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A little learning is a dangerous thing:
Drink deep, or taste not the Pierian spring:
There shallow draughts intoxicate the brain,
And drinking largely sobers us again.
-Alexander Pope, Essay on Criticism
If you have no knowledge at all, you cannot influence anyone. No one will take you seriously if you know nothing, but if you know even a little, those who are easily swayed or have no knowledge of their own will follow you. Those who know only a little do not realize the full extent of what they can do with that knowledge, or may use it for bad. If someone knows a lot, then they know the "whole story" and know all the angles of a certain subject, not just the good or bad parts.
The reason this quote was included in Fahrenheit 451 is because none of the characters are fully informed about anything. Until Montag burned the lady he didn't know any good things about books, only what society had taught him: books are evil, they must be destroyed. When he met Faber at the house he talked about the purpose of books and the deeper meaning behind them and realised that books were a good thing as well. Mildred and her friends only knew that books were bad, and reacted badly when Montag read "Dover Beach" to them. They did not know that words could have a good purpose. Beatty was the only other person who knew everything about books, but the bad things about books stood out to him more and he hated them. However, he was the only person in the book who was justified to hate books because he had read them before.