What is the grammatical role of "either" in the following sentence:
Nevertheless, a goal is seldom destroyed by a delay, so don't destroy your positive attitude, either.
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It's an adverb. Here's what Cambridge Dictionary says about either when it's used as an adverb:
used in negative sentences instead of "also" or "too":
I don't eat meat and my husband doesn't either.
"I've never been to the States." "I haven't either."
They do really good food at that restaurant and it's not very expensive either.
I always think of either when it's an adverb as the negative form of too. For example:
You like it? I like it too!
You don't like it? I don't like it either!
As a general rule, if you want your English to sound proper, when you negate a statement, you have to change too to either.