Joshu's Dog

A monk once asked Joshu, "Does this dog have Buddha-nature?"
Immediately Joshu shouted "Mu!"

Background on this Koan:
Aside from "What is the sound of one hand clapping?", Joshu's dog might be one of the most famous koans. It is also the koan that I worked on in the beginning of my path. I also love dogs. I always say "Dogs are the best people". So I chose to put "Joshu's dog" at the top of this koan list. As I understand it, some monk approached Joshu. I believe Joshu is the name of his temple's location. Joshu's real name was something else. Zen masters tend to live in areas of their choosing and take the name of that area. Monks would test other monks on their level of understanding by challenging them in some way. I believe this is what was happening here. A dog walked by and the monk asked Joshu "Has this dog Buddha-Nature?". Joshu shouted "Mu!" before the first monk could finish his sentence.

Mu, as I understand it, is a Chinese word meaning "No" or "Nothing" (possibly "Void"/"Empty"?). And in Buddhism, it is accepted that we are all one. Everything does have innate Buddha nature whether we are awake to the fact or not. This includes the dog that walked by the two monks. So the puzzle is if the dog has Buddha-nature and both know this fact, why did Joshu shout "No"?

What some people think:
Some think this is about dualistic thought. If you consider "Have" and "Have not" then you missed the point. The dog is. That's it. Dog, d. is. (no need for words).

My thoughts on this koan:
This was very clear to me, even though my interpretation is not 'correct' or 'popular'. Mine was a matter of exact language or symantics.
Does the dog HAVE Buddha-nature? No, of course not. Because a dog IS Buddha-nature. Very simply.
My take on this is the simple fact of "HAVE" indicates the dog is separate from everything else. "IS" indicates the dog is the same as everything, unified.
Now this answer satisfies me, but others might have a hard time accepting my answer. I think it is because I'm not familar with Chinese, their ancient culture, verbal nuances in speech, etc. So modern day English with "HAVE/HAS" and "IS" works for me in the present moment, but I'm not sure if it did then.

My answer to this koan does jive with what some people think, but how I arrived at the conclusion was not jiving with theirs. For me, I was focused on "Have"/"Have not" and "Is".