1. B-Deduction §15-§26

2. §15
Combination (relations between empirical intuitions) cannot be given passively nor generated through active synthesis on the side of sensibility
It is therefore not something given in empirical intuitions (the ‘object as appearance’) but rather must be generated by the subject
Thus, combination “is the representation of the synthetic unity of the manifold” (B131)
That unity cannot arise from combinations since combination presupposes a unity whose form must come from “someplace higher, namely in that which itself contains the ground of the unity of different concepts in judgments…” (B131)

3. §16
The ‘higher unity’ that makes possible combination is the unity that constitutes the form of self-consciousness (apperception) itself: transcendental unity of apperception
The identity of that unified self-consciousness that accompanies all my representations (i.e., analytic unity of apperception) is only possible due to a prior synthetic unity of apperception (=an act of synthesis by means of which all my representations are brought together and represented as unified in one consciousness)
Thus, all combination, and therefore, all general representation depends on an Original Synthetic Unity of Apperception.

4. §17
Just as the pure forms of space and time are conditions of the possibility of empirical intuition, should anything be given to the mind through sensibility, so the synthetic unity of apperception is the form given to all twelve possible thinkable combinations among empirical intuitions, combinations made possible by the understanding whose contribution to cognition lies in the act of thinking/judgment, an act which makes it possible for any given representations to be combined in one consciousness.
Translation: Self-Consciousness (Apperception) has a form, and all acts of thinking bring that form to whatever is thinkable.

5. §17 cont’d
One of the things that the understanding makes possible is cognition, defined as “the determinate relation of given representations to an object.” (B137)
What is an object?
“An object … is that in the concept of which the manifold of a given intuition is united.” (note: here “united” refers directly to the unity among diverse empirical intuitions, a unity that cannot be given in sensible appearances)
This unity presupposes the only source of the form of unity: the original synthetic unity of apperception (OSUA).
Upshot: cognition of objects presupposes OSUA.

6. §18
The unity that makes possible the combination of appearances that constitutes an object is different than the unity of the empirical manifold.
In what does this difference consist?
The unity of the empirical manifold is contingent on what is given in sensibility, and its form is merely subjective (here Kant means to refer to the associations arising from the particular juxtaposition and succession of empirical intuitions, i.e., how a given subject contingently responds to that juxtaposition and succession) whereas
The unity comprised of that combination of appearances that constitutes an object is objective (since it is a unity that is determinate, i.e., grounded in the object of cognition, and thus not contingent in its form (which is given by OSUA through a synthesis according to one or more of the categories, which forms are universal and necessary features of any object given in sensibility.), nor in its matter (because the relations expressed by these combinations are themselves determined, and so constituted, by what is given).

7. What the hell did that last part mean?
Simply this: just as the matter of outer sense intuitions (i.e., sensation) is responsible for the particular spatial and temporal relations any particular sensation is perceived to have with respect to others arising in the stream of consciousness of outer appearances; …
So does the given matter of diverse empirical intuitions determine which combinations are cognized as arising from some object whose nature as an object is constituted by those combinations, the form of which is secured by the application (by means as yet to be explained [this will be given in the Schematism!) of the categories to what is given in sensibility.

8. §19. The logical form of all judgments consists in the objective unity of the apperception of the concepts contained therein.
This is where Kant offers the promised “precisely determined definition of a judgment in general” which was, remember, what he thought was valuable enough to require the revised version of the TD offered in the B Edition. To understand this definition, first we must remember that for Kant, all judgments are “relations among universals” and as such, cannot contain individuals, only concepts. Thus, for Kant, a judgment can only express “the unity, or rule, on the basis of which synthesis in imagination must proceed insofar as it is to be objective and expressible in language” (my lecture, page 14 top) Note: as Kant indicates at B141, the relations among products of reproductive synthesis (which is governed by the laws of the ‘reproductive imagination’) have only “subjective validity”, whereas those relations among “given cognitinos in every judgment” are objective.

9. §19 cont’d
Definition of a judgment in general: “…a judgment is nothing but the manner in which given cognitions are brought to the objective unity of apperception. That is what is intended by the verb of relation ‘is’” (B ) Translation: The business of judgment is to bring concepts to the unity of apperception and thereby give them the relation to an object definitive of cognition. To bring the representations included in a judgment to apperception means to bestow on them the unity of an object, and therefore of something distinct (external and independent) of the manifold appearances of which it is comprised.

10. §19 cont’d
Consequences of this claim: Not all judgments are objective. As Kant makes clear in PFM, judgments of perceptions merely give an intellectual veneer to what are essentially the verdicts of associative imagination. However, judgments of experience order the representations of which the judgment consists in conformity with the objective unity of apperception— the same objective unity which is the basis of the unity of objective time and space, and therefore of all objects in them. Each such judgment in effect gives determinate existence to a particular set of perceptions with relation to the a priori unity of all the manifold in space and time: we say it is so and so, and thereby distinguish an object which necessitates that our representations be so, and not otherwise!

