Why didn't $\Pr \left( A \rightarrow B \right)$ catch on? Students are conditioned to thinking in terms of IF-THEN statements even before high school, and courses offered at the university level often lead to the formalization of material implication. Material implication is a function that two Boolean values and returns/outputs a single Boolean value. One can use 'True' and 'False', but here I will use '1' and '0' respectively. Material implication can be written as a piecewise function (especially since I am not aware of a nice way to format truth tables here).
$$p \rightarrow q \triangleq \begin{cases} 1 & p = 1 \land q = 1 \\ 0 & p = 1 \land q = 0 \\ 1 & p = 0 \land q = 1 \\ 1 & p = 0 \land q = 0 \end{cases}$$
An indicator function of $ A \subseteq \Omega$ is a map $\mathbb{I}_A: \Omega \mapsto \{ 0, 1 \}$, and can be written explicitly in a piecewise fashion.
$$\mathbb{I}_A(\omega) \triangleq \begin{cases} 1 & \omega \in A \\ 0 & \omega \not\in A \end{cases}$$
We will interpret $\Pr(A)$ to be $\int_{\Omega} \mathbb{I}_A(\omega)dP$, and $\Pr(A \rightarrow B)$ to be $\int_{\Omega} \mathbb{I}_A(\omega) \rightarrow \mathbb{I}_B(\omega) dP$. Seeing an implication operator in the integrand is a novelty, but this implication on indicators can be re-written in terms of a new indicator. Thus it is an expression compatible with our existing understanding.
$$\mathbb{I}_A(\omega) \rightarrow \mathbb{I}_B(\omega) = \mathbb{I}_{C}(\omega) \triangleq \begin{cases} 1 & \omega \in A \land \omega \in B \\ 0 & \omega \in A \land \omega \not\in B \\ 1 & \omega \not\in A \land \omega \in B \\ 1 & \omega \not\in A \land \omega \not\in B \end{cases}$$
$$\Pr(C) = \int_{\Omega} \mathbb{I}_C(\omega) dP$$
At some point students encounter the mathematical fact that $\Pr \left( A \rightarrow B \right) \neq \Pr \left( B | A \right)$. Some mathematicians were motivated to develop conditional event algebras that attempt to define these as equal, and I've asked for applications of such algebras in the past. The Goodman–Nguyen–Van Fraasen algebra is an example of a conditional event algebra.
But even without getting into conditional event algebras, $\Pr \left( A \rightarrow B \right)$ is a defined quantity in the standard Kolmogorov treatment of probability that I have yet to see used. Why are we not using it?

Assuming that $A$ and $B$ are events, this seems in effect to be saying $\Pr(A  \rightarrow B)$ would mean $\Pr(A^c \cup B)$,
which is equal to $\Pr(B\mid A)+\Pr(B^c\mid A) \Pr(A^c)$
and  greater than $\Pr(B\mid A)$ unless one of the terms is $0$.
But the question is whether this notation would be useful in any sense.
