What I have been laying before you thus far is the merest sketch of Kant's argument, a full-scale analysis of which, in many stages of development, can be found in Kant's Theory of Mental Activity. Before turning to the question of the nature and role of the Categories, it might be well to try to sum up what I have said thus far in a form easy to grasp. Here is the argument in what I call its Fourth Stage -- quite obviously, I have simply skipped over the three preceding stages in this exposition. One more terminological item -- Kant's use of the word "law." At A113, at the end of the Subjective Deduction, Kant writes: "The representation of a universal condition according to which a certain manifold can be posited in uniform fashion is called a rule, and, when it must be so posited, a law.
With that small clarification, here is the argument as it has evolved thus far in the Critique, laid out in something resembling logical form, with my comments in parentheses.
To Prove: All appearances stand in thoroughgoing connection according to necessary laws. [This is Kant's version of the so-called Law of Cause and Effect, which, in his understanding, is the fundamental principle of all natural science, to be distinguished from the mere fact of subjective association of resembling instances about which Hume writes at such length.]
1. All the contents of my consciousness are bound up in a unity. [This is the premise of the Unity of Consciousness, the assumption Hume cannot consistently deny, in my imaginary debate between Hume and Kant.]
2. The only way to introduce synthetic unity into a manifold of contents of consciousness is by reproducing them in imagination according to a rule. [Recall the analysis of rule-governed activities, and the analysis of the nature of the power of Imagination.]
3. The Categories, taken as a whole, constitute the most general a priori rules of synthesis. [This is the step that has not yet really been demonstrated by Kant, despite his ad hoc invocation of the two Tables. We shall get to it.]
4. If all the contents of my consciousness are bound up in a unity, then they must have been synthesized according to the Categories. [steps 2, 3, and logic.]
5. But, by steps 1, 4, and modus ponens: The manifold of contents of my consciousness must have been synthesized according to the Categories, which is to say:
6. The manifold of contents of my consciousness must have been posited in uniform fashion according to a rule. [steps 5 and 3]
7. But the representation of a universal condition according to which a certain manifold must be posited in uniform fashion is a law. [A113]
8. Therefore, by steps 6 and 7: All appearances [contents of consciousness] stand in thoroughgoing connection according to necessary laws. Q. E. D.
Thus Kant refutes Hume.
As I indicated in my parenthetical comment to step 3, the gap in the argument is anything really connecting up particular Categories with the rest of the analysis. Contrary to what Kant would have us believe, that desideratum appears nowhere in the chapter entitled Deduction of the Pure Concepts of Understanding, but it does appear, brilliantly, in the later chapter entitled The Analogies of Experience, and most particularly in the Second Analogy.
The argument laid out above nowhere appears in that form anywhere in Kant's writings. It is my reconstruction of his argument, based on passages scattered through the first part of the Critique.
And now, I am going to pause to blow my own horn. I offer two excuses for this appalling breach of academic etiquette. First of all, I am old, and ought to be allowed the sort of leeway that tribes have traditionally granted to the old folks sitting around the fire telling stories of their youth. Second, this is a blog, and blogs are quintessentially narcissistic, self-referential, and subjective. So: Kant published the Critique in 1781. I published Kant's Theory of Mental Activity in 1963. In the intervening 182 years, no philosophical text was more studied, puzzled over, anatomized, argued with, refuted, and defended than the Critique. Everyone knew that Kant was trying to "answer" Hume, but no one -- I repeat, no one -- was able actually to present an explicit statement of the argument by which Kant was supposed to have carried out that answer, starting with a clear, simple comprehensible premise and proceeding by logical steps to a conclusion that was recognizably an answer to Hume, whether one thought that it was a successful answer or not. The great German commentators Hans Vaihinger and Erich Adickes did not produce such an argument. The great South African commentator H. J. de Vleeschauwer did not produce such an argument. The great English commentators, Norman Kemp Smith, H. J. Paton, and T. D. Weldon did not produce such an argument. And my old professor, Clarence Irving Lewis, a great logician and the deepest student of Kant in American academic philosophy, never produced such an argument in his immortal course on the First Critique.
I did. You may disagree with my interpretation. You may think the argument as I have reconstructed it is inherently incredible. You may think that Hume actually wins the debate, or even that the entire enterprise is misconceived, as many subsequent philosophers have claimed. But until I published Kant's Theory of Mental Activity, nobody could answer the simple question, "What is Kant's answer to Hume?" This is not what I will be remembered for, if indeed anyone remembers me after I die. That honor will go to the 82 page tract I wrote in 1965 and published in 1970, In Defense of Anarchism. But of all the books I have written since 1963, of none am I more proud than my Commentary on the Transcendental Analytic of the Critique of Pure Reason.