11. The Payoff (Phase I of B-Deduction): The Argument Given in §20
“All sensible intuitions stand under the categories, as conditions under which alone their manifold can come together in one consciousness.”
The argument:
The unity of the sensible manifold necessarily belongs under the OSUA, which stands as the condition of its possibility and source of its form. (B143)
The form of this unity requires an act of the mind/subject which can only be supplied by the understanding, since that faculty is required to bring representations under apperception in general.
“Therefore all manifold, insofar as it is given in one empirical intuition, is determined in regard to one of the logical functions for judgment, by means of which, namely, it is brought to a consciousness in general.”

12. Upshot: the categories, which are the forms of judgment as applied to given empirical intuitions, are the necessary conditions of the possibility of the fact that empirical intuitions fall in one consciousness, and hence, their forms (all 12 of them) MUST NOT ONLY BE APPLICABLE TO SENSIBLE APPEARANCES BUT MUST MAKE ALL COMBINATION AMONG APPEARANCES POSSIBLE!

13. Argument in §20 cont’d
“But now the categories are nothing other than these very functions for judging, insofar as the manifold of a given intuition is determined with regard to them.
Conclusion: “Thus the manifold in a given intuition also necessarily stands under categories.”

14. Transition to Phase II: Why Kant needs the Remark at §21
Given what has been established through §20, what remains to be accomplished if the B-Deduction is to be complete? Answer: a solution to the heterogeneity problem. That is, so far we have only established that the categories must determine our judgments, and associated cognitions, are to be objective and thus constitute an object. But how does sensibility lend itself to this determination, given the radical difference between intuitions and concepts? After all, determination requires judgment and concepts, so intuitions could only be determined (and thus, objects of perception become possible) if the forms of universal/general representations (universals/concepts) be applied somehow to the appearances in sensibility (which are neither conceptual nor universal).

15. Remark at §21
Translation: so far, the deduction of the categories begins by showing that categories arise independent of sensibility in the understanding;
To establish this claim, we had to abstract “from the way in which the manifold for an empirical intuition is given, in order to attend only to the unity that is added to the intuition through the understanding by means of the category.” (B144, my emphasis)
But while we have shown at §20 that the categories are necessary for the unity found in sensible intuitions, this presupposes “the fact that the manifold for intuition must already be GIVEN…prior to the synthesis of understanding and independently from it…..”
The problem is articulated in the continuation of that sentence “…how, however, is here left undetermined.” (B145)

16. §21 cont’d
Upshot: since the understanding only thinks, and thinking requires something be given to the understanding through sensibility in order for thinking to have something to think about, if we don’t find out how concepts come to determine anything in the domain of intuitions, which are perfectly nonconceptual, we cannot explain how objects of experience are even possible.

17. §22 The category has no other use for the cognition of things than its application to objects of experience.
Categories cannot contribute anything to our representations except insofar as they can be applied to objects of experience, and these presuppose and require something given in intuition. Translation: the understanding cannot generate any objects of experience on its own, either by means of the categories or by means of the categories translated according to the pure forms of intuition. Example: even mathematical objects, while thinkable, are not cognizable since they are not objects of experience and hence not real objects at all unless and until they can be shown to have origins in empirical intuitions (=actual cognitions).

18. §23
While intuitions are restricted to the domain of what is given in sensibility according to the pure forms of space and time, and hence, independent of what is given in sensibility with these forms, are nothing at all, …
The pure concepts of understanding are not so limited, but….
This seemingly greater possible application of those pure concepts “does not get us anywhere…[f]or they are then merely empty concepts of objects, through which we cannot even judge whether the latter are possible or not – mere forms of thought without objective reality – since we have available no intuition to which the synthetic unity of apperception, which they alone contain, could be applied, and that could thus determine an object.”[B148]
“Our sensible and empirical intuition alone can provide them with sense and significance.” [B149]

19. §23 cont’d
While we can say what an object drawn from a non-sensible intuition is not, we cannot say anything about what it is. It must be an object that consists in something I know not what, and perfectly so!
Thus, even category of substance could not be used to generate an object judged to be substantial since nothing in this non-sensible intuition would constitute a basis for the determination that it could only exist as a subject but never as a predicate (and the same goes for all the other categories in respect to such an intuition).

20. §24
How the categories come to apply to sensible intuitions: Read B151 thru B152 (blue flag)

21. §24 cont’d
How the categories generate a determinate manifold of intuition: By “determining sensibility internally with regard o the manifold that may be given to it in accordance with the form of its intuition.” [B153] This is accomplished by a “transcendental synthesis of the imagination...” which acts “on the passive subject, whose faculty it is, about which we rightly say that the inner sense is thereby affected.” [B ] This is an application, via the transcendental synthesis of imagination, of the form of apperception (unity) “as the source of all combination” and generates what is determinate in “the manifold of intuitions in general”. This becomes possible “only through the consciousness of the determination of the manifold through the transcendental action of the imagination (synthetic influence of the understanding on the inner sense), which I have named the figurative synthesis.” [B154]

22. §25
Cognition of the subject (the self) requires something be given in sensibility (self-affection), since the transcendental synthesis of the manifold of representations in general is only a thought and as such, absent something given in intuition, entirely empty except for the form of apperception, which only get into actual conscious attention by way of self-affection (since, as with the form of time, something intuitive must be given that has that form, otherwise the form is nothing to me/nothing at all/not real).
Thus, the Subject is an object of experience, but as such, is appearance, not a transcendent existence (=metaphysically real, non- mind-dependent Self).

