In the first part, we left Bruno de Finetti in 1949 as he established that for four distinct individual possibilities (like which team will win a championship), any usual ordering of “tickets” that was “quasi-additive” was also “probabilistic.” He conjectured that this would be true for any finite number of quasi-additively ordered propositions, and invited the community to help prove him right or wrong.

One who accepted de Finetti’s invitation was Leonard Savage, who later developed his own landmark axiomatization of subjective probability. Savage gave the obscure 1949 paper to Charles Kraft, John Pratt and Abraham Seidenberg. They showed in 1959 that de Finetti’s conjecture was wrong if there are five or more basic outcomes.

That’s the “broken” part of the story. De Finetti’s conjecture, scribbled in haste to rebut his friend George Polya, is false. Many people scrambled to fix it. The first were Kraft, Pratt and Seidenberg themselves in 1959. A famous second time was five years later, by Dana Scott. Oddly, neither solution was satisfactory to its authors. Odder still was that De Finetti himself may have come within a whisker of repairing it back at the beginning in 1949. Continue reading