This is the fundamental contradiction, after Paris: Few in the West want to send ground troops to Syria and Iraq to battle the Islamic State group, but it may be even harder to find anyone who thinks airstrikes alone will defeat the radical extremists.

So far, policy makers and experts tend to focus on incremental steps, and indeed the initial French response was more airstrikes. President Barack Obama on Monday insisted that the current strategy "is ultimately is going to work," and rejected any suggestion that American soldiers should be deployed.

But if IS carries through with its threats of further attacks on the West, such an approach may soon be unsustainable, as public pressure would demand action more effective than the combination of airstrikes and ground advances by a mix of local allies.

An international ground operation could become conceivable, and would not necessarily rely on Americans — a constellation of nations, including Egypt, Iran, the Gulf, Europe and Russia, has become increasingly enraged at the jihadis.

That prospect doesn't faze the extremists. In official statements and online chatter, they even taunt the West to launch another doomed crusade in the Middle East.

The narrative of a "holy war" against the infidels is strong in radical Islamic circles. While losing their territorial "caliphate" in Syria and Iraq would be a setback in the short term, a costly and bloody outside intervention also fuels the group's apocalyptic appeal and could serve to convert yet more Muslims to their cause.

And though the speedy liberation of the town of Sinjar in recent days suggests IS forces are far from invincible, a ground war against them would not be easy.

An invasion of Syria's Raqqa and Iraq's Mosul — IS's main urban strongholds in Syria and Iraq, respectively — risks incurring huge losses both among ground troops and civilians in block-by-block, door-to-door, close-quarters combat like that seen in grueling earlier U.S. battles for Ramadi and Fallujah. The militants are known to set up booby traps and IEDs for their enemies and are likely to use civilians as human shields. Fighting would likely involve combat along vast desert highways — areas even the Iraqi and Syrian governments struggle to control — as well as farmland and towns that give the insurgents plenty of opportunity to hide and ambush the invading troops.

It is not unthinkable that truly crushing IS would take longer than the nearly nine-year effort to pacify Iraq after the 2003 U.S.-led invasion.

So would an invasion amount to repeating history? There are some differences.

When the United States pulled together a coalition against Iraq in 2003, world support was wobbly. Now, there is genuine global revulsion at the actions of Islamic State, a greater likelihood than before for a genuinely international coalition with full regional support.

Even before the recent global terrorism, there were mass killings of opposing Sunni tribal fighters, the enslaving and massacring of minorities like the Yazidis in Iraq, summary killings of gay men and captured enemies and random hostages, videotaped beheadings of Western aid workers and journalists, and the cruel subjugation of all who fall into their grasp. That revulsion is unreservedly shared by Russia, which does not often find itself in agreement with the West of late, and essentially all of the governments in the region — including, in a rarity, that of non-Arab Iran.

Some Arab states may themselves still be reluctant to commit ground troops while they focus on priorities like the war in Yemen and placating Sinai. But others might not, and rich Gulf states could be convinced to help pay for an expanded military operation — including more training for Syrian rebels — or ramp up airstrikes.

Arab participation in an actual invasion might minimize the jihadis' ability to present it as an outsiders' crusade. Yet problems would remain. Many Arab governments can be portrayed by the jihadis as "infidels." And any presence of Iranians — non-Arab and non-Sunni — might hinder as much as help.

Another possibility is an invoking of NATO's Article 5 — invoked only once, in the wake of the Sept. 11 attacks — stating that an attack on one member is an attack on all. There is a challenge in directing this against a non-state actor, but NATO's members have made clear the commitment extends to cases of international terrorism.

It is widely thought that NATO would continue to take a hands-off role in the campaign against IS but that the U.S. and other allies would be ready to share intel, killer drones and other assets. "A number of allies are already working with France on their ongoing operations and investigations in the wake of the attacks," NATO said Monday.

Other measures might be taken against the Islamic State:

— Russia or another country — perhaps France itself, if pushed hard enough — may be more willing to step up the bombing campaign in urban areas like Raqqa, despite the risk to civilians.

— Turkey may come under more pressure to seal its border with Syria. The Turks insist it is too hard to stop the supply lines and trade routes that enrich and sustain IS across the porous frontier. But most observers believe Turkey is looking the other way because IS is also fighting the Kurds, whom Ankara views with suspicion.

— To date, special forces attacks have not penetrated the heart of major IS-held cities or attempted to strike at the very top leaders; this, too, can change.

All of these strategies contain risk and difficulty. But all might be considered preferable to an approach that may soon start to look like the acceptance by the West of a theocratic entity, vicious to its subjects and hostile to the world, erasing a Mideast map created by Westerners a century ago.

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Dan Perry is AP's Middle East editor leading text coverage in the region. Follow him on Twitter at www.twitter.com/perry_dan

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Robert Burns in Washington, John-Thor Dahlberg in Brussels, Adam Schreck in Cairo and Zeina Karam in Baghdad contributed to this report.