Final answer:
Lincoln believed the living could honor the soldiers who died at Gettysburg by remaining dedicated to the cause of freedom and ensuring a government 'of the people, by the people, for the people' endures.
Step-by-step explanation:
According to Abraham Lincoln, the living can ensure that the soldiers who sacrificed their lives at Gettysburg "shall not have died in vain" by dedicating themselves to the great task remaining before them. This task, as outlined in Lincoln's famous Gettysburg Address, is to take increased devotion to the cause for which these soldiers gave the last full measure of devotion. The President emphasized the principles of liberty and equality, and underscored the importance of ensuring a "new birth of freedom," and that government of the people, by the people, for the people, shall not perish from the earth.
Lincoln delivered this speech at the dedication of the Soldiers' National Cemetery near the site of the Battle of Gettysburg. The speech commemorated those who had made the ultimate sacrifice and set forth a vision for the future of the United States—not only as a reunified nation after the Civil War but as one that had come to terms with the central issues of emancipation and freedom, thus truly fulfilling the promise of the American Revolution that all men are created equal.