The purpose of the scarlet letter was to let people know how sining and keeping it a secret can kill you and don't hold a grudge fo rthe rest of your life just let it go and forgive.
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The purpose was to publicize Hester's sin . . . as if her pregnancy were not obvious enough. All the people shunned her and were not allowed to forgive and forget. She even had to live outside the city limits. Yes, what she did was wrong. How long can we punish her? And why punish Pearl at all? She is an innocent. More importantly, the father of the child is not branded as Hester is. Ironically, her public "guilt" and ostracism strengthens her, while Dimmesdale's private guilt and the respect of the community eats him up.