I think that the primary function of Candy's dog is to bring out how betrayal between people who love one another can be evident. It is undeniable that Candy loves his dog. Yet, when he does not or cannot defend the dog from the public pressure of killing it, Candy feels that he has subverted an ethical understanding that is intrinsic to the relationship he shares with the dog. This is brought out in his feeling bad about how he needed to have been the one to kill the dog and not surrendering this to someone else. The "silence" that Steinbeck uses to punctuate the moments between when the dog was taken outside and when he was shot was something that helped to bring out the symbolic force of how Candy feels both helpless and responsible both for his predicament and the condition of someone that he loves. The illustration rendered by this situation is that there are times when human beings can do things to those that they love that might not represent love. Human beings have to fully reconcile that betrayal might be part of the experience of love. Steinbeck wishes to bring out complexity and emotional nuances from the situation between candy and his dog. This same multi- layered intricate approach to relationships is something that will be brought out between Lennie and George at the novella's end.
Candy's dog symbolises the tragic nature of those who have outlived their usefulness, or who have become more of a problem than a blessing. Candy loves his dog too much to kill it, but not enough to take responsibility for ending it's suffering. This paradox leads George to the conclusion that he cannot allow anyone else to kill Lennie. The best thing he can do for Lennie, who he genuinely loves, is to allow him to die as peacefully and happily as possible, which is why he talks to him about achieving their dream, whilst surreptitiously raising the gun to the back of his head. George knows that if he spares Lennie, he will be killed in a cruel and terrifying way by Curly's posse, or else incarcerated in a prison for the criminally insane. Lennie is killed in the same manner and with the same gun as Candy's dog, to emphasise the dog's symbolic value.