Conditionals and Conditional Probabilities without Triviality

Abstract

The Adams Thesis holds for a conditional $\to$ and a probability assignment $P$ if and only if $P(A\to B)=P(B\mid A)$ whenever $P\left(A\right)>0$. The restriction ensures that $P(B\mid A)$ is well defined by the classical formula $P(B\mid A)=P(B\cap A)/P\left(A\right)$. 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.