I think that you can define probability theory, use a Dutch Book Argument
to prove, formally, that, under certain conditions, it is the correct way to estimate the probability of an event given a set of data, and then use deductive reasoning, starting from Kolmogorov's Axioms
, to draw conclusions about inductive reasoning.
One problem of this is that probability theory does not give you a prior
— that is a starting probability estimate. One could "solve" this and use a Solomonoff probability distribution*
for that, and prove that this choice will be the best one, up to a multiplicative constant.
That's a fun and interesting idea, but up to a multiplicative constant, I am taller than the Empire State Building — unless you can keep this constant down in some way (I personally know of no way of doing that, but I am no specialist), using a Solomonoff distribution is not necessarily the best choice.