Mill's Proof in Chapter 4 of Utilitarianism

(This reconstruction is taken from Fred Feldman, Introductory Ethics (Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1978), pp. 41-46.)

The Analogical Argument (Paragraph Three)

1. Each person desires his own happiness.
2. If each person desires his own happiness, then each person can desire his own happiness.
3. If each person can desire his own happiness, then each person's happiness is desirable for that person.
4. If each person's happiness is desirable for that person, then each person's happiness is a good for that person.
5. Therefore, each person's happiness is a good for that person.

The Aggregation Argument (Paragraph Three)

5. Each person's happiness is a good to that person.
6. If each person's happiness is a good to that person, then the general happiness is a good to the aggregate of people.
7. If the general happiness is a good to the aggregate of people, then the general happiness is a criterion of morality.
8. Therefore, the general happiness is a criterion of morality.

The Parts and Means Argument (Paragraphs Eight and Nine)

 9. People can desire nothing other than parts of or means to happiness.
10. If people can desire nothing other than parts of or means to happiness, then nothing other than happiness is a criterion of morality.
11. Therefore, nothing other than happiness is a criterion of morality.
12. Therefore, happiness is the sole criterion of morality. (From 8 and 11)