages much of the prevailing Enlightenment conception of reason as an organ for knowledge of the world.

Below I characterize some of the central aspects of Kant’s epistemological framework and how that framework significantly revolutionized our understanding of the possible nature and extent of human knowledge.

The A Priori

A variety of philosophers from the Early Modern period (e.g. Descartes, Leibniz, Hume) argue that there are kinds of knowledge that may be had just by thinking, and that such forms of knowledge are universal and necessary in scope. Hume’s conception of knowledge made this especially clear. According to Hume, knowledge of necessary and universal truths must be purely a function of knowledge of the relations of ideas.

Kant agrees with Hume that a priori knowledge is independent of experience. In fact, he sees it as definitive of ‘pure’ a priori knowledge that it be completely independent of experience in the sense that its content is neither derived from experience not justified by experience in its application. He contrasts such knowledge with ‘empirical’ knowledge or knowledge a posteriori. A priori knowledge may be more or less ‘pure’ according to whether or not the concepts which make it up are themselves a priori knowable. Kant uses the example ‘every alteration has a cause’ as an example of impure a priori knowledge, since the concept <alteration> is empirical.

One important point about Kant’s use of ‘independent’ in describing the a priori. Kant does not think that a priori knowledge is independent of experience in the sense that one need not have any experience in order to have knowledge. On the contrary, Kant thinks that all of our knowledge depends on our having experience of some kind or another, though he doesn’t think this dependence entails that all our judgments are ultimately justified by experience. This is why Kant says that

But although all our cognition commences with experience, yet it does not on that account all arise from experience. (CPR B1)

There seems to be two reasons for Kant’s thinking this. First, we need experience in order for our cognitive faculties to function and develop. Second, we may need particular experiences in order to acquire empirical concepts (e.g. red experiences in order to acquire the concept <red>). This is what distinguishes pure from impure a priori judgments. Impure a priori judgments are partially constituted by concepts whose content is in some way derived from content given by the senses.

If a priori knowledge is, for Kant, knowledge that is (in some sense to be further specified) independent of experience, that fact is not the only mark or indicator that some bit of knowledge is a priori. In addition, Kant argues, any bit of knowledge that is necessary and/or universal in scope is itself a priori.

Necessity and strict universality are therefore secure indications” of an a priori cognition, and also belong together inseparably. But since in their use it is sometimes easier to show the empirical limitation in judgments than the contingency in them, or is often more plausible to show the unrestricted universality that we ascribe to a judgment than its necessity, it is advisable to employ separately these two criteria, each of which is in itself infallible. (CPR B4)

Kant argues that this conception of a priori knowledge is presupposed in many empirical judgments as well as in particular sciences. He specifically points to mathematics (‘5+7=12’) and to physical judgments (‘every alteration has a cause’). One of Kant’s arguments against Humean skepticism is that all of our empirical knowledge (even that knowledge which we think we have unproblematically) presupposes a priori knowledge, which itself requires that there be legitimate use of pure a priori concepts (i.e. concepts which cannot be derived from any sense impression).

Kant thinks that there are many examples of judgments which we claim to know a priori, but he is interested primarily in a specific subset of those which constitute the subject matter of metaphysics—viz. judgments concerning God, the soul (or mind), and immortality. One of Kant’s primary aims is to determine whether metaphysical knowledge is possible, and if it is possible, what the extent and nature of that knowledge might be. This is what it means when Kant says that he wishes to set metaphysics “on the secure path of a science” (e.g. Bxvi). Metaphysical knowledge is problematic, Kant argues, because unlike other forms of a priori knowledge, such as logic and mathematics, it is not at all obvious which metaphysical judgments are in fact correct and thus known, and which are merely thought to be so. This is exemplified, Kant thinks, by the contentious disputes in which philosophers have long been involved. Kant provides the following memorable description of the sad plight of metaphysics.

In metaphysics we have to retrace our path countless times, because we find that it does not lead where we want to go, and it is so far from reaching unanimity in the assertions of its adherents that it is rather a battlefield, and indeed one that appears to be especially determined for testing one’s powers in mock combat; on this battlefield no combatant has ever gained the least bit of ground, nor has any been able to base any lasting possession on his victory. Hence there is no doubt that up to now the procedure of metaphysics has been a mere groping, and what is the worst, a groping among mere concepts. (Bxiv-xiv)

Kant thus hopes that, by giving a critique of reason, he can demonstrate the extent to which metaphysics might count as a science. As a secure science it would rest on a secure set of claims and no longer be a “groping among mere concepts.”

