What are the trade offs of using conditional event algebras?

Question

A conditional event algebra is an algebraic structure that attempts to make

$$Pr(A \rightarrow B) = Pr(B|A)$$

true for all $A,B$ that are events from some probability space. An example of this is the Goodman–Nguyen–Van Fraassen algebra.

When I first learned the basics of probability it seemed intuitive that the above equality would have to be true, but it is not true in general under Kolmogorov's foundations. These days I generally feel that the inequality between conditional probability and probability of implication is fine, and may even add the expressiveness of probability.

But, I would like to hear a more objective and technical analysis of the trade offs. What are the pros/cons of using conditional event algebras vs Kolmogorov's foundations in terms of problems that would change their solvability (or ease thereof)?

Answer 1

Two observations.

1. $A \rightarrow B$ here doesn't mean “Either $B$ or not $A$.” The $\rightarrow$ is a non-Boolean connective or binary operator.
2. It’s not necessarily a question of CEAs versus the Kolmogorov axioms. Boolean CEAs such as G-N-vF algebras support all those axioms. But it’s true more broadly that there are trade-offs, as the Wikipedia article on CEAs notes. There’s an inference rule called import-export, by which $A \rightarrow (B \rightarrow C)$ is equivalent to $(A \wedge B) \rightarrow C)$. The rule is appealing, though not uncontroversial. If we accept it, we would expect $Pr(B \rightarrow C \mathbin{|} A) = Pr(C \mathbin{|} A \wedge B)$. G-N-vF algebras don’t support that identity.