Granger compares society to the phoenix, the legendary bird that destroys itself in a fire only to be reborn from the ashes. His point is that human society keeps becoming wilfully self-destructive, and even afterwards it continues along the same pathways:
"...every time he burnt himself up he sprang out of the ashes, he got himself born all over again. And it looks like we're doing the same thing, over and over, but we've got one damn thing the Phoenix never had. We know the damn silly thing we just did."
(Bradbury, Fahrenheit 451, Google Books)
However, Granger believes that since humans are capable of learning from their mistakes, society is destined to become stronger and more intelligent in the future, instead of regressing as the current generation has. He believes that it his role, and that of men like him, to remember and teach the mistakes of the past, so that future generations won't forget them. Without the framework of prior mistakes, society will continue to destroy itself; with books to remind people of the past, society should be able to become stronger than before.
At the end of the book, after the city has been destroyed by a detonation, Granger says that the "city looks like a heap of baking-powder. It's gone." He then compares the society to a phoenix, who "built a pyre and burned himself up . . . But every time he burnt himself up he sprang out of the ashes, he got himself born all over again." Granger believes society is like the phoenix because it contains the seeds of its own destruction, just as the bird does. The society just destroyed itself by sowing the seeds of its own destruction.
However, Granger believes that the society is different from the phoenix because it can get itself out of the cycle of death and rebirth into which it has put itself. Granger says, "We know the damn silly thing we just did," meaning those who remain are aware of the mistakes of the past. He believes that by having access to books, the people who remain after the blast can rebuild society along better lines. He also believes that if people remember the experiences of who people who came before them and are reflective about the mistakes of the past, the society has a chance to improve itself.