No, because I think it would promote negativity too much.
What I would like instead is something of a "thank you" button, next to the like button. I myself and many with me now use the like button simply as an acknowledgement to the poster. It doesn't necessarily mean I really liked the post. By adding another option for this the like button can be used for actual liking.
Alternatively there could be a rating. Instead of the like and thank you buttons you could give a comment 1-5 stars.