Proof of the Greatest Happiness Principle

Mill's Proof of the Greatest Happiness Principle.
(1) if you see something, this proves that it is visible.
(2) Similarly, desiring something proves that it is desirable.
(3) The only thing that each person truly desires is happiness.
(4) The only thing that is truly desirable for a person is his or her own happiness.
(5) Hence each person should perform the actions that promote the greatest happiness.

Three Problems with this Argument:
First,  (2) does not follow from (1).  Visibility and desirability are not the same kinds of things.
Second,  (4) does not follow from (3).  To assume that we can derive an "ought" from an "is" is to make the naturalistic fallacy, which is a point made by David Hume.
Third,  (5) does not follow from (4).  Just because your own happiness is desirable does not mean that the happiness of other people is also desirable.

Further Objections to Utilitarianism:
(1) We cannot always predict what the consequences of our actions will be.  It is difficult if not impossible to judge the morality of an action based on what the predicted consequences will be.
(2) It is difficult to quantify pleasures.  In other words, it is hard to represent pleasures with numbers in order to determine if happiness will be maximized by an action. Cost-benefit analysis is difficult if not impossible.
(3) Also, the process of trying to determine if an action will maximize happiness is time-consuming and difficult.  It is not practical that we will be able to do such a calculation before we perform an action.
(4) Utilitarianism cannot explain special duties or obligations that we have to people like our friends, family and neighbors.
(5) If we only care about sum total happiness, then there will be actions that are good while these actions do cause a great amount of pain for some people.  For example, say that six of the seven dwarves decide to torture, beat and murder the Dopey for fun.  Even though Dopey experiences a lot of pain, utility is still maximized because the other dwarves are so happy.  Thus a utilitarian must say that it is good for the dwarves to murder Dopey.
(6) Utilitarianism fails to respect individual rights.  The rights of a single person can be violated as long as the greatest good is still maximized.
(7) Utilitarianism does not address the question of what kind of a person we should be.  The focus is entirely on consequences rather than the character of people.  For example, if I am offered a job as an assassin, I might think that killing people for a job would be OK because even if I do not take the job, someone else will.  This means that whether or not I am an assassin, the consequences are the same, hence the moral value of either choice is equal.
(8)  Pain is either entirely subjective (meaning that it does not exist) or perhaps pain is a good thing itself.

The first three objections do not attack the theory but rather the practice of the theory.  Objections (1), (2) and (3) are all objections that the theory is impractical and difficult (if not impossible) to use as a way of making moral decisions.  Objections (4), (5), (6), (7) and (8) object on the grounds that the results of utilitarian theory will conflict with strong moral intuitions or other reasons to consider when deciding if an action is good or bad.