Cox’s Theorem

Cox’s theorem is a celebrated result stating that any system satisfy the Cox’s conditions is isomorphic to probability. But what is more surprising to me is that in the proof of Cox’s theorem, he proved that All associative binary operation with second order derivative is isomorphic to multiplication.

Let f be an associative binary operator, with second order derivative. Then, there exist function g and h, such that
1. h(g(x)) = x for all x
2. f(a,b) = h(g(a)*g(b)) for all a, b

Corollaries:
1. All associative binary operation with second order derivative is isomorphic to addition.
2. All associative binary operation with second order derivative is commutative.

 

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