You are trying to find an answer through logical means. This means you need to find a framework with a clearly defined set of axioms and clear reduction rules that bring you to the correct answer. The world does not work like this. All concepts that seem well-defined break when you look at them too close or too far.

Does the dog has the Buddha nature? The answer to that is neither 'yes' or 'no', because neither will lead you where you want.