The Analytic/Synthetic Distinction

A judgment is known (or knowable) a priori if it is known (or knowable) independently of experience. Kant thinks that this is not the only dimension according to which one can analyze a judgment. Kant also argues that all judgments, in addition to being classifiable as either a priori or a posteriori (or empirical), may also be classified as being either analytic or synthetic.

An analytic judgment is one in which the predicate is ‘contained’ in the concept. One way of understanding this notion of ‘containment’ is via a claim about meaning. Accordingly, analytic judgments are those whose truth is known merely in virtue of knowing the meaning or content of the concepts constituting the judgment. For example, in the judgment ‘All bachelors are unmarried’ the concept <unmarried> is part of the meaning of <bachelor> and so the truth of the judgment is grasped just by knowing the relevant meanings or contents of its component concepts.

Kant also introduces several other markers of analyticity. In total Kant provides us with four different marks of analyticity. He says that in an analytic judgment the predicate is

  1. ‘contained’ within the subject
  2. ‘identical’ with the subject
  3. analytic judgments are ones which are ‘explicative’ rather than ‘ampliative’
  4. analytic judgments are those knowable by means of application of the principle of non-contradiction

Which, if any of these markers is best thought of as the main characteristic of an analytic judgment? This is a disputed issue (cf. (Anderson 2005; Proops 2005)), though certainly, in all cases Kant is thinking of ‘atomic’ judgments of subject-predicate form.

The Synthetic A Priori

Kant argues, in ways similar to Locke, Hume, and Leibniz, that analytic judgments are knowable a priori. Staying with the ‘containment’ metaphor, since the predicate is contained in the subject of an analytic judgment, there is no need to look beyond the judgment to the world (so to speak) in order to determine the truth value of the judgment. In this Kant is obviously in agreement with Locke, Hume, and Leibniz. Kant agrees with his empiricist predecessors in claiming that all a posteriori judgments are synthetic. Since the predicate is adding something new to the subject we must look beyond the judgment to the world—what we can experience—in order that we might determine the relevant judgments truth or falsity. In non-empirical cases (like the bachelor example above) the judgment’s truth is determined by the meanings of the concepts constituting it.

Kant’s main innovation to the a priori/posteriori and analytic/synthetic schemas is to note that the analytic a priori and the synthetic a posteriori do not necessarily exhaust the realm of possible judgments. Here he essentially can be understood to deny that “Hume’s Fork” is an adequate representation of the structure of human knowledge. According to Kant, there are also synthetic a priori judgments that are possible. Kant argues that causal judgments are a clear example.

it is easy to show that in human cognition there actually are such necessary and in the strictest sense universal, thus pure a priori judgments. If one wants an example from the sciences, one need only look at all the propositions of mathematics; if one would have one from the commonest use of the understanding, the proposition that every alteration must have a cause will do; indeed in the latter the very concept of a cause so obviously contains the concept of a necessity of connection with an effect and a strict universality of rule that it would be entirely lost if one sought, as Hume did, to derive it from a frequent association of that which happens with that which precedes and a habit (thus a merely subjective necessity) of connecting representations arising from that association (CPR B4-5)

Take the proposition: “Everything that happens has its cause.” In the concept of something that happens, I think, to be sure, of an existence that was preceded by a time, etc., and from that analytic judgments can be drawn. But the concept of a cause lies entirely outside that concept, and indicates something different than the concept of what happens in general, and is therefore not contained in the latter representation at all. How then do I come to say something quite different about that which happens in general, and to cognize the concept of cause as belonging to it, indeed necessarily, even though not contained in it? What is the unknown=X here on which the understanding depends when it believes itself to discover beyond the concept of A a predicate that is foreign to it yet which it nevertheless believes to be connected with it? (CPR B13-14)

Kant argues here that our judgments concerning events presuppose that they do not just occur but are caused to occur, that we know this to be true necessarily and universally, and that we have no explanation of this fact unless the judgments we make in such cases are synthetic a priori judgments. The question remains, however, just how such synthetic a priori judgments could be possibly true, much less known to be so. What is it that could link the concepts in a subject-predicate judgment such that the truth of the judgment holds necessarily and universally, while its nevertheless being true that the predicate is not contained in the subject of the judgment, and thus that the judgment is not analytic?

