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.
The status of judgments in pure mathematics and in pure natural science
But now it may be objected that the claim that pure mathematics and natural science are bodies of synthetic a priori judgments is far from obvious. In his Introduction to the First Critique, Kant goes to considerable lengths to convince his readers that the axioms or principles of both disciplines are genuine instances of synthetic a priori judgments.
Mathematics
In the case of pure mathematics, he does not consider it incumbent upon him to prove that its judgments are a priori. He takes it as uncontroversial that the judgments of both geometry and arithmetic hold with necessity and universality. When it is said that the internal angles of a (Euclidean plain) triangle add up to 180 degrees, it is implied that this must be true of all triangles. The judgment, therefore, is a priori. And the a priori nature of this particular judgment, Kant plausibly holds, goes for all geometrical judgments. Similarly, in arithmetic, the judgment that 7 + 5 = 12 is a priori; since the judgment implies that whenever the numbers 7 and 5 are added, 12 must be the result. Again, this point can be generalized. The key issue for him is not whether mathematical judgments are a priori, but whether they are synthetic.
He thought that previous philosophers – including Leibniz and Hume – were guilty of a serious oversight in supposing that mathematical judgments are analytic (and hence that the denial of a true mathematical judgment is self-contradictory).This cannot be right, he argues, since we must have recourse to construction in order to determine the truth or falsity of any mathematical judgment. So, in the case of the geometrical question concerning the sum of the internal angles of a triangle, he holds that we need to draw a triangle, either in imagination or e.g. on paper, and proceed to prove the judgment by showing, through the use of the diagram, how the angles must add up to 180 degrees. The specific procedure is well discussed at A 713–24/B 741–52, from which the following is an extract:
[The geometrician] at once begins by constructing a triangle. Since he knows that the sum of two right angles is exactly equal to the sum of all the adjacent angles which can be constructed from a single point on a straight line, he prolongs one side of his triangle and obtains two adjacent angles, which together are equal to two right angles. He then divides the external angle by drawing a line parallel to the opposite side of the triangle, and observes that he has thus obtained an external adjacent
