While there was some speculation that Joe Biden would announce his presidential run during Thursday's appearance on The Late Show with Stephen Colbert, instead the vice president took the opportunity to open about the loss of his son, Beau, who died of brain cancer in May.

"My dad had an expression," Biden said. "He used to say, 'You know you're a success as a parent when you turn and look at your child and realize they turned out better than you.' I was a hell of a success. My son was better than me. And he was better than me in almost every way."

When Stephen Colbert asked Biden if he had anything to announce, Biden joked that he thinks Colbert "should run for president again, and I'll be your vice president." But in all seriousness, Biden explained that he would only run if you could promise voters "you have my whole heart, my whole soul, my energy and my passion."

"I'd be lying if I said that I knew I was there," Biden said. "I'm being completely honest. Nobody has a right in my view to seek that office unless they are willing to give it 110 percent of who they are."

Despite everything Biden is going through, Colbert urged him to run for president. "I know that's an emotional decision you have to make, but it's going to be emotional for a lot of people if you don't run," the host said. "I think that your experience and your example of suffering and service is something that would be sorely missed in the race."