To prove it, suddenly lock your wife and your dog in your trunk for 12 hours, then see which one is happy to see you went you open it.
What's the difference between the wife and the dog though? One has the sense to know that you did this to them while the other only feels the pain of being separated from you. We all know that putting Duke down was a mercy, but what might Duke say about the matter? He had fun at the water park today. Might he not also have fun at the water park tomorrow? That could be the very basis of his argument for living a bit longer.
"I know I'm getting sick, but let's take this day by day. I like getting petted and eating hamburgers today, and that probably won't change by tomorrow. Let me have another day and we'll see".
"Sorry Duke, but at this point it isn't about you anymore. As humans, we can see where you're going. We've already moved on in our minds, and we're ready to conclude this business. We're going to create a montage of pictures from your last day so we can feel like we did our part, and then we're going to adopt another dog that afternoon. You're just a bittersweet memory to us already."
"But I'm still here! I enjoy things! I'd love to have another day with you".
"Sorry Duke. We've already mentally steeled ourselves for your death. Letting you live now would just extend that period of mourning indefinitely".
"Fuck your feelings. I want to live a little longer. Why should I die now so that you don't feel bad for as long"?
"Sorry Duke. This is what it means to be a pet. You are here to serve us, despite what we sometimes tell ourselves. You live and die at our whim. We love you for a while, but when the time comes we're always ready to put you down to protect our own emotions. Them's the breaks. Peace out".