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In the negative form, their incompatibility is denied.
Thus the negative disjunctive denies the alleged incompatibility between the alternatives.
Suppose someone claims, “A person is either a poet or an orator; they cannot be both.” One may deny that exclusion: a person can be both a poet and an orator.
What is denied is the alleged incompatibility, not that every individual actually has both attributes. Denying an exclusion does not by itself prove the two predicates of any particular subject.