Using the basic assumption that "or" means "choose one", and "and" means "then"/"follows", I was trying to reconstruct DeMorgan's Laws without recourse to set theory. You can see how I tried to do this here, with plenty of random constructs. I am afraid that Math depends a great deal on some kind of construct but the ones I have been playing with leave a bad taste of ad-hoc-itude in my mouth.
To analyze "Not A or B" as a narrative we consider that the "or" means choose one of the narratives 'A' or 'B'. Then negate it. This leads to two narratives: A* or B* and the unresolved invocation to choose one. Ad hoc, we say: you create a single narrative from two by appending one to the other, as in A*,B* or "Not A and Not B".
The disturbing "Math" assumption is that you create a single narrative from a choice by appending one outcome to the other. This implies that "A or B" also resolves to "A and B" - pretty distasteful, needing a further exception.
As for the other De Morgan Law: To analyze "Not A and B" in terms of the single narrative (A, B)* we play a law-of-the-excluded-middle game with four possible narratives involving A and B: (A,B), (A*,B), (A, B*) and (A*,B*). At the level of these four artificial narratives (say grouped inside a temp collection), ones that are "Not (A,B)" requires a choice from the other three. By coincidence this is the same outcomes as "A* or B*".
The subtle shifting between use and mention leaves me wanting to abandon this subject.
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