Conditionals and Conditional Probabilities without Triviality

Abstract

The Adams Thesis holds for a conditional → and a probability assignment P if and only if P(A→B)=P(B∣A) whenever P(A)>0. The restriction ensures that P(B∣A) is well defined by the classical formula P(B∣A)=P(B∩A)/P(A). Drawing on deep results of Maharam on measure algebras, it is shown that, notwithstanding well-known triviality results (Lewis, etc.), any probability space can be extended to a probability space with a new conditional satisfying the Adams Thesis and satisfying a number of axioms for conditionals. This puts significant limits on how far triviality results can go.