NATO’s decision last weekend to fight the Afghan war for at least four more years is a serious mistake. It’s a war that cannot be won, on behalf of a government that can’t be trusted to solve a problem that no longer exists — all of it a spectacular waste of money, material and men.

To this day, President Obama claims the war’s aim is to deprive al-Qaeda a haven in Afghanistan. “As we approach our 10th year of combat, we must never lose sight of what’s at stake,” he said.

In truth, not much. As most everyone already knows, Afghanistan is now home to few if any al-Qaeda operatives. All of them long ago fled to Pakistan, and in the nine years since the war began many of them have set up new terror centers in Yemen, Somalia, Iraq, Indonesia and North Africa, among other spots.

The United States and NATO seem unable to attack them directly in any of those other places. But if Western forces pulled out of Afghanistan and al-Qaeda returned, attacking them there would be comparatively easy. What could President Hamid Karzai do?

Now, with all those extremists sitting just over the border in Pakistan, “absent a much better situation in Pakistan, we cannot prevail in Afghanistan,” said Richard Armitage, who served as deputy secretary of state and assistant secretary of defense.

Here’s why: Military leaders are now claiming significant success riding Kandahar of Taliban extremists. While soldiers killed numerous militants, many hundreds of others simply walked away, back over the border to Pakistan — waiting for the Americans to leave. So, what’s the plan for that day?

Afghan soldiers are supposed to begin providing security as NATO forces leave. That’s almost like having no security at all.

This month, the NATO Training Mission in Afghanistan published its first-year assessment of progress. It’s a bleak document. The trainers chartered a new force whose mission is to fill the security vacuum town-by-town nationwide as Western forces pull out. But the instructors are finding they are training ghosts. Once trained, most of these soldiers simply disappear. They walk off of in the night.

During 2009, the report said, 85.2 percent of the soldiers deserted. The rate is improving slightly this year. On average each month, 6.2 percent of the troops are walking away — overall, almost three quarters of them. With considerable understatement, the report concludes: the “level of attrition remains unacceptable and unsustainable.” A BBC reporter visiting a training camp this month found that when the soldiers set out on missions, “they have to lock the doors on the buses” to keep the trainees from jumping off and running away.

But even the few soldiers who remain are problematic. At least 89 percent of them are illiterate; they can’t read either words or numbers. All manner of specialized training requires reading textbooks or instructions. NATO also cannot account for weapons and equipment if soldiers can’t read serial numbers or fill out forms. Equipment theft is rampant.

Literacy “helps prevent corruption,” the report said — a significant problem in Afghanistan. Transparency International’s new corruption-perception index, published last month, shows Afghanistan holding the second-to-last spot, out of 178 nations investigated.

That report came out at about the time Karzai admitted that Iran gives him big bags of cash, $500,000 or so every month or two. He also pardoned the one senior government official who was caught red-handed taking a bribe.

The Afghan military, not surprisingly, is a reflection of its nation. When NATO commanders send Afghan troops on a mission, they have to confiscate their cell phones, to prevent solders who took bribes from the Taliban from calling to tip them off.

Afghanistan is a tribal society, and NATO’s trainers say they are struggling to convince their recruits “to move beyond personal and local loyalties to national loyalties.”

The state also produces 90 percent of the world’s opium, and many thousands of the soldiers are addicted to heroin. It’s also a hierarchical society, so NATO can’t find anyone to serve as officers. Right now it needs 15,000 of them. Many officers already serving are “corrupt, untrained and inefficient.”

The war cannot be won without an effective Afghan army. But to turn the army around means turning this primitive nation around. That can’t be done in four years — or 40.

The whole effort is fraught with enduring, insoluble dilemmas. As Armitage put it, “we can’t keep doing the same thing over and over again and expect a different outcome.”

Joel Brinkley, a professor of journalism at Stanford University, is a Pulitzer Prize-winning former foreign correspondent for the New York Times.