It just occurred to me that it's entirely fitting how often Kowalski refers to the Caudine Forks, for that battle is a metaphor for ping pong arguments:

Suppose we have two statements (Romans), A and B.

A priori, there are four possible situations:

    A & not B | A & B | not A & not B | not A & B

However, as mathematicians, let us play the Samnite. First we close one pass, in proving A → B, which is equivalent to ruling out A & not B. Then we close the other pass, in proving B → A (ruling out not A & B).

Now our romans, err, statements are confined to A & B | not A & not B, and (with no truth values having been harmed) must therefore pass under the yoke of equivalence: A ↔ B.