I've studied wild elephants for almost ten years now, and I've been asked many times about how we study these kinds of abilities, especially how elephants reportedly grieve.
It's a hard kind of question to answer through science: even for our same species where we share the benefit of language, it's difficult to say you feel exactly the same as I do when I express the combination of sadness, denial and loss that make up my (personal) sense of grief.
Some scientists are exploring elephant intelligence using captive elephant subjects. To do this, they are scaling up (on a grand scale) the kinds of tests and puzzles developed to explore the cognitive abilities of non-human primates. These include huge mirrors to explore self-recognition and tasks where individuals have to cooperate to solve a food task.
So far, Asian elephants (the species most extensively tested) are doing pretty well in convincing researchers that they're smart.
But what does that have to do with elephant grief?
Well, it's obviously hard to study these kinds of phenomena systematically. But despite that difficulty, it's important wherever possible to use evidence that comes from wild animals, because studies in captivity inevitably involve individuals in abnormal (i.e. man-made) settings and who often have traumatic histories.