Protocol: IntroductionAllen Jeffrey Gurfel

Leibniz sorted all propositions into two mutually exclusive categories: analytic or synthetic. Hume too divided beliefs into two categories: a priori beliefs, involving relations of ideas; and a posteriori beliefs, involving matters of fact. Kant takes up these distinctions but makes a novel claim. He claims that we possess synthetic a priori knowledge. The CPR is concerned with answering the question How is synthetic a priori judgment possible? The Introduction serves to lay out Kants schematic of judgment and to present candidatesfor example, mathematical propositionsfor the title of synthetic a priori knowledge.

Kant immediately and unequivocally acknowledges that, as far as time is concerned, all our cognition begins with experience. It does not follow, however, that all cognition arises from experience. It may be that impressions from without and the set-up of our cognitive faculties contribute to a composite experiential cognition. This suggestion is not original to Kant. Lockean secondary properties produce qualitative experiences that are functions of both the object and the subject. For example, color is a function of the object perceived and the visual apparatus and mind of the perceiver.[footnoteRef:1] And Spinoza argued that our perceptions of external objects are a function of the nature of those objects and the nature of our own bodies. [1: From 13 of Kants Prolegomena: "Long before Locke's time, but assuredly since him, it has been generally assumed and granted without detriment to the actual existence of external things, that many of their predicates may be said to belong not to the things in themselves, but to their appearances, and to have no proper existence outside our representation. Heat, color, and taste, for instance, are of this kind. Now, if I go farther, and for weighty reasons rank as mere appearances the remaining qualities of bodies also, which are called primary, such as extension, place, and in general space, with all that which belongs to it (impenetrability or materiality, space, etc.)no one in the least can adduce the reason of its being inadmissible."]

The a priori / a posteriori distinction concerns the origin of cognition. Kant defines a priori cognitions as those that are absolutely independent of experience and sense impressions. A priori judgments have two identifying features: necessity and universality. Empirical cognitions, on the other hand, are a posteriori, possible only through experience, and thus always contingent, having at best merely comparative universality.

The analytic/synthetic distinction concerns the content of judgment. A judgment in which a subject is put in relation to a predicate must be one of two types, analytic or synthetic. The analytic relation is one of containment; that is, if the predicate is already contained in the concept of the subject, then the judgment is analytic. Such a judgment may clarify a concept, breaking it down in some way, but it does not add to knowledge. If the negation of a proposition leads to a contradiction, then the proposition is analytic. All other judgments are synthetic. In a synthetic judgment, a judgment of amplification, something new is predicated of the subject; that is, the predicate is not already contained in the concept of the subject. Judgments arising from experience are always synthetic. Neither a synthetic judgment nor its negation lead to any conceptual contradiction.

What is synthetic a priori judgment? First, the truth of such a judgment cannot be decided merely by content, since the subject of a synthetic judgment doesnt contain the predicate. Second, such judgments are not derived from experience, as experience is in principle incapable of delivering universality and necessity. Third, such judgments are necessary, universal, and ampliative (i.e. informative). Consider the judgment every event has a cause. This judgmentas opposed to the similar every effect has a causeis synthetic, as the concept of an event does not logically entail any notion of causation. It is not contradictory to imagine a non-caused event. Yet if this judgment is true, it is not made so by experience. Its necessity is non-logical and it is known absolutely prior to experience.

Kant provides an example of synthetic a priori knowledge: mathematical judgments, the fundamental propositions of arithmetic and geometry are synthetic a priori. That mathematical judgments are a priori is not very controversialthey are both necessary and universal, the two infallible markers. But what does it mean to say that they are synthetic? It is worth looking at Kants own example from arithmetic, 5 + 7 = 12.

Kant begins by observing that mathematicsthe inferences of mathematicshas been built up from some basicsthe principlesin accordance with the law of contradiction. It does not follow, however, that the principlesthat is, the basics themselvescan also be cognized from the principle of contradiction. Kant writes that a synthetic proposition can be comprehended in accordance with the principle only insofar as another synthetic proposition is presupposed from which it an be deduced, and never in itself. What does he mean by in accordance with the principle of contradiction? What does it mean to show that 5 + 7 = 12 can be derived from the concept of a sum of seven and five in accordance with the principle of contradiction? Presumably, this would be to demonstrate the analyticity of 5 + 7 = 12 by showing that the negation of 5 + 7 = 12 leads to a contradiction. It is obviously true that the sum of five and seven is twelve, but this is not what Kant is after, for as he writes at B205, that I ought to think this is the addition of the two is not here at issue; in the case of an analytic proposition the question is only whether I actually think the predicate in the representation of the subject. As Kant notes, the point is more evident with greater numbers, say 13,492,112.922 + 1,299,093,283.0242. It is certainly true that, as far as what is actually thought, the sum (1,312,585,395.95) is not part of the concept of either addend or of the mere concept of the sum of the addends. The solution requires the assistance of an intuition. The intuition seems to be one of sequence (and incrementation), as Kant describes the step-wise process of addition. But such an intuition presupposes as a condition the sensible intuitions of space and time. A sequence cannot be conceived in the absence of a spatial or temporal dimension. These intuitions, Kant will argue, are provided from our side, so to speak; that is, they arise from our cognitive faculties and are the transcendental conditions of any and all possible experience as such, and are thus necessary and universal.

So mathematical knowledge is knowable a prioriabsolutely without experiencebut requires a contribution from that aspect of our cognition which does not arise from experiencealthough it may be first set in motion, as a temporal matter, by experience. Mathematical knowledge is synthetic, as it is informative and its truth cannot be derived from a mere logical analysis of the concepts.