Now the entire final aim of our speculative a priori cognition rests on such synthetic, i.e., ampliative principles; for the analytic ones are, to be sure, most important and necessary, but only for attaining that distinctness of concepts which is requisite for a secure and extended synthesis as a really new acquisition (CPR B13-14)

Kant argues that we need to explain how synthetic a priori judgments are possible, and that the explanation of the possibility of significant portions of our knowledge rests on this, including mathematics and natural science, as well as the very possibility of metaphysics.

Mathematics and the Synthetic A Priori

Recall that for Hume, mathematical knowledge was a function merely of knowledge of relations of ideas, in other words, that math is analytic and a priori. Kant disputes this. Our mathematical knowledge is certainly a priori, he thinks, but it is not explained in terms of relations of ideas or concepts (i.e. in terms of containment or any of the other marks of analyticity). Instead, Kant argues that mathematical knowledge must be synthetic, but since it is necessary and universal, also a priori. Here’s how Kant puts the point in his Prolegomena:

The essential feature of pure mathematical cognition, differentiating it from all other a priori cognition, is that it must throughout proceed not from concepts, but always and only through the construction of concepts (Critique, p. 713). Because pure mathematical cognition, in its propositions, must therefore go beyond the concept to that which is contained in the intuition corresponding to it, its propositions can and must never arise through the analysis of concepts, i.e., analytically, and so are one and all synthetic. (Prolegomena 4:272, p. 20)

Kant uses two examples in his argument concerning mathematics. The first is the judgment ‘7+5=12’ and the second is the geometric judgment that ‘the shortest distance between two points is a straight line’. Though both judgments are knowable a priori, Kant thinks that in neither of these two cases can we explain the truth of the judgments analytically.

One might well at first think: that the proposition 7 + 5 = 12 is a purely analytic proposition that follows from the concept of a sum of seven and five according to the principle of contradiction. However, upon closer inspection, one finds that the concept of the sum of 7 and 5 contains nothing further than the unification of the two numbers into one, through which by no means is thought what this single number may be that combines the two. The concept of twelve is in no way already thought because I merely think to myself this unification of seven and five, and I may analyze my concept of such a possible sum for as long as may be, still I will not meet with twelve therein. (Prolegomena 4:268-9, p. 18)

Kant’s argument here is that while it might be analytically true that the sum of 7 and 5 is a number (and also that it must be a natural number), it is not clear from analysis alone that the sum is equal to 12, or any other pair of natural numbers which might sum to 12.

Kant seems to think that to the extent one finds the answer twelve obvious one is adverting, either explicitly or implicitly, to some particular experience of adding units, such as counting on one’s fingers, or adding up objects (e.g. apples, chairs, blocks, etc.).

Also, if ‘7+5=12’ is known analytically, then in thinking it one is equivalently thinking of any or all the numbers which are equal to 12, including very large numbers (e.g. any numbers m and n which might yield 12). This certainly doesn’t reflect our experience when calculating such sums, which may yield further evidence that the judgment isn’t analytically true.1


Allison, Henry E. 1973. The Kant-Eberhard Controversy. Baltimore: The Johns Hopkins University Press.

———. 2004. Kant’s Transcendental Idealism: Revised and Enlarged. New Haven: Yale University Press.

Beck, Lewis White. 1955. “Can Kant’s Synthetic Judgments Be Made Analytic.” Kant-Studien 47 (1-4): 168–81.

———. 1956. “Kant’s Theory of Definition.” The Philosophical Review 65 (2): 179–91.

———. 1969. Early German Philosophy. Cambridge: Belknap Press of Harvard University Press.

———. 1978. “Analytic and Synthetic Judgments Before Kant.” In Essays on Kant and Hume, 82–100. New Haven, CT: Yale University Press.

Gardner, Sebastian. 1999. Kant and the Critique of Pure Reason. London: Routledge.

Hogan, Desmond. 2013. “Metaphysical Motives of Kant’s Analytic\textendash synthetic Distinction.” Journal of the History of Philosophy 51 (2): 267–307.

Proops, Ian. 2005. “Kant’s Conception of Analytic Judgment.” Philosophy and Phenomenological Research 70 (3): 588–612.

Van Cleve, James. 1999. Problems from Kant. Oxford: Oxford University Press.

  1. For further discussion see also (Allison 1973, 2004; Beck 1955, 1956, 1978, 1969; Gardner 1999; lovejoy1906a; Van Cleve 1999; Proops 2005; Hogan 2013). ↩︎