Consider pairs of meaning such as "undo" vs. "not do", "be uninterested" vs. "be disinterested", "untie" vs. "not tie", etc – and other negations along other distinctions as well.
Now, we could go for lexically determined negation - i.e. "not do" is more likely than "undo", but "untie" is more likely than "not tie". Therefore "not do" and "untie" both get the same, "simple" negation marker. "Not tie" and "undo" get an inverse negation marker instead, which is slighly more complicated (i.e. maybe requires an auxiliary or something).
We could also have a third and maybe a fourth negation marker that go for negating even more unusual parts.
Another venue where inverse negation could happen is in contexts like this:
I don't like all flavours of icecream(, but I do like some) vs.
I dislike all flavours of icecream
This gets into scope and such, and how the negation interacts with subjects and objects could also be interesting for a language with this property. Maybe the "regular" negation has different syntactical properties from the inverse negation, giving the same semantic sense of negation different syntactical properties depending on the verb, forcing distinct constructions into existence in order to express the same ideas with regards to scope and all for verbs of different classes.
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