What connotations does the word disorder have for a native speaker? Does it sound very negative or rather neutral?
It's still better to describe phenomena such as ADHD as mental illness, but it still sounds too negative to me.
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It depends significantly on the context.
In terms of ADD and ADHD, the fact that they are abbreviations generally removes any connotation that disorder may hold because ADD and ADHD are very common terms that people have tended to directly associate with the condition in question.
If, however, disorder were to be spelled out in terms of a clinical diagnosis, there might be a small negative connotation, but generally speaking, I do not think it would be very significant.
On the other hand, if you were to say he has a mental disorder, that is very negative and might well be considered to be bordering on rude.
In general, the word "disorder" does carry a very negative connotation. For something to be a "disorder", it must not follow the correct "order". This can suggest that it is somehow "unnatural", since it does not follow the "natural order of things". And no one likes to be told they are unnatural.
This is especially the case when referring to a medical disorder. I get the impression that the phrase "mental disorder" is slowly disappearing from medical language, the same way previous phrases thought to be offensive have, such as "mentally retarded".
The word “disorder” has so many uses in so many contexts, it’s hard to say what kind of connotation it has. In a specific context, though, it can have a specific positive or negative connotation. In the phrase “disorderly conduct”, it suggests drunkenness or being a deadbeat. In the sentence “Entropy is a measure of disorder”, it can suggest interestingness as well as death: zero entropy means adhering to a dull, rigid pattern, too dull and rigid to support life; but death is the result when entropy gets too high, because too much disorder is also incompatible with life.