The dual of the negation operator on a structure is just the negation operator on the dual of that structure. This characterization of the dual of negation yields some simple results, as we noted in Chapter 12: If N̂ is the dual of N, then N̂N̂(A) ⇒ A for all A, but the converse does not hold in all structures; if A ⇒ B, then N̂(B) ⇒ N̂(A) in all structures, but not conversely; even if N(A) exists for some A, N̂(A) need not exist (if the structure is not classical); N̂ is a logical operator. These simple consequences are part of the story about dual negation, but only a part. There is an interesting story about this little-studied operator that deserves telling.

Theorem 17.1.Let I = 〈S, ⇒〉) be an implication structure for which disjunctions of its members always exist. Then D(A, N̂(A)) [that is, A ∨ N̂(A)] is a thesis of I for all A in S.

Proof. For every A in S, A, N̂(A) ⇒̂ B for all B in S, since N̂ is the negation operator on the dual of I. Since disjunctions always exist in the structure, it follows by Theorem 14.8 that B ⇒ D(A, N̂(A)) for all B in S. Consequently, D(A, N̂(A)) is a thesis of I for all A in S.

Since N and N̂ are logical operators on any structure, it is possible to compare N and N̂ with respect to their implication strengths.