Hamlet is clearly intelligent, as he understands other people and their motives well. He can see behind their actions and understand their true thoughts and feelings. For example, he realizes that his mother, Gertrude, is faking her sympathy and grief about her husband's death. Hamlet says, "a beast, that wants discourse of reason / would have mourn'd longer." He means that even a lowly animal would have mourned his or her mate longer than she has. He does not accept surface impressions but understands people's deeper emotions.
Later in the play, he believes that actors can move the audience to show their true emotions, and he devises a plan to put on a play that replicates his father's death so that his uncle, Claudius, shows his guilt and repentance about killing Hamlet's father (who is Claudius's brother). Hamlet says, "the play 's the thing / Wherein I'll catch the conscience of the king." This plan shows his intelligence, as Hamlet is a wily manipulator of others.
Hamlet is indeed very intelligent. Perhaps the best example of his intelligence is the fact that he convinces others that he has become mad when in fact he has not. The whole act of madness by Hamlet is to confuse his uncle and disguise the fact that Hamlet knows the truth about the death of his father.
Hamlet reveals his plan to act mad to Horatio and asks him not to give his plan away in Act I Scene II:
How strange or odd soe'er I bear myself,
As I perchance hereafter shall think meet
To put an antic disposition on,
That you, at such times seeing me, never shall,
With arms encumber'd thus, or this headshake,
Or by pronouncing of some doubtful phrase,
Or by pronouncing of some doubtful phrase,
As 'Well, well, we know,' or 'We could, an if we would,'
Or 'If we list to speak,' or 'There be, an if they might,
Or such ambiguous giving out, to note
That you know aught of me: this not to do,
So grace and mercy at your most need help you, Swear.
The other clear sign of Hamlet's intelligence is his clever use of words with double meanings about the people around him, whether it's his uncle, his mother or Ophelia's father Polonius. These people think that the things Hamlet says to them are a result of his illness and do not think deeply. But Hamlet is in fact stating ugly truths about them and using some very strong words to describe them. It requires intelligence to be able to use double entendres so successfully (figure of speech that uses multiple meanings of words.)
For example, in Scene II Act II, Polonius says that Hamlet thought he was a "fishmonger." In that time frame, this term was also used to refer to pimps:
He knew me not at first; he said I
was a fishmonger: he is far gone, far gone: and
truly in my youth I suffered much extremity for
See
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truly in my youth I suffered much extremity for
love; very near this.
As you can see, not only is Polonius unaware that Hamlet has called him an unkind name but he even feels sorry for Hamlet, saying that he has gone completely mad due to love.
I do believe Hamlet is intelligent. He seems to be the only one in the play who recognizes the situation between the King and the Queen for what it is. I think the best examples of Hamlet's intelligence are the scenes when he appears to be mad to the other characters in the play, but his language is full of significance, symbolism, and double meanings. One scene where this occurs is Act IV, Scene 2. Hamlet is talking with Rosencrantz and Guildenstern after he murders Polonius, and he toys with them by running in circles with what he is saying but making significant sense to the audience.
A similar effect is created during the scene with the players (Act III, Scene 2). Here Hamlet skillfully sets a trap for his uncle while also bantering with other characters in the scene. The other characters believe this reveals his madness, but the audience once again can detect the significance of his actions and words.