Again, it would have to depend on how one defines language. If one considers language as something that can express very specific actions such as the classic “pass the salt”, dance specifically probably would not be able to convey that message. Dance seems far more general in that you would be able to capture emotions on anger, loneliness, happiness and other relatively simple emotions. From my perspective language is something that can express any sort of emotion, therefore I would have to say that dance is a language.
Professional dancers are extremely capable. While watching the Alvin Ailey performance, there was a lot of grace and tension when needed which shows that dance can effectively communicate emotion. Dance definitely tells a story, which is the basis of the dances we saw. From the dances performed, each one tried to tell a story, and to certain extents they (the dancers/ choreographers) were successful. Therefore, I think that dance is indeed a language despite having limitations such as specificity.
