What does Proctor's decision to confess his involvement with Abigail reveal about his character and why does Elizabeth lie to Danforth in Act III?
Proctor's decision to admit to the crime of adultery illustrates that he is a man who is in search of the truth. He is determined to bring to light, in the court, the real reason behind the accusations and Abigail's authority to make them. He is determined to prove that the reasons behind the accusations have nothing to do with actual witchcraft, but are an attempt to get rid of his wife. Abigail conveniently is using the witch trials to get what she wants, John Proctor as her husband.
Proctor's confession, he hopes, will shock the court into looking at Abigail in a different way. Up until now, she has successfully captivated the court and been made a supreme authority on accusing others of witchcraft. Proctor is trying to break this spell that she has over the court by confessing.
Elizabeth lies to the court for two reasons. 1. Because she is trying to protect her husband's reputation. 2. Because she feels guilty about the affair, feeling partly responsible for John turning to another woman for warmth and love, she admits later on to him that she was a cold wife.
What Proctor's confession says about his character is that he has a strong sense of right and wrong, and will not succumb to the mob mentality found in the village at this time. Elizabeth lies to Danforth solely to protect her husband. You see, they are truly united in their relationship, and refuse to go along with the hysteria.