Q: If space is an a priori form of intuition, how can Kant claim that it has empirical reality as at A28/B44? A: Space is a priori in that its source is human sensibility, not experience. But space is empirically real in the sense that it is a component of the world of experience, as the mind's contribution to experience.
Q: What does Kant mean when he says that space is an infinite given magnitude (A25/B39)? A: Kant uses the infinitude of space to show that space is not a general concept, but an intuition, and hence a "given" magnitude. General concepts have an infinite number of possible things under them (in their "sphere"). The concept 'body' comprehends potentially infinitely many bodies. But the infintude of space is that of infinitely many spaces together making up one space.
Q: If space is an intuition, why does Kant refer to it as a concept? A: Space is indeed an intuition, not a general concept. When Kant speaks of the concept of space, he means the conception we have of what space is, or the way we characterize space generally.
I resume the textual narrative. We have completed our study of Transcendental Aesthetic, the doctrine of sensibility and its intuition, and we now turn to Transcendental Logic, the doctrine of the understanding and its concepts. Logic was understood by Kant as the study of the rules of the understanding. Construing logic in this way is contrary to the contemporary view introduced by Frege in the late nineteenth century. Frege is said to have exorcised "psychologism" from logic and placed it on an objective footing. (Frege wrote, "If people consider, instead of things themselves, only subjective representations of them, only their own mental images -- then all the more delicate distinction in the things themselves are naturally lost, and others appear instead which are logically quite worthless" (Grundgesetze der Arithmetik, Vol. 1).
Kant begins his discussion of the understanding by making a number of distinctions between types of logic. The first is between general and special logic. General logic holds for all objects whatsoever, while special logic applies only to objects of a certain type. Some examples of special logic taken from contemporary philosophy might be quantum logic (restricted to the peculiar realm of quantum mechanics) and denotic logic (restricted to the concepts of obligation and permission). This logic is an "organon" of the specific "science" to which it applies, according to Kant. This means that it contains substantive principles of that science.
The next distinction is that between pure and applied general logic. Applied logic is overtly psychological. "It treats of attention, its impediments and consequences, of the source of error, of the state of doubt, conviction, etc." (A54/B78-9). To some extent, this is reminiscent of an epistemological view known as externalism. According to this view, whether one knows depends on whether one is related properly to the facts to be known. Pure general logic, on the other hand, presents rules for the formal employment of the understanding (it is a "canon" for the understanding).
The final distinction is between general logic "transcendental logic." Whereas general logic is not concerned with the origin of our representations, treating all of them alike, transcendental logic would contain rules for a priori thought of objects. According to Kant, some of our cognitions, e.g. space and time, are a priori. The representation of them as being a priori is a transcendental representation. Thus a transcendental representation is a meta-cognition. "The term 'transcendental' . . . signifies such cognition as concerns the a priori possibility of cognition, or its a prioriemployment" (A56/B81).
Transcendental logic, then, consists in rules for the employment of concepts originating a priori in the understanding. Both general and transcendental logic are divided by Kant into analytic and dialectic. The more modest analytic shows how a priori concepts and the principles associated with them provide a negative criterion of truth. Truth or falsehood of cognitions is the distinction between those cognitions to which an object agrees or corresponds, and those to which one does not. For example, a principle of general logic is that of contradiction, so that any concept which is self-contradictory is thereby "false," in the sense that it does not apply to any object whatsoever. A principle of transcendental logic is that of causality, according to which no event occurs without being the consequence of a previous state of affairs and a law of nature. So a representation of an uncaused event is "false," because it conflicts with transcendental logic.
Dialectic purports to give a positive criterion of truth, which Kant thought to be impossible. General logic would attempt to parlay its rules into an organon, stating exactly what truths hold. Kant gives no examples of such a procedure, but he declares that it produces nothing but illusion, producing "mere talk." Transcendental logic attempts to extend the principles of the understanding beyond experience, passing judgment even "upon objects which are not given to us, any, perhaps cannot in any way be given (A63/B88). This "hyperphysical" employment of the understanding will be criticized in the long section of the Critique, the Transcendental Dialectic.
The task of transcendental analytic is to "dissect" the faculty of the understanding into its elements. This complicated task is divided into two parts. The first is to show which pure concepts the understanding generates and to show by what right the understanding applies them to objects. The second shows how specific principles ssociated with the pure concepts are justified.
Kant lays out for "concerns" to be met in the Analytic: to show that the concepts are pure and not empirical, that they belong to the understanding, that they are fundamental, and that the system of such concepts be complete. I will devote a little time to the last concern first, because Kant's insistence on completeness and systematicity have been criticized quite severely.
Kant claimed that the categories, the pure concepts of the understanding, must be generated in a systematic way, from a single principle. His reason for so insisting is that the understanding itself is a unity, so its concepts must be unitary. But this seems to be a flat-out non-sequiter. There is no reason to think that the product of something unitary must itself be unitary.
The critics maintain that this kind of arguing is mere subtrefuge, covering up Kant's personal prediliction for orderliness. In many cases, Kant's views on a subjects are summarily dismissed as due to his "passion for architectonic." But in fairness, we must point out two things. First, Kant was working in the tradition of Leibniz, who tried to reduce all principles of reason to two: contradiction and sufficient reason. Second, that same tradition (following Wolff) generated a bewildering array of categories with no apparent justification for the arrangement. Concepts got into the system because they had some use within the system or simply were traditional. For example, the concepts "one," "true," "perfect" are at one point introduced arbitrarily under the concept of "being." That these concepts should have any place at all was due to their long-standing presence in scholastic philosophy (or so Kant claimed).
This said, we shall now examine the "single principle" that Kant proposed as means to generate his table of categories. Before introducing it, Kant supplies us with a "clue" to discovery of the system of categories. This clue is a similarity between concepts and judgments. Both are "higher" or meta-representations: representations of representations.
Jugments combine cognitions into a unity. The concepts 'body' and 'divisible' are combined in the judgment 'all bodies are divisible.' Thus the two concepts are represented as belonging to each other. The former concept applies to objects (bodies, which are appearances), while the latter applies more widely. The judgment applies the concept of divisiblity to the objects comprehended under the concept of a body. But the concept body does not represent objects directly, either. To apply to something, it must be combined in a further judgment, say one which predicates it of metals, as in 'Every metal is a body.' And the concept of metal also represents indirectly. Ultimately, every concept is indirectly representative, because only intuitions represent objects directly. There are no "lowest species" or "individual concepts" which are so specific as to pick out only one thing.
So just as judgments represent objects indirectly through concepts, concepts themselves represent objects indirectly. Both are "functions of unity" wherein several items are combined into one. The "clue" to the production of the table of categories is that the combination of representations effected by the judgment is subject to certain "forms" of judgment. In turn, the combination of intuitions in concepts is subject to concepts which are analogous to these forms. But there is much work to do before we can reach this point.
In class I discussed the forms of judgment themselves, as laid out in the table of judgment. These are Kant's reworking of traditional characterization of propositional forms in Aristotelian logic. My presentation of this material can be found in the lexicon, under 'judgment.'