WASHINGTON — What was the Islamic State thinking? We know it is sophisticated in its use of modern media. But what was the logic of propagating to the world videos of its beheadings of two Americans (and subsequently a Briton) — sure to inflame public opinion?

There are two possible explanations. One is that these terrorists are more depraved and less savvy than we think. They so glory in blood that they could not resist making an international spectacle of their savagery and did not quite fathom how such a brazen, contemptuous slaughter of Americans would radically alter public opinion and risk bringing down upon them the furies of the U.S. Air Force.

The second theory is that they were fully aware of the inevitable consequence of their broadcast beheadings — and they intended the outcome. It was an easily sprung trap to provoke America into entering the Mesopotamian war. Why?

Because they’re sure we will lose. Not immediately and not militarily. They know we always win the battles but they are convinced that, as war drags on, we lose heart and go home.

They count on Barack Obama quitting the Iraq/Syria campaign just as he quit Iraq and Libya in 2011 and is in the process of leaving Afghanistan now. And this goes beyond Obama. They see a post-9/11 pattern: America experiences shock and outrage and demands action. Then, seeing no quick resolution, it tires and seeks out leaders who will order the retreat. In Obama, they found the quintessential such leader.

As for the short run, the Islamic State knows it will be pounded from the air. But it deems that price worth paying, given its gains in propaganda and prestige — translated into renown and recruiting — from these public executions.

Understanding this requires adjusting our thinking. A common mantra is that American cruelty — Abu Ghraib, Guantanamo, “torture,” the Iraq War itself — is the great jihadist recruiting tool. But leaving Iraq, closing Abu Ghraib and prohibiting “enhanced interrogation” had zero effect on recruiting. In fact, jihadi cadres from Mali to Mosul have only swelled during Obama’s outstretched-hand presidency.

Turns out the Islamic State’s best recruiting tool is indeed savagery — its own. Deliberate, defiant, triumphant. The beheadings are not just a magnet for psychopaths around the world. They are choreographed demonstrations of its own unbounded determination and of American helplessness. In Osama bin Laden’s famous formulation, who is the “strong horse” now?

We tend to forget that at this stage in its career, the Islamic State’s principal fight is intramural. It seeks to supersede and supplant its jihadi rivals — from al-Qaida in Pakistan to Jabhat al-Nusra in Syria — to emerge as champion of the one true jihad.

The strategy is simple: Draw in the world’s great superpower, create the ultimate foil and thus instantly achieve supreme stature in radical Islam as America’s nemesis.

It worked. A year ago, the world had never heard of this group, then named ISIS (the Islamic State in Iraq and Syria). Now it is the subject of presidential addresses, parliamentary debates and international conferences. It is the new al-Qaida, which itself has been demoted to JV.

So eclipsed and upstaged is al-Qaida that its leader, Ayman al-Zawahiri, scrambled to reveal the creation of a new India/South Asia branch. It announced itself this month with its first operation — a botched attack on a Pakistani frigate that left 10 al-Qaida fighters dead and the ship intact.

While al-Qaida was being humiliated, a huge Paris conference devoted entirely to the Islamic State was convened by Secretary of State John Kerry. Like his other conferences, it failed. Obama’s “broad coalition” remains a fantasy.

It’s more a coalition of the unwilling. Turkey denied us the use of its air bases. The Sunni Arab states are reluctant to do anything militarily significant. And not a single country has volunteered combat troops. Hardly a surprise, given that Obama has repeatedly ruled that out for the U.S. itself.

Testifying Wednesday to the Senate, Kerry said that the Islamic State “must be defeated. Period. End of story.” Not the most wisely crafted declaration: The punctuational emphasis carried unfortunate echoes of Obama’s promise about health care and the word “must” carried similar echoes of Obama’s assertions that Bashar al-Assad had to go.

But Kerry’s statement remains true for strategic and even moral reasons. But especially because when the enemy deliberately brings you into combat, it is all the more imperative to show the world that he made a big mistake.

Email letters@charleskrauthammer.com.