Few have given as much thought as Barack Obama to the pitfalls of waging open-ended war on an abstract noun. On top of its impracticalities – how can you ever declare victory? – fighting a nebulous enemy exacts an insidious toll. Mr Obama built much of his presidential appeal on such a critique – the global war on terror was eroding America’s legal rights at home and its moral capital abroad. The term “GWOT” was purged the moment he took over from George W Bush. In his pledge last week to “degrade and ultimately destroy” the Islamic State in Iraq and the Levant, known as Isis, he has travelled almost full circle. It is precisely because Mr Obama is a reluctant warrior that his legacy will be enduring.

The reality is the US war on terror has succeeded where it was supposed to. Mr Bush’s biggest innovation was to set up the Department of Homeland Security. If you chart domestic terror attempts in the US since September 11 2001, they have become increasingly low-tech and ineffectual. From the foiled Detroit airliner attack in Mr Obama’s first year to the Boston marathon bombings in his fifth, each attempt has been more amateur than the last. The same is true of America’s allies. There has been no significant attack in Europe since London’s July 7 bombings nine years ago. Western publics have acclimatised to an era of tighter security.

If this is the balance sheet of the US war on terror, why lose sleep? Chiefly because it understates the costs. The biggest of these is the damage an undeclared war is doing to the west’s grasp on reality. Myopic thinking leads to bad decisions. Mr Obama pointedly avoided using the word “war” last week. Although there are more than 1,000 US military personnel in Iraq, and more than 160 US air strikes in the past month, he insisted on calling his plan to destroy Isis a “campaign”. Likewise, the US uniforms are those of “advisers” and “trainers”. These kinds of euphemism lead to mission creep. If you embark on something with your eyes half-open, you are likelier to lose your way.