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Kant identifies three possible types of judgment:

1 Analytic a priori judgments;

2 Synthetic a posteriori judgments;

3 Synthetic a priori judgments.

In order to explain this threefold division, he further distinguishes between analytic and synthetic judgments, on the one hand, and a priori and a posteriori judgments, on the other. These further distinctions can be explained as follows.

An analytic judgment is one in which the meaning of the predicate term is included in the meaning of the subject term. Example: ‘All bachelors are unmarried (men).’ Kant notes that the denial of an analytic judgment is self-contradictory (as in ‘It is not the case that all bachelors are unmarried’).

A synthetic judgment is one in which the meaning of the predicate term is not contained in the meaning of the subject term. Example: ‘All men are mortal.’ Accordingly, the denial of a synthetic judgment is not self-contradictory. (The judgment ‘It is not the case that all men are mortal’ is doubtless false; but it is not self-contradictory, given the meaning of ‘men’ and ‘mortal’, etc.)

An a priori judgment is a judgment that is thought of as holding independently of experience. Kant says that there are two ‘sure criteria’ of, two infallible ways of identifying, an a priori judgment. If a judgment claims to hold either with necessity or strict universality, then it must be an a priori judgment. For no judgment that depends on experience can be thought of as holding either necessarily or with strict universality. Experience can show that some judgment is or is not the case, but not that it necessarily is or is not the case (or must or must not be the case). Similarly, while experience can show that all instances so far examined of a particular unrestricted class are such-and-such, it cannot show that all past, present and future instances of that class are such-and-such. The most that our evidence to date can entitle us to claim, assuming it is wide-ranging and that no counter-examples have been encountered, is what Kant calls ‘comparative universality’. That is, we may employ an inductive argument on the basis of our experiential evidence to date, and claim that all instances are probably such-and-such. But cases of comparative universality are not cases of strict universality (where ‘no exception is allowed as possible’ (B 4)). Throughout his critical works, whenever Kant discusses universality, he means strict, not comparative, universality unless he says otherwise.

It is worth noting that Kant does not define an a priori judgment as one that claims to hold with necessity and/or universality. For him, an a priori judgment is defined as one that is thought of as holding independently of experience. But if we can discover a judgment that does claim to hold with necessity and/or universality, then we can be sure that it is a priori. Necessity and universality are infallible means of recognizing an a priori judgment.

Given his definition of an a priori judgment, we can understand Kant’s claim that besides a priori judgments (or principles) there may also be – indeed, are – a priori concepts. For example, he holds that the concept of cause is an a priori concept. By this he means that it is not derived, or formed, from any experience; it is, rather, a concept that we possess independently of experience. Such concepts may well be applied to what can be experienced, although this is not an essential feature of an a priori concept. What is essential is that an a priori concept is a concept that we possess independently of any experience.

An a posteriori (or empirical) judgment is a judgment that is thought of as holding on the basis of experience. It cannot, therefore, claim to hold with either necessity or (strict) universality. ‘Copper dissolves in sulphuric acid’ is an a posteriori judgment. We need to consult experience in order to confirm or disconfirm it. And the judgment that is, in fact, thereby confirmed carries only comparative universality, based on our past and present experience of the behaviour of copper in sulphuric acid.

Just as there can be a priori concepts, so there can be a posteriori ones.As one would expect, these are dependent (at least in part) upon experience. Our concept of copper is a posteriori, since it requires experience in order to be formed.

We can now examine Kant’s division of judgments into three categories. In particular, we need to ask how each type of judgment can be established.

(1) Analytic a priori judgments. Since every analytic judgment depends, for Kant, only on the meaning of the terms involved, such a judgment can always be established by determining that the meaning of the predicate term is included in the subject term, i.e. by showing that the denial of the judgment is self-contradictory. All analytic judgments, therefore, must be a priori, since no recourse to experience is necessary to establish them. (The subject and the predicate terms themselves may require experience in order to be formed, as in the case of ‘bachelor’ and ‘unmarried’. But the point is that once the meaning of the subject and the predicate terms is understood – whether they are themselves expressive of a priori or a posteriori concepts – there can be no need to consult experience in order to establish any analytic judgment. All that is required is an analysis of the meaning of the terms involved, together with the application of the principle of contradiction.)

