I don't think dogs or cats "grieve" in the sense that they can articulate "hey someone I "know" is gone and I'm sad". However, they do have senses that can't be explained so we tend to label them with our words-hence "grief". However, I believe they react to stuff-such as new surroundings, threats, hunger, pain, illness and we can never get inside their brain so the only way we can explain it is by using our words. If I say to my dog-"lets go to the park" she will become so excited she pees just waiting for me to leave the house. Her body sense something-in our words "good" -like eating or running or whatever-and it causes a physical reaction that is easily seen.
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