WASHINGTON — Why did they do it? What did the Islamic State think it could possibly gain by burning alive a captured Jordanian pilot?

I wouldn’t underestimate the absence of logic, the sheer depraved thrill of a triumphant cult reveling in its barbarism. But I wouldn’t overestimate it either. You don’t overrun much of Syria and Iraq without having deployed keen tactical and strategic reasoning.

So what’s the objective? To destabilize Jordan by drawing it deeply into the conflict.

At first glance, this seems to make no sense. The savage execution has mobilized Jordan against the Islamic State and given it solidarity and unity of purpose.

Yes, for now. But what about six months hence? Solidarity and purpose fade quickly. Think about how post-9/11 American fervor dissipated over the years of inconclusive conflict, yielding the war fatigue of today. Or how the beheading of U.S. journalists galvanized the country against the Islamic State, yet less than five months later, the frustrating nature of that fight is creating divisions at home.

Jordan is a more vulnerable target because, unlike the U.S., it can be destabilized. For nearly a century Jordan has been a miracle of stability — an artificial geographic creation led by a British-imposed monarchy, it has enjoyed relative domestic peace and successful political transitions with just four rulers over four generations.

Compared to Lebanon, Syria and Iraq, similarly created, Jordan is a wonder. But a fragile one. Its front-line troops and special forces are largely Bedouin. The Bedouin are the backbone of the Hashemite monarchy but they are a minority. Most of the population is non-indigenous Palestinians, to which have now been added 1.3 million Syrian refugees.

Most consequential, however, is the Muslim Brotherhood with its strong Jordanian contingent — as well as more radical jihadist elements, some sympathetic to the Islamic State. An estimated 1,500 Jordanians have already joined the Islamic State in Iraq and Syria. Others remain home, ready to rise when the time is right.

The time is not right today. Jordanian anger is white hot. But the danger is that as the Jordanians attack — today by air, tomorrow perhaps on the ground — they risk a drawn-out engagement that could drain and debilitate the regime, one of the major bulwarks against radicalism in the entire region.

We should be careful what we wish for. Americans worship at the shrine of multilateralism. President Obama’s Islamic State strategy is to create a vast coalition with an Arab/Kurdish vanguard and America leading from behind with air power.

The coalition is allegedly 60 strong. (And doing what?) Despite administration boasts, the involvement of the Arab front line — Jordan, Saudi Arabia, Bahrain and the United Arab Emirates — has been minimal and symbolic. In fact, we’ve just now learned that the UAE stopped flying late last year.

The Obama policy has not fared terribly well. Since the policy was launched, the Islamic State has doubled its Syrian domain. It’s hard to see a Jordanian-Saudi force succeeding where Iraq’s Shiite militias, the Iraqi military, the Kurds and U.S. airpower have thus far failed.

What’s missing, of course, are serious boots on the ground, such as Syria’s once-ascendant non-jihadist rebels, which Obama contemptuously dismissed and allowed to wither. And the Kurds, who are willing and able to fight, yet remain scandalously undersupplied by this administration.

Missing most of all is Turkey. It alone has the size and power to take on the Islamic State. But doing so would strengthen, indeed rescue, Turkey’s primary nemesis, the Iranian-backed Bashar al-Assad regime in Damascus.

Turkey’s price for entry was an American commitment to help bring down Assad. Obama refused. So Turkey sits it out.

Why doesn’t Obama agree? Didn’t he say that Assad must go? The reason is that Obama dares not upset Assad’s patrons, the Iranian mullahs, with whom Obama dreams of concluding a grand rapprochement.

For Obama, this is his ticket to Mt. Rushmore. So in pursuit of his Nixon-to-China Iran fantasy, Obama eschews Turkey, our most formidable potential ally against both the Islamic State and Assad.

What’s Obama left with? Fragile front-line Arab states, like Jordan.

But even they are mortified by Obama’s blind pursuit of detente with Tehran, which would make the mullahs hegemonic over the Arab Middle East. Hence the Arabs, the Saudis especially, hold back from any major military commitment to us. Jordan, its hand now forced by its pilot’s murder, may now bravely sally forth on its own. But at great risk and with little chance of ultimate success.

Charles Krauthammer is a columnist with the Washington Post Writers Group.