The Transcendental Deduction
Some brief and very crude remarks.
I. What is the problem that the transcendental deduction is supposed to solve?
The problem is how we can know that the categories apply to experience -- or, put differently, how we can know that experience must conform to the categories.
There's no problem about how empirical concepts (red, heavy) apply to experience. They are derived from experience in the first place, so of course they apply to it.
However, Kant thinks that the categories (substance, causation, etc.) are completely a priori; they are not derived from experience but rather from the way our minds work. If Kant is right, then we can't think about the world except by using these concepts. They provide the form of thought in something like the same way that space and time provide the forms of intuition. But just because we must use them in order to think at all does not guarantee that our experience will be such as to permit us to use them.
II. What is a transcendental deduction? It's basically just a justification of the use of these concepts. Empirical concepts could have an empirical justification, but a priori concepts (the categories) can't be justified this way, so they need a "transcendental" deduction (i.e. a justification in terms of the necessary conditions for having experience at all).
III. How does the argument go?
Great question. Kant wrote very different versions of the deduction in the first and second (A and B) editions of the Critique of Pure Reason, and both are remarkably obscure. My outline below is not intended to be a serious interpretation of the text, but I hope it at least gives a broad sense of the direction of the argument.
1. Our experience is unified. (It's not just a lot of independent ideas or perceptions, but a coherent whole.)
What accounts for the unity of experience? What unifies experience is not anything empirical in the experience itself. (Think of Hume's problems trying to determine what could tie together the "bundle of perceptions" that he thought constituted the self.) Nor is it that all our representations are attributes of a thinking substance, as Descartes for instance held. As Locke pointed out, even if this is true, it doesn't explain why our experience seems unified: in principle there's no reason the same substance couldn't support multiple identities, or why multiple substances couldn't support a single one. Rather, Kant's view is that:
2. What unifies experience is that we can think of each of our representations as being ours. (As Kant puts it, it is possible for "the 'I think'" to accompany all of my representations.)
The most important step in the reasoning is the next one. Kant asks what is necessary for us to be able to think of all our representations as belonging to a single self. His answer is that this is only possible if we can also think of all our representations as representations of an objective world, a world that exists outside us and is distinct from our subjective experience of it.
Why does he think this? The main point is that we need to interpret our mental states as representations of an external world in order for them to make sense. Try to imagine what experience would be like if you didn't regard it as representing an objective world. Consider your visual experience. Now try to subtract the idea that your experience of colored shapes is experience of a world outside you. Every time you turn your head, move your eyes, or blink, the colored shapes you see move and change. It's hard to see how you could make any sense of the play of experience if you didn't think of it as an experience of something objective. So Kant thinks:
3. We can think of our representations as being ours only if we can also think of them as representing an objective world.
Now, what could make that possible? Kant thinks that there's no way you could ever get the idea of an object from the raw material of sensation. If we didn't already have the idea of an object, and use it to organize our experience, there's no way we could ever get it from the play of our sensations.
More specifically, for reasons described by the empiricists, Kant thinks that we couldn't get the concept of substance from experience; and for reasons described by Hume in particular, that we couldn't get the concept of causal relations (necessary connections) between objects merely from our experience of those objects. But the notions of substance and causation are important parts of the concept of an object.
So we have:
4. We can only think of our representations as representing an objective world if we can use the concept of an object in general to organize them.
5. The concept of an object in general is precisely what the categories provide. (To concentrate on only two of the twelve categories, the general concept of an object is the concept of substances that stand in causal relations with one another.)
So, 6. A unified experience would only be possible if we were able to apply the categories to experience. Since we actually have a unified experience, we must be able to apply the categories.
To recap briefly:
1. unified experience
2. unified experience --> can apply 'I think' to representations
3. can apply 'I think' to representations --> can think of representations as representing objects
4. can think of representations as representing objects --> can apply the idea of an object in general to experience
5. idea of an object in general = the categories
6. can apply the categories to experience
One question to ask about this reconstruction (which should make you suspect that either (a) it misses something important about Kant's argument, or (b) Kant's argument doesn't actually get him as far as he wants it to):
The argument may show that we are in fact able to apply the categories to our experience. But how does it imply Kant's transcendental idealism? That is, how do we get from the fact that we can apply the categories to the idea that the empirical world is largely our own construction? Kant thinks that we construct the empirical world (the world of "appearances") by applying the categories to our experience (or rather, using the categories in the very construction of experience), and that causation, substantiality, etc. are not features of things in themselves.
But why shouldn't we conclude instead that we've gotten lucky -- that in fact our a priori concepts do apply to the things themselves, and that's why we can successfully use them to make sense of experience? (And in fact post-Darwin we don't even need to appeal to luck: there could be an evolutionary explanation of why the ways we need to organize our experience correspond to the way the world actually is.)