What is one positive of wearing a veil in The Minister's Black Veil?
There are several benefits to wearing a veil, some of which apply in this case and some of which do not.
In several cultures, a veil is used to signify and symbolize innocence and purity. You can see this in the tradition of the bridal veil, and in the Islamic tradition of women wearing veils. Since the minister in this story is Christian, that would definitely apply.
In the cultures practicing this kind of veil use, the innocence is also related to a kind of privacy. To put it bluntly, if you are wearing a veil, no one can see your face or features, or tell your expressions. That definitely applies here.
Veils and other ceremonial clothing works to isolate the wearer from the rest of society: they mark the wearer as other, special, or distant from the rest of life. That too would apply here.
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