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It all depends on how the LW feels towards the cat. If they don't have any particular emotional attachment they should definitely put the cat down.
Why are animals that are traditionally used as pets given more rights than a cow or a pig? Why the fuck does a cat have a right to life that a chicken doesn't?
If you eat meat but think this cat shouldn't be killed if the LW wants to than you have a preconceived notion that is causing negative effects (overcrowding shelters; filling the streets with strays).
I was surprised Cary completely removed killing the cat from the options, and didn't even ask how attached they were to it. I completely disagree that killing the cat would amount to any kind of "failure". Who in their right mind would hire a veterinary behaviorist for an animal they didn't like?
All vegans please don't respond to this. You are ideologically sound in my mind.