Disjunction: the simple characterization

We saw in preceding chapters how the duals of various logical operators can be defined. In the case of the hypothetical, H, its dual, Ĥ, has rarely been studied, if at all. In the case of negation, as we pointed out, the dual N̂ has an important role to play in the study of generalizations of classical results within nonclassical implication structures. Likewise with conjunctions: The dual of conjunction, Ĉ, is just conjunction on the dual structure Î and is, as we noted, a logical operator on I as well. But contemporary logicians think of the dual of conjunction not merely as the dual of an operator, but as an important logical operator in its own right. There is a direct characterization of Ĉ as a logical operator on I, since Ĉ is, as we indicated in the preceding chapter, a well-known logical operator on I in its own right, namely, disjunction on I.

Let I = 〈S, ⇒〉 be an implication structure. For any A and B in S, the logical operator of disjunction is a function that satisfies the following two conditions:

1. D1. For any T in S, if A ⇒ T and B ⇒ T, then D(A, B) ⇒ T, and

2. D2. D(A, B) is the weakest member in S to satisfy the first condition. That is, for any U in S, if [for all T in S, if A ⇒ T and B ⇒ T, then U⇒ T], then U ⇒ D(A, B).