"The expression P(a_x [ ]-> o_i) is meaningless...."

Causal decision theory \(the current standard\) says we should instead use the counterfactual conditional $\\ \\mathbb P(a\_x \\ \\square \\! \\! \\rightarrow o\_i).$

The expression P(ax [ ]-> oi) is meaningless. Probability theory is an extension of classical propositional logic (CPL) to express degrees of certainty; probabilities are assigned to CPL formulas. "ax [ ]-> oi" is not a CPL formula, as there is no counterfactual conditional operator in CPL.