Observations
1. I strongly suspect that there is a profound connection between Godel’s incompleteness theorem and the existence of emergent properties. If we treat our standing knowledge of a thing’s parts as a sort of axiom system from which conclusions/consequences are drawn, we can readily see how emergent properties might correspond to the Gödel statements in that axiom system.
2. Psych-meds provide a strange prospective on one’s personhood. They way in which behavior changes merely because of an alteration in your brain’s chemistry provides one with an experience of his own personhood being undermined. I always thought I was in control of my temper---the reality is that I did not have a temper because I was depressed and didn’t feel anything.
3. The reason inductive logic may have proven so tricky to justify and formalize is that it is inherently non-monotonic in character: Something might be true if we have premises A and B but not true with A, B, and C all being true. What this means is that in non-monotonic logics, what you don't know affects your conclusions. The inability to include this feature until recently in models of inductive logic may be responsible for the notorious difficulty of the problem. Between non-monotonic logic and probability theory (and possibly category theory), I believe a full formalization of inductive logic is possible.