On the stump, Mr. Biden says he is running to achieve three main things, none of them distinguishable from what he might have said when he ran in 1988 and 2008: 1) to lead the “battle for the soul of America,” 2) to restore the middle class and 3) to unite the country. He also says that he was inspired to run for president in 2017 after seeing how Mr. Trump responded to violent clashes involving white nationalists in Charlottesville, Va.

When you ask people close to Mr. Biden why he’s running, they tend to dispense with any attempt at a higher purpose beyond dislodging President Trump. “It’s that old Vince Lombardi line: Winning isn’t everything, it’s the only thing,” said former Senator Christopher Dodd, Democrat of Connecticut. “And that is the prevailing view of an awful lot of people within the Democratic Party.”

Friends of Mr. Biden suggest that he decided to run for president in large part because he could not have lived with himself if he did not. “Joe has this thing called the ‘Look in the Mirror’ test,” said Ted Kaufman, Mr. Biden’s longtime chief of staff who served out Mr. Biden’s term in the Senate after he became vice president in 2009. “Clearly he has the best chance of beating Trump. And if he did not run this time and Trump won, then what would he think of himself when he looked in the mirror?”