Fifty years ago the writer and philosopher Hannah Arendt witnessed the end of the trial of Adolf Eichmann, one of the major figures in the organisation of the Holocaust. Covering the trial Arendt coined the phrase "the banality of evil", a phrase that has since become something of an intellectual cliche. But what did she really mean?

One thing Arendt certainly did not mean was that evil had become ordinary, or that Eichmann and his Nazi cohorts had committed an unexceptional crime. Indeed, she thought the crime was exceptional, if not unprecedented, and that as a result it demanded a new approach to legal judgment itself.

There were at least two challenges to legal judgment that she underscored, and then another to moral philosophy more generally. The first problem is that of legal intention. Did the courts have to prove that Eichmann intended to commit genocide in order to be convicted of the crime? Her argument was that Eichmann may well have lacked "intentions" insofar as he failed to think about the crime he was committing. She did not think he acted without conscious activity, but she insisted that the term "thinking" had to be reserved for a more reflective mode of rationality.

Arendt wondered whether a new kind of historical subject had become possible with national socialism, one in which humans implemented policy, but no longer had "intentions" in any usual sense. To have "intentions" in her view was to think reflectively about one's own action as a political being, whose own life and thinking is bound up with the life and thinking of others. So, in this first instance, she feared that what had become "banal" was non-thinking itself. This fact was not banal at all, but unprecedented, shocking, and wrong.

By writing about Eichmann, Arendt was trying to understand what was unprecedented in the Nazi genocide – not in order to establish the exceptional case for Israel, but in order to understand a crime against humanity, one that would acknowledge the destruction of Jews, Gypsies, gay people, communists, the disabled and the ill. Just as the failure to think was a failure to take into account the necessity and value that makes thinking possible, so the destruction and displacement of whole populations was an attack not only on those specific groups, but on humanity itself. As a result, Arendt objected to a specific nation-state conducting a trial of Eichmann exclusively in the name of its own population.

At this historical juncture, for Arendt, it became necessary to conceptualise and prepare for crimes against humanity, and this implied an obligation to devise new structures of international law. So if a crime against humanity had become in some sense "banal" it was precisely because it was committed in a daily way, systematically, without being adequately named and opposed. In a sense, by calling a crime against humanity "banal", she was trying to point to the way in which the crime had become for the criminals accepted, routinised, and implemented without moral revulsion and political indignation and resistance.

If Arendt thought existing notions of legal intention and national criminal courts were inadequate to the task of grasping and adjudicating Nazi crimes, it was also because she thought that nazism performed an assault against thinking. Her view at once aggrandised the place and role of philosophy in the adjudication of genocide and called for a new mode of political and legal reflection that she believed would safeguard both thinking and the rights of an open-ended plural global population to protection against destruction.

What had become banal – and astonishingly so – was the failure to think. Indeed, at one point the failure to think is precisely the name of the crime that Eichmann commits. We might think at first that this is a scandalous way to describe his horrendous crime, but for Arendt the consequence of non-thinking is genocidal, or certainly can be.

Of course, the first reaction to such an apparently naive claim may be that Arendt overestimated the power of thinking or that she held on to a highly normative account of thinking that does not correspond to the various modes of reflection, self-muttering, and silent chatter that goes by that name.

Indeed, her indictment of Eichmann reached beyond the man to the historical world in which true thinking was vanishing and, as a result, crimes against humanity became increasingly "thinkable". The degradation of thinking worked hand in hand with the systematic destruction of populations.

Although Arendt focuses on Eichmann's failure to think as one way of naming his ultimate crime, it is clear that she thinks the Israeli courts did not think well enough, and sought to offer a set of corrections to their way of proceeding. Although Arendt agreed with the final verdict of the trial, namely, that Eichmann should be condemned to death, she quarreled with the reasoning put forward at the trial and with the spectacle of the trial itself. She thought the trial needed to focus on the acts that he committed, acts which included the making of a genocidal policy.

Like the legal philosopher Yosal Rogat before her, Arendt did not think that the history of anti-semitism or even the specificity of anti-semitism in Germany could be tried. She objected to Eichmann's treatment as a scapegoat; she criticised some of the ways that Israel used the trial to establish and legitimate its own legal authority and national aspirations. She thought the trials failed to understand the man and his deeds. The man was either made to stand for all of nazism and for every Nazi, or he was considered the ultimately pathological individual. It seemed not to matter to the prosecutors that these two interpretations were basically in conflict. She thought that the trial necessitated a critique of the idea of collective guilt, but also a broader reflection on the historically specific challenges of moral responsibility under dictatorship. Indeed, that for which she faulted Eichmann was his failure to be critical of positive law, that is, a failure to take distance from the requirements that law and policy imposed upon him; in other words, she faults him for his obedience, his lack of critical distance, or his failure to think.

