Does my dog know he is a dog?

No.

Sorry to disappoint all the answers that say yes, then proceed to comically describe how their dog views the world.

An animal knowing what it is is called self-awareness.

Humans are self-aware because we know we are humans.

One test of this is to put a mirror in front of the animal and watch how they respond. A human will see the image in the mirror and know that the image is them. Dolphins have been known to use a mirror to get something off their back, implying that they are self-aware. Basically, the mirror self-recognition test is a positive test for self-awareness.

Dogs do not display any kind of behavior that would suggest they are self-aware; in fact, they show the complete opposite, thinking the dog in the mirror is a different dog, and they’ll start barking and growling at it.

Some even go as far to say that since the dog lacks the ability to recognize itself in the mirror, it lacks a level of intelligence required in order for it to think of itself as an individual and unique being. I think that’s a bit excessive.

Source: https://www.quora.com/Does-my-dog-know-he-is-a-dog

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