(2) Synthetic a posteriori judgments. These judgments are not analytically true, and are established by recourse to experience. There is clearly no difficulty in grasping how there can actually be such judgments. When a judgment cannot be determined in virtue of the meaning of the terms involved (and so is synthetic), it is an entirely familiar, and frequently a successful, procedure to seek to establish it a posteriori, i.e. by consulting experience. The judgment ‘All men are mortal’ is synthetic. It is also a posteriori, since it is established on the basis of past experience and induction (the universality claimed is only comparative). Note that all empirical judgments – judgments that make recourse to experience – are synthetic a posteriori. For if a judgment requires experience to be established (and so is a posteriori), it cannot be true merely in virtue of the meaning of the terms involved. Hence, it must be synthetic as well as a posteriori.

(Granted what is said under (1) and (2), it is clear that there cannot possibly be any analytic a posteriori judgments. If such judgments existed, they would have to be true merely in virtue of the meaning of the terms involved yet require experience to be established. But since any judgment whose truth or falsity depends wholly on the meaning of its constitutive terms can always be established without consulting experience (a priori), it cannot require experience to be established.)

(3) Synthetic a priori judgments. These are not analytically true yet require to be established independently of experience. Undoubtedly, it is this class of judgments in which Kant is principally interested. Now although such judgments are not ruled out ab initio (as are analytic a posteriori judgments), it is not at all obvious how any claimed synthetic a priori judgment could ever be established. In order to establish it, we evidently cannot consult experience, otherwise the judgment would not be a priori but a posteriori. On the other hand, we plainly cannot seek to establish the judgment merely on the basis of the meaning of the terms involved. Only analytic judgments can be established in this manner; and ex hypothesi we are interested in establishing a synthetic, not an analytic, judgment. But if the judgment cannot be established either in virtue of the meaning of the terms involved or by consulting experience, how is a connection between the subject and the predicate of any supposed synthetic a priori judgment to be established?

Let us return to our earlier example. In the judgment ‘Every change of state must have a cause’, the concept cause is not included in the concept change of state.As Hume has shown, the denial of the judgment is not self-contradictory. So the judgment cannot be analytic. It is, therefore, a synthetic judgment. But the judgment also claims necessity (‘must have a cause’) as well as strict universality (‘Every change of state’). So it is an a priori judgment: one that cannot be dependent on experience. But how can we hope to establish a genuine connection between the subject and the predicate in such a judgment?

It is, of course, just this question, when generalized to include all the axioms and principles of pure mathematics and natural science, as well as the significant judgments in metaphysics, that not only awoke Kant from his dogmatic slumbers but led him to propose his own revolutionary response (his Copernican revolution). Putting the point in his own terminology, Kant holds that all the fundamental judgments in these areas are synthetic a priori. Accordingly, the key question for him is to discover how such judgments can ever be established.

Let me reiterate the point that I made in my introduction, though without employing there the terminology of the First Critique. In raising the question, ‘How is it possible to establish synthetic a priori judgments?’, Kant is not questioning the legitimacy of such judgments in two areas: namely, pure mathematics and natural science. On the contrary, he believes that it is ‘incontestable’ that, in these two areas, we are already in possession of bodies of synthetic a priori knowledge of objects. Rather, his point is that until Hume’s questioning of the synthetic a priori principle ‘Every change of state must have a cause’, it had not occurred to him (Kant) – or, he thinks, to anyone else – that this species of judgment forms the kernel not only of two bodies of undoubted scientific knowledge (mathematics and natural science), but of metaphysics as well (whose claim to be a science is by no means undoubted). Now that he has recognized the centrality of synthetic a priori judgments, he also realizes that the way to answer, if at all, the central questions of metaphysics is not essentially through any mere analysis of metaphysical assertions (the common pursuit of metaphysicians to date). How could it be, granted that these assertions are synthetic, not analytic? Instead, the central questions of metaphysics are to be answered, if they can be answered by theoretical reason at all, through first coming to understand how the fundamental synthetic a priori judgments are established in the two areas of science where it is incontestable that they do exist. Once this essential preliminary investigation has been achieved, Kant believes that we should be in a position to see whether the central questions of metaphysics can also be established.