But more than this, she faults him as well for failing to realise that thinking implicates the subject in a sociality or plurality that cannot be divided or destroyed through genocidal aims. In her view, no thinking being can plot or commit genocide. Of course, they can have such thoughts, formulate and implement genocidal policy, as Eichmann clearly did, but such calculations cannot be called thinking, in her view. How, we might ask, does thinking implicates each thinking "I" as part of a "we" such that to destroy some part of the plurality of human life is to destroy not only one's self, understood as linked essentially to that plurality, but to destroy the very conditions of thinking itself.

Many questions abound: is thinking to be understood as a psychological process or, indeed, something that can be properly described, or is thinking in Arendt's sense always an exercise of judgment of some kind, and so implicated in a normative practice. If the "I" who thinks is part of a "we" and if the "I" who thinks is committed to sustaining that "we", how do we understand the relation between "I" and "we" and what specific implications does thinking imply for the norms that govern politics and, especially, the critical relation to positive law?

Arendt's book on Eichmann is highly quarrelsome. But it is probably worth remarking that she is not only taking issue with the Israeli courts and with the way in which they arrived at the decision to punish Eichmann to death. She is also critical of Eichmann himself for formulating and obeying a noxious set of laws.

One rhetorical feature of her book on Eichmann is that she is, time and again, breaking out into a quarrel with the man himself. For the most part, she reports on the trial and the man in the third person, but there are moments in which she addresses him directly, not on the trial, but in her text. One such moment occurred when Eichmann claimed that in implementing the final solution, he was acting from obedience, and that he had derived this particular moral precept from his reading of Kant.

We can imagine how doubly scandalous such a moment was for Arendt. It was surely bad enough that he formulated and executed orders for the final solution, but to say, as he did, that his whole life was lived according to Kantian precepts, including his obedience to Nazi authority, was too much. He invoked "duty" in an effort to explain his own version of Kantianism. Arendt writes: "This was outrageous, on the face of it, and also incomprehensible, since Kant's moral philosophy is so closely bound up with man's faculty of judgment, which rules out blind obedience."

Eichmann contradicts himself as he explains his Kantian commitments. On the one hand, he clarifies: "I meant by my remark about Kant that the principle of my will must always be such that it can become the principle of general laws." And yet, he also acknowledges that once he was charged with the task of carrying out the final solution, he ceased to live by Kantian principles. Arendt relays his self-description: "he no longer 'was master of his own deeds,' and … he 'was unable to change anything'."

When in the midst of his muddled explanation, Eichmann reformulates the categorical imperative such that one ought to act in such a way that the Führer would approve, or would himself so act, Arendt offers a swift rejoinder, as if she were delivering a direct vocal challenge to him: "Kant, to be sure, had never intended to say anything of the sort; on the contrary, to him every man was a legislator the moment he started to act; by using his 'practical reason' man found the principles that could and should be the principles of law."

Arendt makes this distinction between practical reason and obedience in Eichmann in Jerusalem in 1963 and seven years later she began her influential set of lectures on Kant's political philosophy at the New School for Social Research in New York City. In a way, we can understand much of Arendt's later work, including her work on willing, judgment and responsibility, as an extended debate with Eichmann on the proper reading of Kant, an avid effort to reclaim Kant from its Nazi interpretation and to mobilise the resources of his text precisely against the conceptions of obedience that uncritically supported a criminal legal code and fascist regime.

In many ways, Arendt's approach is itself quite astonishing, since she is, among other things, trying to defend the relation between Jews and German philosophy against those who would find in German culture and thought the seeds of national socialism. In this way, her view recalls that of Hermann Cohen, who argued tragically in the early part of the 20th century that Jews would find greater protections and cultural belonging in Germany than in any Zionist project that would take them to Palestine.

Cohen thought universality belonged to German philosophy, rather than considering internationalist or global models that might provide an alternative to both nation-states. Arendt lacks Cohen's naivete, and sustained an important critique of the nation-state. She reformulates Cohen's project in a new social and political philosophy: truly staying with Kant or, rather, reformulating him for a contemporary social and political philosophy in a true sense would have stopped Eichmann and his cohorts, would have produced another kind of trial than the one she saw in Jerusalem, and would have redeemed the German-Jewish philosophical vocation – one that she tried to bring with her to New York. What had become banal was the attack on thinking, and this itself, for her, was devastating and consequential. Remarkable for us, no doubt, is Arendt's conviction that only philosophy could have saved those millions of lives.