Specifically Lennie and Candy's dog.
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Specifically Lennie is not "fit" because he is mentallly challenged and the dog is old. When Carlson shoots Candy's dog, he does it quickly and painlessly, but the dog would not have survived much longer because he was not fit. Still, Candy wishes he had put the dog out of its misery because he loved the dog and would have seen the killing as a kind of act of mercy. Lennie cannot survive in the world without George; more importantly, he cannot survive in a world where he cannot control his strength and kills people and animals with that strength. When George decides to shoot him, he does it out of a kind of love for Lennie, knowing that Lennie is not fit to survive in the world in which they live and that the other ranch hands will, in fact, kill him and probably kill him brutally.