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I am aware that Kant addressed Hume's skepticism on causality, but I don't see anything in his CPR that solves the problems of induction and uniformity of nature in other contexts like everyday life, where Hume's question, "will the sun rise tomorrow?" seems to persist. Exactly how does/would Kant solve the problems of induction (particularly uniformity of nature) in non-causal contexts, if he did at all?

I am aware that there has already been a question like this, but I thought the answer to it only addressed induction in the context of causal formulations about the world; in this respect, I think Kant only dodged the problem of induction in causal formulations by solidifying causality as an intuition. Rationalizing your belief that the sun will rise tomorrow and rationalizing your belief that C-fiber firing causes pain seem to be two different things.

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    Critique of the Power of Judgement. Half of the book pretty much handles this question. – Philip Klöcking Jun 11 '16 at 8:32
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If I understand what you're saying, your argument might be expressed as follows:

  1. If induction is possible for a certain type of facts, there are causal relations which are knowable with respect to those facts.
  2. There is no causal relation with respect to any fact about noumena that is knowable.
  3. Therefore, induction is impossible for any fact about noumena.

If that's what you're saying, I imagine that Kant would fully agree. However, it seems that you're suggesting that an adequate theory of induction would somehow allow for knowledge about noumenal facts. It that case, it might be helpful to note that the second point of the argument only applies in virtue of the fact that we have to rely on our senses:

"But, after all, the possibility of such noumena is quite incomprehensible, and beyond the sphere of phenomena, all is for us a mere void; that is to say, we possess an understanding whose province does problematically extend beyond this sphere, but we do not possess an intuition, indeed, not even the conception of a possible intuition, by means of which objects beyond the region of sensibility could be given us, and in reference to which the understanding might be employed assertorically." (Critique of Pure Reason, A255/B311, emphasis added)

Because of that, there's no way to empirically acquire facts about noumena, so for all practical purposes, such facts don't exist for us. Therefore, Kant's philosophy would only suffer in comparison if there was such a thing as legitimate theories of induction which could claim extrasensory access to facts.

However, assuming that no such theories exist, we have to be satisfied knowing that the rising of the sun will always remain for us a phenomenal fact. Since causal relations with respect to phenomenal facts are knowable, there seems to no real obstacle to our drawing inductive conclusions concerning such phenomena. And that's what science has always been about.

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