Haimon is the voice of reason in the play. His argument to his father is a model of persuasion. He begs Creon to listen to reason:
It is not reason never to yield to reason!
Haimon argues that Antigone should be rewarded for trying to bury her brother, not punished. It was not wrong, Haimon asserts, for Antigone to cover her dead brother's body so that the vultures would not eat it.
He tells Creon that he does not have to be so rigid in his beliefs, so stubborn, that he cannot believe that he "alone can be right." Haimon uses the analogy of trees in a storm. Those trees that are able to bend survive the storm; but those that cannot bend are destroyed.
When Creon is still not convinced, Haimon tells Creon that it is his duty to listen to his people. Creon argues that the "king is the state," but Haimon counters with the state is the people, and that a king should govern according to the desires of his people.
So Haimon's argument rests on three principles: (1) Antigone's action should not be regarded as a crime. She was acting out of compassion; she was obeying God's law, (2) the people are in agreement with Antigone, and a king's duty is to listen to his people, and (3) it is not wrong for a king to change his mind, to be merciful.
In Antigone, Haemon argues with his father, Creon, over the harsh sentence he has meted out for Antigone's blatant disregard of his decree. Haemon is engaged to Antigone, but he also loves his father, and he simply tries to speak some sense to Creon. He tells his father what others are saying, giving Creon every opportunity to soften his position and relent on his pronouncement of death. He tells Creon that the town sympathizes with a sister who loved her brother enough to die doing the right and proper thing by burying him the best way she could. What he says is this:
the city weeps for this girl, says she's the least worthy of all women to die so badly for such noble deeds. “She didn't let her brother, who had fallen in combat, lie unburied, to be devoured by some ravenous dog or bird. They ought to give her an award!”
Haemon's argument doesn't work, of course, but he lets his father know his people would find it endearing and just if he waived the death sentence. Creon stubbornly stays the course, though, and loses far more than the respect of the town.