The logical operators have been defined over all implication structures. It is possible to develop a concept that is the generalization of the notion of a truth-value assignment that is also applicable in all structures, despite the fact that most of them do not have truth-bearers as members. It is a notion that reduces to the familiar truth-value assignments on those structures that are associated with the usual logical systems, and it yields a general framework within which one can study the extensionality (or nonextensionality) of the logical operators on arbitrary structures.

Extensionality and bisection implications

Suppose that S is a nonempty set, and T = 〈K, L〉 is a bisection on it. Form the bisection implication relation “⇒T” that is associated with T, and let IT = 〈S, ⇒T〉. Since “⇒T” is an implication relation, we can study the behavior of the logical operators with respect to it.

Consider the structure IT. Since CT(A, B) ⇒TA and CT(A, B) ⇒TB, and is the weakest to do so, it is possible to compute the distribution patterns for CT(A, B) in the sets K and L, given the distributions of A and B in K and L. Conjunctions. Suppose that A is in K. Then, since CT(A, B) ⇒TA, CT(A, B) cannot be in L.