In a recent blog post I commented that the basic principle of utilitarianism is Jeremy Bentham’s axiom: “it is the greatest happiness of the greatest number that is the measure of right and wrong.”
We can construct the following hypothetical society: imagine a country where the majority take great delight in watching the torture of the minority. When individual members of the minority are tortured, hundreds of members of the majority come to watch and they have a seemingly insatiable appetite to view such a spectacle. The sum total of the additional happiness of those watching the torture is so great that it more than offsets the pain of the member of the minority being tortured. The utilitarians who follow Bentham’s axiom would know what to do: open torture theatres up and down the country and have numerous daily torture shows to maximise happiness.
The thought is horrific. Karl Popper was aware of such problems with standard utilitarianism. He wrote the following:
I suggest… to replace the utilitarian formula “Aim at the greatest happiness for the greatest number”, or briefly, “Maximise happiness”, by the formula “The least amount of suffering for all”, or briefly, “Minimize suffering”. Such a simple formula can, I believe, be made one of the fundamental principles (admittedly not the only one) of public policy. 
He went on to say:
A … criticism of the Utilitarian formula “maximise pleasure” is that it assumes, in principle, a continuous pleasure-pain scale which allows us to treat degrees of pain as negative degrees of pleasure. But, from the moral point of view, pain cannot be outweighed by pleasure, and especially not one man’s pain by another man’s pleasure. Instead of the greatest happiness for the greatest number, one should demand, more modestly, the least amount of avoidable suffering for all; and further, that unavoidable suffering—such as hunger in times of an unavoidable shortage of food—should be distributed as equally as possible…. It adds to clarity in the field of ethics if we formulate our demands negatively, i.e. if we demand the elimination of suffering rather than the promotion of happiness. 
Professor Popper’s negative formulation (negative utilitarianism or NU for short) of the standard utilitarian principle has a fundamental problem pointed out by R.N. Smart in 1958:
Suppose that a ruler controls a weapon capable of instantly and painlessly destroying the human race. Now it is empirically certain that there would be some suffering before all those alive on any proposed destruction day were to die in the natural course of events. Consequently the use of the weapon is bound to diminish suffering, and would be the ruler’s duty on NU grounds.
If the human race were to continue in existence for an infinite amount of time, no doubt there would be an infinite amount of suffering to go with it. According to this ideology, it is far better to have a finite amount of suffering and end the world now.
Bernard Williams concluded his critique of utilitarianism with the following words:
The important issues that utilitarianism raises should be discussed in contexts more rewarding than that of utilitarianism itself. The day cannot be too far off in which we hear no more of it.
There are those who predict that the world will end tomorrow, December 21, 2012. NASA has denied that this will occur. While those putting forward the prediction are not blaming negative utilitarians with a world ending weapon for this disaster, the negative utilitarians might look forward to its occurrence. I have been trying to think of any good that will come of the end of the world and have finally come up with something: Bernard Williams’ prediction that the day will come when we will hear no more of utilitarianism will finally be fulfilled.
 K.R. Popper, The Open Society and Its Enemies, Volume 1: The Spell of Plato, (Fifth revised edition, Routledge, 1966), p.235n6(2).
 Ibid., pp284-285n2.
 R.N. Smart, “Negative Utilitarianism,” Mind, New Series, Vol. 67, No. 268 (October, 1958), pp. 542-543. A copy of this is available freely on line.
 Bernard Williams, “A critique of utilitarianism,” in J.J.C. Smart & Bernard Williams, Utilitarianism: for and against, (Cambridge University Press, 1973,) p.150.