An Inconsistency in Kassler's First Twelve-Note-Class System

Abstract

In a paper1 entitled Toward a theory that is the twelve-note-class system Michael Kassler constructs several formal systems, of which the first, denoted by R1, has twelve primitive symbols, one formation rule, an axiom, and three rules of inference. In an effort to translate the theory from the language of symbolic logic, in which it is couched, into the algebraic language of mappings and other relations, I have discovered an apparent inconsistency in the system. The algebraic study is contained in my thesis2 for the M.A. degree. Below I offer a short version, in Kassler's language, presenting the primitive basis of R1, together with the relevant definitions, and exhibiting the inconsistency in the form of a counter-example of Kassler's principal metatheorem. The twelve primitive symbols are: 0 1 2 3 4 5 6 7 8 9 a b. A formula is defined to be a non-empty finite string of occurrences of primitive symbols. The formation rule is: r is a wff (well-formed formula) if every primitive symbol occurs in r. The axiom is: 0123456789ab. In order to state the rules of inference, several preliminary definitions are needed. First, the dimension of a formula is the number of tokens (i.e., of occurrences of primitive symbols) in it; e.g., the dimension of 2332 is four. Given a formula r of dimension c and a formula A of dimension d, r is a subformula of A if there is a set S of c positive integers, each less than or equal to d, such that if 1 ? i ? c then the ith token in r is the jth token in A, where j is the least element of S that is greater than exactly i 1 elements of S; such a set S is called a determining set of r from A. Kassler gives this example: if r is 918132 and A is 9.13891338725 then (6, 4, 1, 2, 8, 11) is a determining set of the subformula r from the formula A.