Perhaps President Obama had no choice but to say he’d begin a withdrawal from Afghanistan in 18 months. Perhaps a war-weary American public would have rejected an open-ended commitment of troops.

But in trying to mollify his domestic audience, Obama may have lost the audience most crucial to the success of the policy. I mean the Afghan people, and the Pakistani military — without whose cooperation this strategy cannot work.

Setting this deadline may have made sense from a domestic perspective. It will calm some Americans who fear another Vietnam (even though the analogy is faulty). Moreover, as Obama indicated late in his speech, there is a caveat to this date certain: It will be based on “conditions on the ground,” meaning it can be pushed back if the Afghan situation doesn’t improve as fast as the president hopes.

But I fear that the mention of a specific date for starting a pullback — July 2011 — sends the wrong signals to Afghans and Pakistanis. They won’t grasp caveats and nuance. They are more likely to deduce that the Americans have their eyes on the exits, and that the Taliban will reign in 18 months’ time.

Some say a specific deadline was necessary to pressure Afghan President Hamid Karzai to reform his government, whose rampant corruption and ineptitude has fueled Taliban gains.

Yet, in reality, U.S. military and civilian officials were already planning to do an end run around Karzai by working more intensely with provincial and local officials. Secretary of Defense Robert Gates has repeatedly cited Afghanistan’s history as a decentralized state with a weak central government.

Indeed, Gen. Stanley McChrystal’s strategy depends on securing and winning the confidence of ordinary Afghans. If the U.S. troop surge convinces Afghans that the tide has turned against a Taliban victory — and that the Americans won’t leave them hanging — many positive things can happen.

For one, villagers or tribesmen may decide to form their own local defense forces to keep the Taliban out. As Gates indicated in congressional testimony last week, such local forces may prove as important in pushing back the Taliban as the Afghan National Army, which is likely to take years to train and cohere.

For another, many mid- and lower-level Taliban may decide to come in from the cold, encouraged by U.S. and Afghan programs to reintegrate them back into society. Such reintegration is a crucial part of McChrystal’s plan.

Yet the rise of local defense forces and any plans for reintegration depend on Afghans’ perception of the overall situation. Intimidated by the Taliban, many rural Afghans want to ally with the side they believe will emerge as the winner. If villagers think that the U.S. commitment is fleeting, they will hesitate to reject Taliban fighters. And those fighters will think twice about changing sides.

Moreover, another component of U.S. strategy could be undermined by the perception that U.S. troops are on short time. The best long-term hope for ending the Afghan conflict may lie with a political settlement in which some senior Taliban leaders break with al-Qaida, and rejoin the Afghan political process. This is a goal Afghan leaders seek, their public wants, and U.S. officials haven’t ruled out.

There is no chance of such talks, however, while Taliban leaders believe they can win by fighting. If the Taliban think Americans are leaving by a certain date, they will have no incentive to compromise. “With this timeline, it makes it even more unlikely that their leadership will talk,” says Ahmed Rashid, a top expert on the Taliban and author of Descent Into Chaos. “They will think they can hang on.”

So I wish Obama could have hinted at a withdrawal target without being so specific. All the more so, because of the impact the July date may have inside Pakistan.

The administration is asking Pakistan, whose military finally attacked its own Taliban, to crack down on Afghan Taliban leaders hiding in its southern province of Baluchistan. From there, Mullah Mohammed Omar and his men are believed to direct attacks in the Afghan provinces of Helmand and Kandahar, just across the border, where most of the U.S. troop surge is headed.

The Pakistani military is unlikely to crack down on the Afghan Taliban, however, if it believes NATO troops are leaving too quickly. Pakistani brass want a friendly government in Kabul, and they have a deep fear of post-NATO chaos. Obama’s exit date may convince them that their best option is to help the Afghan Taliban, who were once their allies, retake power next door.

So Obama is caught in an incredible bind. He must convince Afghans and Pakistanis that we won’t leave so quickly and carelessly as to abandon them to chaos. Otherwise he won’t get the cooperation necessary to make his strategy work. But he must convince Americans that the troops will be coming home soon.

If he can square that circle, his strategy may still succeed (and I believe he has no option but to try it). But he may come to regret setting that very specific exit date.