How do we prove that animals are not self-aware?There is no way to prove that a particular species is not self-aware. The classic mirror self-recognition test (also known as the "mark test") can indicate that an animal is, indeed, self-aware. It cannot, however, indicate that an animal lacks self-awareness. It is only applicable to visually oriented animals who don't mind conspicuously examining various parts of themselves in front of a mirror, in the presence of observers.
Even with such limitations, it has already demonstrated self-awareness in birds, mammals, and insects. (magpies, orcas, dolphins, elephants, rhesus macaques, gorillas, orangutans, chimps, bonobos, humans, and ants, to be more specific.) Page on journalofscience.net
Here's a beautiful quote:
"Self-awareness is like gravity," Johns Hopkins's Roma says. "We can't touch it directly, so if we want to measure it, scientists must develop valid techniques to directly observe its effects. Currently, mirror mark tests are the best-known and most accepted method, but the absence of an effect does not necessarily mean the absence of the thing we're trying to measure. Ultimately, evidence from multiple techniques should converge on the truth, whatever it may be. Such is the beauty of how scientific advances turn controversy into common knowledge."
Kids (and Animals) Who Fail Classic Mirror Tests May Still Have Sense of Self
(I've given some more explanation here: Susan Cho's answer to Is there a test that can prove that animals are self-aware?)