Pets aren't considered juridical persons, rather they're viewed more as property under the law. To the extent a veterinarian caused dame you can sue to recoup costs incurred as a result. Consult a local attorney, but I'm not sure you'll be able to pursue damages for pet.
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Veterinarians are subject to malpractice actions just as other licensed professionals are. There is the same difficulty in proving a vet caused the harm as there is in any other medmal case.
Further, damages are limited to the value of the animal and the expense of treatment necessitated by the malpractice. There is no pain and suffering or loss of consortium. This limit on damages generally makes such cases economically unviable.
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Humankind has not yet evolved to recognize other species as sentient beings, so they are still chattel, like your television set or bicycle (how else do we justify factory animals?). Anyway, you can sue for the value of the pet and possibly for the wasted medical expenses if they were incurred as a result and exceed the value of the pet.
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