  According to  Ashworth,[22]  serious and sophisticated investigation of logic ended at about the third decade of the sixteenth century. The Port Royal Logic of the following (seventeenth) century seems typical in its approach: its authors frequently suggest that logic is trivial and unimportant. Its doctrine includes that of the square of opposition, but the discussion of the O form is so vague that nobody could pin down its exact truth conditions, and there is certainly no awareness indicated of problems of existential import, in spite of the fact that the authors state that the E form entails the O form (4th corollary of chapter 3 of part 3). This seems to typify popular texts for the next while. In the nineteenth century, the apparently most widely used textbook in Britain and America was Whately’s Elements of Logic. Whately gives the traditional doctrine of the square, without any discussion of issues of existential import or of empty terms. He includes the problematic principles of contraposition (which he calls “conversion by negation”):