Why is "well" used with linking verbs instead of "good"?
As any grammar handbook, English teacher, or parent correcting a child will tell you, you're supposed to say "I don't feel well" instead of "I don't feel good." Well rather than good seems to be used with other linking verbs, too (e.g., "Are you well?"). However, the rule for other words is that adjectives, not adverbs, are used with linking verbs. We say "I feel sad," not "I feel sadly," and "He feels tired," not "He feels tiredly." So why is well, an adverb, preferred over good, an adjective, when used with linking verbs? What makes good/well the exception?
Solution 1:
So why is well, an adverb, preferred over good, an adjective, when used with linking verbs?
It's well as an adjective that is preferred over good as an adjective.
Though that well is also an adverb is a factor in two ways.
The first is that since good is sometimes used as an adverb, and this sense is considered incorrect, some of the cases where good should be corrected to well is one of those cases:
*I didn't play good.
The other is that the adjective sense of well grew out of the adverbial sense.
This adjective sense well is more specifically about health and well being, but it probably does originate in an adverbial sense whereby the Old English "ic eom swiðe wel" which word-to-word translates as "I am very-much well" was likely first understood as an adverb modifying the verb am in the existential sense (a bit like "I exist" so "I am existing very well").
Conversely the opposing adverb evil of ic wæs swiðe yfle meant the opposite ("I was very-much evil" meaning you aren't doing so good at being, because you are sick or otherwise beset with misfortune).
The well of this "I am very-much well" then came to be understood as an adjective, giving us the adjective form of well ("I am well" being hence comparable in structure to "I am tall"). The adverb form of evil meanwhile largely died out except perhaps in the expression "speak evil of him".
The other adverbial meanings of well did not become adjectives in the same way ("He is very well at science" is not generally accepted, though "He is very good at science" or "He is doing very well at science" are).
Now, it's perfectly logical to say "I don't feel good" etc., but since well is more specifically about health, that is the form that people keep using for that context, and "I don't feel good" hence sounds wrong to many people.
Not to everyone, and some would see nothing wrong with "I don't feel good" or think it wrong but use it anyway and "I don't feel so good" seems even more reasonable.
When it comes to comparing "I am well" to "I am good" the value of keeping to well for matters of health is more apparent; "I am good" could refer to moral or other qualities while "I am well" is immediately understood as referring to well-being.
So with "I am well/good" there's definitely a strong value in choosing well. With "oh, I really don't feel too well/good" the value is weaker and opinions will begin to differ; sticklers for rules insisting on well to be consistent with everything else as well as because that's a sort of use the word came to us serving, while others would just consider it understandable, logical, and clear.
Solution 2:
Well is an adjective.
well adjective (better, best)
[predicative]
- In good health; free or recovered from illness
[ODO]
It just happens to have the same form as the adverb of good; and its comparative/superlative forms happen to be the same too. But it's an adjective.
Solution 3:
Just to expand upon Andrew's answer, which I upvoted, the following are all legitimate, grammatical uses of well/good. (I started to put these in a comment but it got too elaborate for the medium.)
I feel well. [I am not sick.]
I feel good. [I am feeling buoyant or optimistic.]
I feel ill. [I am sick.]
I feel bad. [I am sick or I have a negative feeling about some circumstance or condition.]
I feel good about that. [I have a favorable view toward the thing in question.]
I feel bad about that. [I have a negative feeling toward the thing in question.]
Note that in the last case you often hear people say "I feel badly about that," possibly because they think that they need an adverb when they actually need an adjective. To "feel badly" about something would actually mean to perform the function of feeling in a substandard way.