One of the many script weaknesses in III is the ambiguity in the scene with Michael, Kay and Anthony at Michael's party. Michael wants Anthony to finish law school; Anthony is determined to be a professional singer. Michael seems to denigrate his prospects for success as a singer. He tells Anthony that having a law degree "is insurance...After that you can do anything you want. You can work for me." Anthony replies that "I will never be a part of your business." He says no to continuing law school and walks out.
The implication is that Michael wants Anthony to be his successor in the "legitimate" Corleone business empire. But, he also said Anthony could do anything he wanted after he got the "insurance" of a law degree." We might infer that he's advising Anthony in a fatherly way that the odds of success in music are slim compared with a career in law. But he also says Anthony "throws his life away...he throws greatness away."
What did he want for Anthony?
