The Core of Logics

Abstract

Like mathematics so often logic is taught to introductory students in a very mechanical way, the emphasis being on memorization and working problems. Particularly egregious is that the logic taught in philosophy departments is devoid of philosophy. Students rarely encounter the deep philosophy underpinning the structures. Logic is the theory of innate order in the universe and is the language of that order. More explicitly the foundation of that order is binary, based on the most fundamental law of all: dialectics. Something is apprehended because of what it is not. This chapter summarizes the development of thinking underpinning this idea of the innate binary structure. It is an ordered binary space starting in one dimension and progressing through three, and beyond. The philosophical basis of single, two (Table of Functional Completeness), and three (three-dimensional hypercube) dimension space provides coherency to ideas like deduction, induction, and inference, in general. The ordering in these spaces is founded on the same thinking giving rise to numbers and arithmetic. An exposition of how binary logical space develops sets the stage for discussing foundational ideas like the relationship between arithmetic (and its follow-on, mathematics) and logic, pattern recognition, and even whether we may be a simulation, a conjector made by Nick Bostrom. Research directions are proposed such as questioning the nature of axioms, exploring the insufficiency of Peano's postulates, proof theory, and ordering of operators based on intellectual complexity.