  We propose a formalization of the three-tier causal hierarchy of association,
intervention, and counterfactuals as a series of probabilistic logical
languages. Our languages are of strictly increasing expressivity, the first
capable of expressing quantitative probabilistic reasoning -- including
conditional independence and Bayesian inference -- the second encoding
do-calculus reasoning for causal effects, and the third capturing a fully
expressive do-calculus for arbitrary counterfactual queries. We give a
corresponding series of finitary axiomatizations complete over both structural
causal models and probabilistic programs, and show that satisfiability and
validity for each language are decidable in polynomial space.
