An Arithmetization of Logical Oppositions

Abstract

An arithmetic theory of oppositions is devised by comparing expressions, Boolean bitstrings, and integers. This leads to a set of correspondences between three domains of investigation, namely: logic, geometry, and arithmetic. The structural properties of each area are investigated in turn, before justifying the procedure as a whole. To finish, I show how this helps to improve the logical calculus of oppositions, through the consideration of corresponding operations between integers.

Appendix: A Constructive Geometry of Logical Relations

In the first section of the paper, a historical reference has been made to Shao Wong’s ordering of the 64 hexagrams. Its striking feature is that it also respects the central symmetry of contradictory oppositions between the Boolean bitstrings—and their corresponding blue integers, here below, as is the case in all contemporary gatherings of logical geometry.

We propose in the following a similar constructive representation of logical oppositions: all are decreasing quadrangles, of length L and width l. Each progression of a given 2ⁿ quadrangle consists in duplicating it either horizontally (from left to right) when n is odd, or vertically (from top to bottom) when n is even. The resulting figure is either a rectangle, such that L = 2 l whenever n is odd, or a square, such that L = l whenever n is even. Each quadrangle is a complete set of bitstrings from the minimal value 0 to the maximal value 2ⁿ – 1, and the new ordering also preserves the properties of vectors (except Chasles’ relation).