In the novel 'Of Mice and Men' by John Steinbeck, the author uses the dog to illustrate some of the main fears in America in the time of the great depression. These fears of joblessness and poverty were much worse for people (like Candy and some of the others) who were approaching old age with no retirement money and no-one close to them who cared that they lived out their old age in dignity and comfort. Everyone hates the fear of being a burden, even - or especially - on their loved ones, so most workers would rather keep their health and strength so they can be, and feel that they are, useful. Candy's dog is probably incontinent and getting close to the time when it can't do anything for itself - there is a horrible message in Carlson's solution to being a 'senior.'
Carlson gives three of main reasons for wanting Candy's dog shot. You can argue that his reasons are partly good and partly just selfish and mean.
First of all, he says that the dog stinks. He hates to smell it. The smell bothers him and so he wants the dog dead.
Second, he says that the dog is no longer worth anything. He says that it really cannot do anything. Because it is not good for anything, it might as well be dead.
Finally, he argues that the dog is not really enjoying life anyway so it is no harm to kill it.