It should be clear by now, but apparently it isn’t: al-Qaeda is an idea, a way of thinking. Al-Qaeda thinks the world is divided between believers and nonbelievers, and the believers are divinely obliged to destroy the nonbelievers. It is a simple idea that has attracted tens of thousands of Muslims, but it is neither a political prescription nor the makings of an army. The Sunni Arabs who drifted into Iraq after the invasion and the Iraqis who embraced al-Qaeda were never an organization. They were never an army. They were never the main enemy. They numbered, what, a couple of thousand? They nearly triggered a civil war, but even that they failed to accomplish.

The success we’re seeing today in Iraq has nothing to do with rooting out terrorist cells. What we’re seeing instead is a shriveling of grassroots support, Sunni Muslims turning against al-Qaeda and its messianic, dualistic way of looking at the world. It hasn’t gone unnoticed in the Middle East that al-Qaeda has killed more Muslims than nonbelievers. Or that al-Qaeda has failed to take an inch of ground in the name of Islam. With this kind of record how could the Iraqis not turn against al-Qaeda?

None of this, of course, is to take away from the turnaround in Iraq. Last month we saw the fewest American casualties since the invasion in 2003. Basra, Sadr City, and Mosul are coming back under Baghdad’s control. Many Iraqis feel safe enough to move back into their houses. And none of it should take away from General Petraeus; our troops, who are bleeding and dying to hold together a country vital to American interests; or the Iraqis, who have backed away from civil war. So why should we now mischaracterize the enemy?

The tendency will be to leave it at the lie: We fought and beat al-Qaeda in Iraq. But it’s a lie we’ll pay for later. By mischaracterizing the enemy in Iraq, we mischaracterize the enemy in Pakistan. Whether the car bomb that destroyed the Danish embassy in Pakistan on Monday was the work of an actual member of al-Qaeda or not does not matter – what does is that al-Qaeda’s way of thinking is not defeated.