The problem of dirty hands concerns whether political leaders or other like actors can ever be justified in committing even gravely immoral actions when "dirtying their hands" in this way is necessary for realizing some important moral or political end, such as the preservation of a community's continued existence or the prevention of imminent societal catastrophe. If political actors can be so justified, a paradox or contradiction seems to emerge because it appears that these actors can, or even must, carry out actions that are, ex hypothesi, morally impermissible. Classic examples of situations in which the problem of dirty hands could arise include ticking time bomb scenarios of the kind popularized by the television series, 24. The problem of dirty hands lies at the point where moral philosophy, political philosophy, and political ethics all intersect.
Walzer and Williams on dirty hands
Though the discourse on dirty hands goes back as far as Machiavelli, contemporary philosophical interest in the problem of dirty hands had been revitalized by the works of American political theorist Michael Walzer and other thinkers. The term itself comes from Jean-Paul Sartre's 1948 play Dirty Hands, in which Hoederer speaks of having dirty hands up to his elbows, then asks, "But what do you hope? Do you think you can govern innocently?"
Walzer argued that, in cases of "supreme emergency" in which a political community's continued existence is in imminent danger, its leaders might be obligated to dirty their hands and sanction gravely immoral actions for the sake of saving the community. Discussing the British bombing campaigns against German cities from 1940-1942, Walzer wrote:
[I]t does seem to me that the more certain a German victory appeared to be in the absence of a bomber offensive, the more justifiable was the decision to launch the offensive. It is not just that such a victory was frightening, but also that it seemed in those years very close; it is not just that it was close, but also that it was so frightening. Here was a supreme emergency where one might well be required to override the rights of innocent people and shatter the war convention.
Given the view of Nazism that I am assuming, the issue takes this form: should I wager this determinate crime (the killing of innocent people) against that immeasurable evil (a Nazi triumph)?
Elsewhere, the late British philosopher Bernard Williams explored the problem of dirty hands in less hyperbolic situations, more the everyday necessities of political life than the extraordinary undertakings of defending one's community from outright destruction: "[I]t is a predictable and probable hazard of public life that there will be these situations in which something morally disagreeable is clearly required. To refuse on moral grounds ever to do anything of that sort is more than likely to mean that one cannot seriously pursue even the moral ends of politics" 
The problem of dirty hands: paradox or confusion?
If the problem of dirty hands is a genuine problem, then it poses a serious challenge to certain theories of ethics insofar as those theories seem (at least at first blush) ill-equipped to provide a satisfactory account of how it might be possible for a political actor to be either permitted or obligated to commit actions that are ex hypothesi morally impermissible.
On the one hand, utilitarianism is (roughly) the moral theory that the goodness or badness of any given moral action is essentially a matter of that action's consequences, whether it promotes more in the way of pleasure or pain. (Jeremy Bentham is one example of a utilitarian philosopher.) On this view, the problem of dirty hands seems to become unintelligible because, under the utilitarian calculus, the act of dirtying one's hands (say, for the sake of saving one's political community) seems to count as a moral action. For the problem of dirty hands to arise, we must understand the act of dirtying one's hands as not simply unseemly, but also immoral.
On the other hand, deontological theories of morality, such as those of Immanuel Kant, hold that the correctness of a given moral action depends in no small part on the nature of the action itself, rather than on the consequences of those actions. The expression "Let justice be done, though the world perish", for example, is firmly grounded in a deontological perspective. On the deontological view, it is not readily clear how it could be possible (as the problem of dirty hands would have it) for a political agent to ever have a permission or a duty to dirty her hands.
For his part, Bernard Williams regarded as peculiar those morality systems that categorically privilege moral considerations over and above all others: "Ethical life itself is important, but it can see that things other than itself are important. It contains motivations that indeed serve these other ends but at the same time can be seen from within that life as part of what make it worth living" 
The problem of dirty hands also dissolves in the hands of political realists who maintain that politics operates largely outside the domain of morality.
- State of exception (Agamben)—a related concept, the state of exception involves the suspension of the rule of law, which can be compared to the suspension of the rule of morality in "supreme emergencies"...
- Jean-Paul Sartre, Dirty Hands, in Three Plays by Jean-Paul Sartre, trans. Lionel Abel (New York: Alfred A. Knopf, 1949), 121.
- Michael Walzer, Just and Unjust Wars: A Moral Argument with Historical Illustrations, 2nd ed. (New York: Basic Books, 1992), 259.
- Bernard Williams, "Politics and Moral Character", in Public and Private Morality, ed. Stuart Hampshire (Cambridge, UK: Cambridge University Press, 1978), 62.
- Bernard Williams, Ethics and the Limits of Philosophy (Cambridge, MA: Harvard, 1985), 184.
- Coady, C.A.J., "The Problem of Dirty Hands". The Stanford Encyclopedia of Philosophy (Spring 2014 Edition), edited by Edward N. Zalta.
- Coady, C.A.J., "Tony Coady on Dirty Hands in Politics", on Philosophy Bites. Interview by Nigel Warburton. October 25, 2009.
- The International Encyclopedia of Ethics's entry on "Dirty Hands"