It will prove useful in this study to split the operators into two groups: the hypothetical, negation, conjunction, and disjunction in one group; quantification and identity in the other. In studying the features of these operators it will be convenient as well to begin with a simple characterization of each of them. Later (Chapter 15) we shall describe the official, parameterized forms for them. This division into the simple and the parameterized forms segregates the features of the logical operators in a way that makes for a more perspicuous picture, but separates the features in a way that has its own theoretical interest. We begin with the simple, unparameterized story.

The simple characterization

If I = 〈S, ⇒〉 is an implication structure, then the hypothetical operator H (on I) is the function H⇒ that assigns to each pair 〈A, B〉 of members of S a special subset H(A, B) of S. The special character of the subset is given by a condition that characterizes the function H– just as the special characters of the other logical operators are brought out by certain conditions that characterize them in turn. In fact, as we shall see, all the members (if there are any) of H(A, B) are equivalent to each other with respect to the implication relation “⇒” of S, so that often we shall simply refer to the hypothetical H(A, B) as if it were a member of S rather than a subset of S.