The simple characterization

Our aim here, as in the preceding chapter, is to characterize the negation operator as a special kind of function that can act on all implication structures, general and special.

Let I = 〈S, ⇒〉 be an implication structure. For any A in S, we shall say that N(A) is a negation of A if and only if it satisfies the following two conditions:

1. N1. A, N(A) ⇒ B for all B in S, and

2. N2. N(A) is the weakest member of the structure to satisfy the first condition. That is, if T is any member of S such that, A, T ⇒ B for all B in S, then T = ⇒ N(A).

As with the hypothetical, the negation operator on the structure is supposed to sort out, for any A, those elements in the structure that are the negations of A. Strictly speaking, then, the negation operator assigns to each A of S a set of members of S that will satisfy the preceding conditions. That set may be empty, for there is, as we shall see later, no guarantee that negations always exist. However, if there are several members, then they will be equivalent under the implication relation of the structure. Thus, as long as there is only one implication relation that is being studied on a set S, there will not be any confusion if we treat N(A) as if it were an element of S.