Further to yesterday's post, I'll continue with my defense of Shakespeare's use of puns by having a look at a scene from Hamlet which shows how they are often used in argument. In this case (1.3), Polonius is questioning Ophelia about the nature of her relationship with the prince, and receives the reply: "He hath, my lord, of late made many tenders/ Of his affection to me". Both the answer and the word "tenders" infuriate him, and he becomes fixated on it and attempts to use it as a way into Ophelia's way of thinking. Thus we get the following, in which he puns on the word three times, while showing that he's aware of the device (almost apologizing for it), and repeating another word of Ophelia's ("think") which he also attempts to twist to his advantage:

LORD POLONIUS

Do you believe his tenders, as you call them?

OPHELIA

I do not know, my lord, what I should think.

LORD POLONIUS

Marry, I'll teach you: think yourself a baby;

That you have ta'en these tenders for true pay,

Which are not sterling. Tender yourself more dearly;

Or--not to crack the wind of the poor phrase,

Running it thus--you'll tender me a fool.

I'd argue that it's a mistake to think of punning as no more than a stylistic choice. Rather, it's an integral part of conversation and argument, and happens all the time.