There is no doubt that Polonius proves himself to be the fool of Shakespeare’s Hamlet. Just look at the difference between the advice from his son, Laertes, to Ophelia and the advice Polonius gives. Polonius gets mad at Laertes for dawdling ("Aboard, aboard, for shame!") but then detains Laertes even longer with a speech giving silly advice.
Even today, if a person strings cliché after cliché together, that person simply ends up looking (at best) like a copycat. Such is the situation here. The irony is that some of this advice became memorable allusions to Hamlet found elsewhere, including, "neither a borrower or a lender be" and "This above all, to thine own self be true."
Also, Polonius uses Reynaldo to spy on his own son in Paris. A further example of Polonius’s foolish nature is his conversation with Gertrude during which Polonius does in fact prove that “brevity is the soul of wit” by being the example of the contrary. Another way to say that is to say that the number of words one should use to signify wisdom should be the very smallest number. Ironically, Polonius is neither wise nor brief here. How can a reader tell? Polonius promises not to use any “art” in his explanation and then rambles on and on about how Hamlet is acting: "'tis true: 'tis pity, / And pity 'tis 'tis true -- a foolish figure."
Even Gertrude gets tired of Polonius’ rambings, saying, "More matter, with less art." Polonius proves himself the fool once again by saying again and again that he will heed her advice and "use no art at all." Polonius, of course, fails miserably.
Probably the crowning glory of Polonius as the fool is the manner in which he is killed. Seemingly obsessed with spying, Polonius now seems to get a thrill out of figuring out what everyone else is doing.