Leonard, Goodman, and the Development of the Calculus of Individuals

This paper investigates the relation of the Calculus of Individuals presented by Henry S. Leonard and Nelson Goodman in their joint paper, and an earlier version of it, the so-called Calculus of Singular Terms, introduced by Leonard in his Ph.D. dissertation thesis Singular Terms. The latter calculus is shown to be a proper subsystem of the former. Further, Leonard’s projected extension of his system is described, and the definition of an non-extensional part-relation in his system is proposed. The final section discusses to what extent Goodman might have contributed to the formulation of the Calculus of Individuals. 1 The Calculus of Individuals In 1936, Henry S. Leonard and Nelson Goodman presented a joint paper at the meeting of the Association for Symbolic Logic which was held at the meeting of the Eastern Division of the American Philosophical Association in Cambridge, Massachusetts. Eleven years later, they published an elaborated version of this paper under the title “The Calculus of Individuals and its Uses” [12]. The calculus they introduce in this paper is today usually taken as a basis for the study and use of formal part-whole relations (often called “mereology”) in analytic metaphysics, sometimes mediated by Goodman’s presentation of the calculus in his The Structure of Appearance [7]. Goodman used the Calculus of Individuals in his Ph.D. dissertation thesis A Study of Qualities of 1940 [6], which eventually became The Structure of Appearance. As in his joint paper with Leonard, he used the calculus as an addition to set theory to solve a problem known as the difficulty of imperfect community in Rudolf Carnap’s Aufbau [1]. Only in Structure Goodman abandoned set theory and presented a nominalistic construction that used only the Calculus of Individuals.1 The focus of this paper, however, will be an investigation of the system that Leonard presents in his Ph.D. dissertation thesis Singular Terms [10] of 1930, which is the first See [2], 121–139, and [3], §3.2, for a discussion.