The severed arm of an octopus can crawl on its own and grab food. Elephants can distinguish between people based on age, language and gender. Crows are adept at using tools.

In “Are We Smart Enough to Know How Smart Animals Are?” -- a new book by Frans de Waal -- the primatologist argues that the mental powers of animals are far more complex than often generally assumed. Many experts have been asking the wrong questions about animal intelligence, he argues, blaming the creature in the cage for poor test results when it’s the...