The Gateless Barrier, Case 1, Joshu’s Dog
A monk asked Joshu “Does a dog have Buddha Nature?”
Joshu said “Mu” [no]
Case 18 in The Book of Serenity has a longer version of this story. In that version, Joshu is asked the same question by two monks. To the first he answers ‘Yes’. To the second, he answers ‘No’.
You can see in this a characteristic way of talking about Emptiness, similar to the apparent negations that appear in the Heart Sutra.
In early buddhism in India, Buddha Nature, the potential to become enlightened, is restricted to human and similar beings. Dogs don’t have it. When buddhism develops in China, there’s a change. All living things have Buddha Nature, and, eventually, all things have Buddha Nature, which is taken to its logical end point in Dogen’s reworking of ‘All things have Buddha Nature’ to ‘All existence is Buddha Nature’. Enlightenment ceases to be a personal quality or possibility, and becomes universal. Every window springs open.
On the one side people, insects, birds and grasses. On the other, the Universal Body of the Buddha. On the one side your karmic consciousness, on the other side Buddha. On the one side form, on the other side emptiness. On the one side the complete exertion of a single thing, on the other the complete dynamic functioning of all things.
And although these two sides are the same, they don’t encounter each other. When one side is illuminated, the other is dark.
So, the dog doesn’t have Buddha Nature.