Longman Dictionary of Contemporary English homepage

rightly

adverb
     
right‧ly
1 correctly, or for a good reason [≠ wrongly]:
I was, as you rightly said, the smallest boy in the class.
As you so rightly pointed out, things are getting worse.
They have been treated badly, and they are rightly upset.
quite rightly British English
There's a lot of talk, quite rightly, about the dangers of smoking.
This photo was taken in Paris, if I remember rightly.
2

rightly or wrongly

used to emphasize that someone else thinks that something they did was right, but you think it was wrong:
The prime minister was widely judged, rightly or wrongly, to be an honest man.
3

and rightly so

spoken used to say that a decision or action you have just described is fair and morally right, in your opinion:
A lot of people round here were furious, and rightly so.
4

I can't rightly say/don't rightly know

spoken used to say that you are not sure whether something is correct or not

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