I am much more interested in people than in animals, but to Anna, my daughter, animals are the centre of her life. When she was seven years old she would canter around like a centaur and, although she enjoyed eating meat, she gave that up to respect animal rights. She now runs a horse training and therapy centre in Dorset, and three recent events at the centre helped me to see animals in a very different light.

I’ll start with the two pigs. They lived in a small house in a walled garden, rootling up the massive weeds that smthered the land. Peppa was ill for a few days and she stayed in the house, whereas her sister, George, as usual hurried towards Anna whenever she arrived with food. Yet George just ate her share, leaving the other half uneaten for Peppa. She seemed to show not only kindness and restraint, but also an idea of dividing the food into two fair portions.

My next example is Kahn, a rather irritable, slightly lame horse. After trying many remedies for Kahn’s problem, Anna decided to ask “horse whisperer” Gaynor Davenport to visit the centre.

Gaynor began by warning Anna not to tell her anything about the horse. She approached him gently, and after reassuring him she stood with her head close to his, as if his thinking and her thinking were almost touching. After a long silence, she began to repeat what she had learnt from Kahn. Kahn had told her that he had a heart murmur and that he was eleven years old. Anna knew about the heart murmur but thought he was nine. When she checked his records she found that he was correct.

Kahn also told Gaynor that a black and white horse in the stable opposite his had been taken away the previous day, that he would be killed and that Kahn was very afraid that he would be the next horse to be sent away to his death. Anna was able to reassure Kahn that the black and white horse had gone to a new life in a happy home, and that neither he nor Kahn would ever be sent off to be killed.

Gaynor discovered from Kahn where the pain in his leg muscle was, lower down than Anna had thought. With the new insights and Gaynor’s medicine Kahn’s lameness was healed and, although he did not wholly change from his rather irritable ways, he became calmer and happier.

Gaynor’s work made us wonder how much horses overhear and actually understand human speech and seemingly abstract concepts such as numbers, colours, anatomy, time and death. There is the question about how they imagine future events and human intentions, and how much their emotions are reactions to past memories and future hopes and fears as well as to present events. Gaynor has been particularly helpful with listening to horses who have been ill-treated in the past about their memories, injuries and fears, in order to inform the plans for their healing care at the centre.

My third example is Flash, a German Shepherd. Throughout his thirteen years, a long life for a large dog, Flash was Anna’s closest and most loyal companion. He became very ill, blind with cataracts, in pain, and scarcely able to move. When the vet visited, Anna very sadly arranged for him to call again two days later to give Flash a fatal injection.

She talked with her friends about where to bury Flash, wanting him to be near the house, and chose a place beneath two trees in the walled garden. The pigs had only recently moved away to another weedy plot, and the dogs had never gone into the walled garden while they were there.

When the vet returned two days later, Flash could not be found. Anna and her friends searched around the fields and barns, wondering how he could possibly have walked so far. At last they looked in the walled garden and found him lying under the two trees. He raised his head and laid it on Anna’s lap as if to say, more clearly than words could, “I trust you, Anna. Yes, this is the right place and the right time.”

Like Kahn, Flash seemed to have understood complex, partly abstract ideas, including possibly his own death. Perhaps he wanted to ease Anna’s distress and make his demise a little less agonising for her. And he saved her from the hard task of carrying his body to the garden.

These may all simply be slightly odd coincidences, though Kahn’s information to Gaynor could only have come from him. She lived many miles away and had never visited the centre before. Sceptics may say that they need very many similar examples before they can be convinced. But that is not really the point. It is not to claim that all larger mammals understand human speech, or even that a certain proportion understands. It is simply to suggest that some dogs and horses seem to understand more of what humans talk about, perhaps especially humans who are close to them, than is usually assumed to be the case.

Primatologist Frans de Waal and others have long argued that morality is not a social veneer, wholly learned and acquired to cover our innate selfishness. It is deeply innate within evolved human nature, and is shared with other animals, such as the bonobo who tried to help a wounded bird to fly (as described in de Waal’s book Our Inner Ape). The horses at the riding centre showed friendly support to one another as well as rivalry. Newly arrived horses who were very distressed and volatile calmed down within days, suggesting that their new herd, as well as the humans, gave them much reassurance and support and seemed to share with them moral feelings of trust and empathy.

When psychologists observed the reactions of children aged about one year to relatives who were asked to feign sadness, pain and choking distress, not only were the children very concerned, but so too were the household pets, who tried to give comfort. Carolyn Zahn-Waxler and colleagues gave many examples of the youngest children learning from their pet dogs and cats how to relate with them carefully and tenderly.

The more these kinds of personal moral relationship are understood and encouraged, perhaps the more hope there is that at a global level humans and other species will live in greater respectful interdependence, to their mutual benefit.

Priscilla Alderson is Professor of Childhood Studies at University College London.