The real threat posed by last week's brutal bombings in London is that the Muslim terrorists who apparently planted the bombs still think they can win. Breaking that psychology is the fundamental challenge for responsible leaders in the West and the Muslim world.

Compare the first statements from both sides, and you can see the essential battle of wills. A group calling itself the Secret Organization of al Qaeda in Europe posted a triumphal statement immediately after the London attacks last Thursday, claiming that "Britain is now burning with fear, terror and panic." That was untrue, as anyone watching Londoners on television could see. But the attacks showed the determination and resourcefulness of the terrorist underground.

British Prime Minister Tony Blair's first statement after the bombings also went to the heart of the matter: "Our determination to defend our values and our way of life is greater than their determination to cause death and destruction." Later in the day Blair put it more succinctly: "We shall prevail, and they shall not."

But what does "winning" mean, and how does it fit public sentiments on both sides of the battle? That's the subject of two fascinating studies that surfaced just before the London bombings. Taken together they help clarify some of the strategic issues facing the Bush administration and its allies.

The terrorists' motivation is outlined in a disturbing new book by political scientist Robert A. Pape, "Dying to Win: The Strategic Logic of Suicide Terrorism." He analyzed the 315 suicide attacks that occurred around the world from 1980 to 2003 and concluded that in nearly every case terrorists were resisting what they regarded as foreign occupation. Their goal was "to compel modern democracies to withdraw military forces from territory that the terrorists consider to be their homeland," Pape writes. They turned to suicide attacks because, in their judgment, they worked against democratic societies, which have difficulty absorbing the pain the terrorists can inflict.

Pape quotes a 2003 sermon by Osama bin Laden that focused on what bin Laden saw as America's vulnerability to such attacks: "America is a great power possessed of tremendous military might and a wide-ranging economy, but all this is built on an unstable foundation which can be targeted, with special attention to its obvious weak spots. If America is hit in one hundredth of these weak spots, God willing, it will stumble, wither away and relinquish world leadership."

But what is the real psychological base line of America and its allies? Here I turn to a new paper by three Duke University political scientists -- Christopher F. Gelpi, Peter D. Feaver and Jason Reifler -- titled "Casualty Sensitivity and the War in Iraq" and available on the Internet. They argue that it isn't casualties per se that drive U.S. public opinion about war. Instead, it's the public perception of whether a war is winnable.

"When the public believes the mission will succeed, then the public is willing to continue supporting the mission, even as costs mount. When the public thinks victory is not likely, even small costs will be highly corrosive," the authors write. (Feaver is currently working for the Bush National Security Council staff.) Putting these two studies together, we can see that the challenge for the United States and its allies is to define "winning" in an achievable way, so that public support can be maintained. To win, says Pape, the United States must "defeat the current pool of terrorists now actively planning to kill Americans" and at the same time "prevent a new, potentially larger generation from rising up." As long as a war can be characterized as resistance to foreign occupation, the terrorists will maintain support from their public.

That's the puzzle the Bush administration confronts in Iraq. There aren't any easy solutions, but it seems to me that the administration is on the right track with its plans, reported in a leaked British memo last week, to reduce troop levels in Iraq by more than half by early next year and turn over 14 of 18 provinces to Iraqi control. That will allow the United States to focus on training Iraqi security forces, which are the only ones that can stabilize the country in the long run. The administration is also wise to seek a political settlement with Iraq's Sunni Muslims, in the recognition, as Deputy Secretary of State Robert Zoellick said in Amman, Jordan, this week, that "military means alone are not capable of defeating the insurgency."

The most reassuring fact, a week after the London bombings, is that the terrorists clearly failed to achieve their goals. The West is not terrorized, and Western governments are more united now than before. The West isn't going to lose. But what will "winning" look like?