What does "I must have you dance" mean? What kind of a sentence is it?
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The only way I can make sense of that sentence is someone expressing his or her strong desire to induce the dancer to dance for him or her. Although not all may agree, I think of this sort of construction as a part of high-society small talk.
The question here is on the polysemy of the verb have.
"I'll have you do your homework, whether you like it or not." (says mum to son)
(King of Hearts to Mad Hatter)