Hamlet tries to reason out whether or not his Uncle Claudius actually did murder his father. Although he has seen his father's ghost, he uses his intellectual reasoning. Perhaps, the ghost that he sees is an evil apparition. Perhaps, the ghost is not really his father:
Angels and ministers of grace defend us!
Whether you are a spirit of health or a goblin damned,
Bringing with you airs from heaven or blasts from hell,
Whether your intentions are wicked or charitable,
You come in such a questionable shape
That I will speak to you. I’ll call you Hamlet!
King! Father! Royal Dane! O, answer me! (Act 1, Scene 4)
From this text, we can see that Hamlet is questioning whether the apparition is a spirit or a goblin. He is using his intellect to reason out what he sees. He is overriding his emotions and is trying to intellectually decide if the ghost is a true message from his father.
Another time when Hamlet uses his intellectual reasoning is when he has the actors reenact his father's murder. Hamlet wants to be sure that his Uncle Claudius has indeed committed the murder. He is intellectually questioning everything that has occurred, as he should. Hamlet speaks with Horatio and asks him for his help:
There is a play tonight before the king.
One scene of it comes near the circumstances,
Which I have told you, of my father's death.
I beg you, when you see that act begin,
Even with the very criticism of your soul
Watch my uncle. If his hidden guilt
Don’t show itself in that one speech,
It is a false ghost that we have seen,
And my imaginations are as unstable
As god of fire’s anvil. Watch him carefully,
Because I will have my eyes riveted on his face,
And, after the play, we will combine our observations
In condemning of his actions. (Act 3, Scene 2)
In this selection, we can see that Hamlet is carefully thinking about a way to prove Claudius' guilt. Hamlet is using careful reasoning. He is not sure that Claudius is guilty. After the actors reenact his father's murder, Hamlet tells Horatio that they will meet and combine their observations to determine Claudius' guilt. Clearly, Hamlet is not making a rash decision. He is using clever thinking to determine if Claudius has indeed committed the murder. Hamlet is not allowing his emotions to rule him. He is using careful reasoning before actually condemning his Uncle Claudius.