The dual of the converse of the inverse of the logical statement $p \to (q \to r)$ is equivalent to...

The given statement is $p \to (q \to r)$.

The inverse of a conditional statement $A \to B$ is $\sim A \to \sim B$.

The converse of $C \to D$ is $D \to C$.

Thus, the converse of the inverse of $A \to B$ is $\sim B \to \sim A$, which is the contrapositive of the original statement.

For $p \to (q \to r)$, the converse of the inverse is:

$\sim (q \to r) \to \sim p$

We can simplify this using the equivalence $X \to Y \equiv \sim X \vee Y$:

$\sim (q \to r) \to \sim p \equiv (q \to r) \vee \sim p$

$\equiv (\sim q \vee r) \vee \sim p$

$\equiv \sim p \vee \sim q \vee r$

To find the dual of a logical expression, we replace $\vee$ with $\wedge$ and $\wedge$ with $\vee$ (while keeping the literals unchanged).

The dual of $(\sim p \vee \sim q \vee r)$ is $(\sim p \wedge \sim q \wedge r)$.

Now, we check the given options to find which one is equivalent to $\sim p \wedge \sim q \wedge r$.

Evaluating the first option:

$\sim [p \vee (r \to q)] \equiv \sim [p \vee (\sim r \vee q)]$

Using De Morgan's laws:

$\equiv \sim p \wedge \sim (\sim r \vee q)$

$\equiv \sim p \wedge (r \wedge \sim q)$

$\equiv \sim p \wedge \sim q \wedge r$

This matches the dual of the converse of the inverse.

Answer: $\sim [p \vee (r \to q)]$