In the opening of Paradise Lost, Milton asks his muse for poetic assistance:

That to the highth of this great Argument
I may assert Eternal Providence,
And justifie the wayes of God to men.

Here, Milton uses "argument" to mean "subject" or "theme." He asks his muse to help him explain his subject—the Biblical story of the Fall—in a way that "justifie[s] the ways of God to men."

The word "argument" ends up being a pun, however, since one of the main ways Milton builds on his theme of the Fall is by weaving argumentative and persuasive techniques into the scene in which Eve accepts the serpent's deception and eats the Fruit of the Tree of the Knowledge of Good and Evil. This scene appears in Book IX.

Book IX actually encapsulates several "arguments." The first is Satan's assertion that he will find a way to deceive and undermine Adam and Eve, which ends with Satan's occupying the form of the serpent in lines 180–185.

When looking for evidence on argument and persuasion in Book IX, studying Satan's various arguments carefully is a must; Milton has made him a master of persuasion, using logic (logos), influence (ethos) and emotion (pathos) in turn to make his case and get his way.

Meanwhile, Eve is also practicing the art of persuasion. Eve suggests that she and Adam work on their gardening tasks in separate parts of the garden. Adam objects, fearing that something bad will happen to Eve if she's off alone; but Eve persuades him to let her go, on the grounds that she wants to test her own ability to work and resist temptation without Adam around to help her.

Even before the serpent and Eve meet, Book IX is already aligning Eve with the serpent/Satan by making them both the "arguers." They both assert a specific position, defend it, and ultimately win their arguments by getting to do what they argued in favor of doing.

Starting at about line 455, Satan focuses all his attention on persuading Eve to eat from the tree. One way to approach this subject would be to compare the argument between Satan and Eve with the argument between Eve and Adam. With Adam, Eve was the one asserting a point; with Satan, she's the one playing defense.

In particular, look at Satan's words from lines 582-610, as well as Eve's reply following. These lines contain a wealth of examples of persuasive techniques and counterarguments in Paradise Lost.