What is the converse of $P \implies (Q \implies R)$?

Question

Is the mathematical converse of $P \implies (Q \implies R)$

either

$(Q \implies R) \implies P$

or

$R \implies Q \implies P$

In other words, do I apply the converse for the nested sentence and then apply the converse to the outer sentence?

Thank you for any and all help.

Answer 1

It's the former. The converse of $P \implies (Q \implies R)$ is

$(Q \implies R) \implies P$.

Answer 2

The converse of a conditional sentence $A \implies B$ is $B \implies A$. Note that this definition applies only to sentences whose main connective is $\implies$, and it does not depend on any other connectives that appear in the sentence. So, to find the converse we can ignore any connectives besides the main $\implies$.

In your case, just let $(Q \implies R) =: A$. Then your sentence becomes $P \implies A$. The converse of this is $A \implies P$, which by the definition of A is $(Q \implies R) \implies P$.