In his ch. 3, Mill responds to someone who asks, “How is the promotion of the happiness of all (and so primarily of others) my duty?,” and he takes the relevant answer to be an indication of the sanction (i.e., the stick, not the carrot) for not fulfilling this duty. After a brief acknowledgment of the influence of external sanctions, he focuses on the internal sanction of moral “feeling” (pp. 28f). A summary description of its character appears in the last paragraph of the chapter (p. 34), but notice also his claim that this feeling of sympathy is natural and grows with the development of society (pp. 32f).

On the other hand, Mill’s focus in his “proof” of the utilitarian principle in ch. 4 is not its implications of concern for the happiness of others but rather his claim that happiness (in his sense) is “desirable, and the only thing desirable, as an end” (p. 35). His argument that it is desirable (pp. 35f) has been criticized, but his main focus is the claim that only it is of value. Note that he argues, not that all other things of value are of value only as means to it, but rather that those things (like virtue) that are valuable not merely as means are valuable because they are (or have become) part of happiness (pp. 37f). The two components of his argument are summarized (and the first perhaps clarified) on p. 39. (The remainder of the chapter is also of some interest because it can read as a response to the Kantian idea of a will acting free from inclination.)