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DanielLC comments on Looking for proof of conditional probability - Less Wrong
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I feel like independence really is just a definition, or at least something close to it. I guess P(A|B) = P(A|~B) might be better. Independence is just another way of saying that A is just as likely regardless of B.
P(A|B) = P(A|~B) is equivalent to the classic definition of independence, and intuitively it means that "whether B happens or not, it doesn't affect the likelihood of A happening".
I guess that since other basic probability concepts are defined in terms of set operations (union and intersection), and independence lacks a similar obvious explanation in terms of sets and measure, I wanted to find one.