Yueh-Shan Hung-tao was sitting [in zazen], a monk asked him, “When you are sitting immovably, about what do you think?” The master replied, “I think about not thinking.” The monk responded, “How does one think about not-thinking?” The master replied, “Without thinking.”
This has been translated in various ways to show the distinction between thinking, not-thinking and without thinking (sometimes translated as nonthinking).
I want to highlight the correlation between self-inquiry and thinking-inquiry.
In thinking: the object content is whatever shows up. Thinking of what to make for dinner, how to calm down a friend, and so on.
In not-thinking: the object content is thinking itself. This is the attempt to stop thinking, which is a common aim of some meditators, under the idea that not-thinking is a correct and useful path to awakening.
In without thinking: there is no position taken with respect to everyday content, or with thinking itself. It is a meta stance to thinking and not-thinking (which are not a polarity, for operating on different levels, as mentioned above.)
Self behaviour is akin to Thinking. It operates at the level of everyday, conditioned content.
Not Self/No Self is akin to Not-Thinking. It’s focus is not on everyday content, but on the negation of self as a persistent separate entity. Hence its operating on a different level.
Without Self is akin to Without Thinking: It holds no position with regard to self content, or a negation of self as no-self. Life is experienced as it is, which includes the positive activities of self, and the negation of self called no-self. It is outside the game. All questions of self/no-self are moot.
Championing No Self is to remain inside the paradigm of self/no-self.