Non-X is neither X nor Not-X

Non-X would normally be considered logically separate than X in common parlance, e.g. take X to be "sense" or "profit". However it is not quite Not-X either and can be a meaningful construction in the following way that is different than Not-X:

Say we want to refer to an activity or thing not by an affirmation, but by a negation. This is generally not necessary but sometimes can be useful. For example when I do not desire to give the activity or thing an affirmative name, yet I still want to refer to it somehow for my own mental organization and categorization. By using the non-X construction, I can indicate to myself that it is not X, but it lies in the vicinity of the X concept.

For example, when I want to refer to my activities that require hard work and thought, but they are not quite part of my professional sphere or even part of my "serious" work. Then I refer to it is non-work these days. This tells me that it is not standard "professional" or "serious" work and is more in the "fun" or "recreation" or "hobby" category. However the actual amount of serious work required to make the activity work for me as a meaningful activity (excuse the puns and self-referentiality) also pushes me to not categorize it as not-work since it requires a good amount of work! Thus non-work. The construction invokes the idea of serious work without having to be too serious about it nicely for me :) An example of not-work is for example banter over dinner and drinks with friends. Such relaxed conversations can of course fluidly shift categories like anything in lived experiences, go from not-work to non-work to work, etc. as many of us would recognize. Other cases where I have found the Non-X construction useful are "non-seeking", "non-professional" which is basically same as "non-work", and "non-theorems" for the time being.

Also the starting point of the logical separateness of X and Not-X is an axiom only in classical Boolean logic which is not an absolute concept or framework to understand our lived experiences. For example, are dreaming, coma, etc. states of living beings active states or inactive states? Say taking a break from work, is it not working or is it a recovery part during work, e.g. when a professional footballer is not sprinting and regaining breath during a match or even a break from intense training for general recovery? Of course the axiom is clearly useful in standard arithmetic and mathematics more generally, and in fact needed for progess.

None of the ideas above are not new. Zen koans, the use of the dreaming state to illustrate many things in advaita vedanta, etc. show how binary logic is not the only way to appreciate our lived experiences. Even from the view of classical logic say coming from western philosophical traditions, it has been realized that classical logic and natural language are not always in tandem. More currently, the subject of logic itself is moving beyond classical X and Not-X notions of truth values of statements prompted by the state of affairs for which quantum physics is absolutely necessary to make progress.