bad, badly

1. After the verb feel, bad is an adjective complement (meaning either 'guilty, ashamed' or 'unwell') rather than an adverb:

To be absolutely honest, what I feel really bad about is that I don't feel worse—Michael Frayn, 1965.

After to be and most other verbs, badly is required: Things started to go badly for them / They sank seventeen ships and badly damaged eight more / Russia badly needed Finish products / We are not too badly off. In these cases, bad is used only informally or in dialect:

I only came cause she's so bad off—L. Hellman, 1934.

2. In a slang (principally youth slang) use originating in US Black English, bad means the very opposite of its traditional meaning, i.e. 'excellent, very good', and in this meaning even has special degrees of comparison badder, baddest:

She said that part of the problem was that they often wanted to distinguish themselves by being badder than their rivals. Badder? Surely the comparative of bad is worse? But then worse has connotations of being less good at doing something.—Spectator, 1993.

This is a good example of how one of the most basic words can be twisted and pulled in all directions, rather as wicked and mean have been in the past.

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