) The paper gives a set of 7 axioms, based on Savage's axioms, which is necessary and sufficient for an agent's preferences in a dynamic decision problem to be represented as expected utility maximization with Bayesian belief updating.This helps us see in exactly which situations Bayesian updating works and why.Probability theory is the main mathematical tool for Carnap's inductive logic as well as for Bayesian confirmation theory.
I won't try to reproduce the mathematical notation here (see the page numbered 88 in this ungated PDF), but here's the informal explanation given in the paper: This axiom is clearly violated in Vladmir Nesov's Counterfactual Mugging counter-example to Bayesian updating.
Then we jump to the middle of the twentieth century and Hempel's pioneering work on confirmation.
After looking at Popper's falsificationism and the hypothetico-deductive method of hypotheses testing, the notion of probability, as it was defined by Kolmogorov, is introduced.
It lay in the surprisingly slippery nature of the P value, which is neither as reliable nor as objective as most scientists assume.
“P values are not doing their job, because they can't,” says Stephen Ziliak, an economist at Roosevelt University in Chicago, Illinois, and a frequent critic of the way statistics are used.