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In Shakespeare's Hamlet, irony reigns supreme in the court of Polonius—in terms of his advice. Shakespeare uses this character to utter some of the most recognizable (and valuable) pieces of advice that we may know—though some of us may be surprised to learn that they don't come from the Bible, and Ben Franklin had nothing to do with these gems of wisdom. It was Shakespeare's Polonius. What makes it so ironic (and sadly humorous) is that Polonius gives excellent advice, but cannot see his way to using it himself. Herein lies the irony.
For example, his first bit of advice to Laertes is don't tell people what is on your mind, or act hastily.
Give thy thoughts no tongue,
Nor any unproportion'd thought his act… (I.iii.63-64)
Certainly this is ironic because Polonius is unable to act as he instructs his son to act while away at school. Polonius cannot keep his mouth shut. If he has a thought, he must share it. If he does not have a thought, he still must be saying something even if it makes no sense whatsoever.
In fact, Polonius gets so caught up in his ramblings that Gertrude finally says to him...
More matter, less art. (II.ii.102)
In other words, don't worry so much about being fancy in the way you speak. If you have something to say, get on with it.
Polonius also does not act cautiously, but impetuously. When he hides behind the curtain in Gertrude's room to listen to what Hamlet has to say, he does so impulsively. Everyone is sure that Hamlet is mad (except Horatio). It seems impetuous that Polonius, having first-hand experience of Hamlet's lack of sanity, would agree to spy on him without some careful thought. When Hamlet hears Polonius cry out, the Prince thinks it is his uncle, and he stabs and kills Polonius by mistake.
Polonius offers other advice, and tells his son...
Give every man thy ear, but few thy voice;
Take each man's censure, but reserve thy judgment. (I.iii.72-73)
He is once more telling his son to be careful of what he says, but encouraging to listen carefully to what is said to him and around him, but to hold his judgement on what he hears.
Polonius would have to be a man of depth and more intelligence to pay attention to this advice. Once again, he is impulsive in what he does. He listens to what goes around him, but offers an opinion immediately. He does not weigh both sides of a situation, but offers his baseless judgment first. With Hamlet, he does not observe Hamlet with his friends, then with Ophelia, and then with his family, think on it, and then make a judgment. To Polonius, it matters little to him if he is making a sound judgment; he wants to be within the "inner-circle" regarding what is going on in the castle, and his wish to be "needed" by the new King causes Polonius to judge too quickly and forget to listen more, and reason things out.
Polonius has wonderful advice. Shakespeare uses Polonius' inability to follow his own advice to lead Polonius to a sad end.