23. Phase II— The Argument @ §26
Synthesis of apprehension = “composition of the manifold in an empirical intuition, through which … empirical consciousness of it (as appearance) .. Becomes possible” [B160]
“We have forms of outer as well as inner sensible intuition a priori in the representation of space and time, and the synthesis of apprehension of the manifold of appearance must always be in agreement with the latter, since it can only occur in accordance with this form.” [B160]
“But space and time are represented a priori not merely as forms of sensible intuition, but also as intuitions themselves (which contain a manifold), and thus with the determination of the unity of this manifold in them (see the Transcendental Aesthetic).” [B160]

24. §26 cont’d
“Thus even unity of the synthesis of the manifold, outside or within us, hence also a combination with which everything that is to be represented as determined in space or time must agree, is given a priori along with (not in) these intuitions, as condition of the synthesis of all apprehension." [B161]
“But this synthetic unity can be none other than that of the combination of the manifold of a given intuition in general in an original consciousness, in agreement with the categories, only applied to our sensible intuition.” [B161]
“But this synthetic unity can be none other than that of the combination of the manifold of a given intuition in general in an original consciousness, in agreement with the cagegories, only applied to our sensible intuition.” [B161]

25. §26 cont’d
(vii) “Consequently all synthesis, through which even perception itself becomes possible, stands under the categories, and since experience is cognition through connected perceptions, the categories are conditions of the possibility of experience, and are thus also valid a priori of all objects of experience.” [B161]

26. §27
“We cannot think any object except through categories; we cannot cognize any object that is thought except through intuitions that correspond to those concepts….[since] all our intuitions are sensible, … this cognition … is empirical … [which means it] is experience. Consequently no a priori cognition is possible for us except solely of objects of possible experience.”[B ]
Question: does this mean all objects of thinking must be possible objects of experience? Kant’s answer: No! (read B166 footnote)

27. §27 cont’d
TD of Categories yields: necessary agreement betw experience and the concepts of objects of experience.
But, “there are only two ways in which a necessary agreement of experience with the concepts of its objects can be thought: either the experience makes these concepts possible or these concepts make the experience possible.” [B166]
The first way is not available (why not?).
Only the second way remains: “a system of the epigenesis of pure reason”: “the categories contain the grounds of the possibility of all experience in general from the side of the understanding.” (read FN 48)

28. §27 cont’d
Kant finally reveals why his transcendental psychology cannot be empirical psychology: “[if the categories] were rather subjective predispositions for thinking, implanged in us along with our existence by our author in such a way that their use would agree exactly with the laws of nature along which experience runs (a kind of preformation-system of pure reason [cf. fn 49]), then … this would be decisive against the supposed middle way: that in such a case the categories would lack the necessity that is essential to their concept.”
That is, “I would not be able to say that the effect is combined with the cause in the object (i.e., necessarily), but only that I am so constituted that I cannot think of this representation otherwise than as so connected, which is precisely what th skeptic wishes most, for then all of our insight through the supposed objective validity of our judgments is nothing but sheer illusion…” [B168]

29. Doctrine of the Elements, Part II, Div. I, Bk
Doctrine of the Elements, Part II, Div. I, Bk. II Introduction On the transcendental power of judgment in general.
Key: “If the understanding in general is explained as the faculty of rules, then the power of judgment is the faculty of subsuming under rules, i.e., of determining whether something [an object of cognition] stands under a given rule (casus datae legis [case of the given law]) or not.” While “general logic can give no precepts to the power of judgment” (and hence, cannot settle whether an object of cognition stands under a given rule [because general logic “abstracts from all informative content of cognition” {B }])… “…things are quite different with transcendental logic”. “…the peculiar thing about transcendental philosophy is this: that in addition to the rule, which is given in the pure concept of the understanding, it can at the same indicate a priori the case to which the rules ought to be applied [and hence, do apply]”[B175]

30. First Chapter On the schematism of the pure concepts of the understanding
First: if an object is subsumable under a concept, its must be homogeneous with it (that is, “the concept must contain what is represented in the object that is to be subsumed under it”)
This leads to a problem: “…pure concepts of the understanding … in comparison with empirical (…general sensible) intuitions, are entirely unhomogeneous, and can never be encountered in any intuition.”
JP: This is the heterogeneity problem come back to haunt us!
How can this problem be overcome? Answer: transcendental schemas!

31. But what are transcendental schemas???!?!?!
Answer: roughly, they are “a third thing, which [stands] in homogeneity with the category on the one hand and the appearance on the other, and makes possible the application of the former to the latter.”
More of the answer: a transcendental schema is a “mediating representation [which] must be pure (without anything empirical) and yet intellectual on the one hand and sensible on the other.”[B177]
JP: But IK…YOU said nothing intellectual can be sensible and nothing sensible can be intellectual. THAT’S ON YOU, BROTHER!