Thus far we have suggested that our account of the logical operators is, in broad outline, a two-component theory. One component is a theory of implication structures consisting of a number of conditions that, taken together, characterize implication relations. Implication structures, which the first component studies, are sets together with implication relations defined over them. The second component consists in descriptions of the various logical operators with the aid of implication relations. The operators are taken to be functions that to any given implication structure assign members (or sets of members) of that structure.

There are two other programs that may be familiar to the reader, that of G. Gentzen and P. Hertz, on the one hand, and that of N. Belnap, Jr., on the other. Each of them, however, is different in scope and aim from the program described here.

The Gentzen–Hertz program

The allusion to a functional account of the logical operators, to their roles with respect to implication relations, will strike a familiar note for those who have studied the logical insights of Gentzen and Hertz. It was Gentzen (1933) who developed and extended the work of his teacher Hertz (1922, 1923, 1929) and adumbrated the view that the logical connectives were to be understood by their roles in inference (Gentzen, 1934). The Gentzen–Hertz insight, as it developed, isolated those aspects of inference against which the “meanings” of various logical connectives were to be explained or understood: There was a division of labor.