The Interpretation of Implication

Abstract

When the student of logic is first initiated into the problem of the relation of implication as expressed by the hypothetical proposition, he enters into a portion of logic which seems to him an impenetrable jungle. Elementary statements of the meaning of implication, as he is acquainted with them, are so meagre that they have failed to give any indication of the problems and paradoxes involved. Adequate discussions, on the other hand, are so abstruse that the initiate is lost in their incomprehensibility, for these are issues discussed by the technical logicians largely among themselves. When one considers the important bearings implicit in the interpretation of implication, such a state of affairs is greatly to be deplored. Through a simple but adequate presentation of the meaning of a hypothetical proposition one may be led not only to an understanding of a fundamental paradox of logic but to the whole question of modal logic and of the philosophy of logic. It is my purpose in what follows to present as simply as possible the various interpretations of the meaning of implication and to show their relations to modal logic. Having done this I believe it will be possible to state the case in favor of material implication in a new light.