Chapter 10

Languages as complete and distinct systems of reference

Section 1 is an extended commentary on Edward Sapir’s formulation nearly a century ago of what he considered the two most fundamental properties of human language, first that each one is a formally complete system of reference to experience and second that each one is formally distinct from every other. Section 2 considers some aspects of the development of these formulations, noting that they have been considered separately and not integrated as fully fleshed out systems of reference, as Sapir envisioned. Section 3 examines more closely what such an integration looks like in a case involving simple arithmetic. Section 4 begins with a brief review of the accomplishments of Greco-Roman logic and more recent developments in the theory of logic, leading to a consideration of what may be needed to fulfill Sapir’s program. Section 5 summarizes some of my own recent research on extending first-order logic by replacing the unordered set of individuals with a specific ordering of a set of sets of individuals that is isomorphic to an ordering of sets of sets of numbers that contain no pairs of divisible numbers, which was investigated by Richard Dedekind shortly before the turn of the twentieth century.