Parametric conditions for the operators

The preceding chapters describe the various logical operators, each without reference to any of the others. The descriptions are uniform in format: the weakest to satisfy a certain condition that is characteristic for the operator under discussion. We now wish to supplement that account with an additional requirement, a “parametric” requirement. The description of each operator still will be independent of the others, and the uniform format will be preserved. However, without the additional condition, the system would be weaker than the system studied thus far; without parameterization, some of the logical operators would fail to have some of the familiar features we normally expect them to have. The need for the additional condition needs a bit of explanation.

When we described the conjunction operator, we required that C(A, B) imply A as well as B and that it be the weakest member of the structure to imply A as well as B. It certainly seems as if that is the entire story to be told about conjunctions. Yet, surprisingly enough, that description will not guarantee that in all implication structures, A together with B implies their conjunction.

Let I = 〈S, ⇒〉 be an implication structure.