The Calculus of Individuals and its Uses
Leonard (Henry S.) & Goodman (Nelson)
Source: Journal of Philosophical Logic 5.2, June 1940, pp. 45-55
Paper - Abstract

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Authors' Introduction

  1. An individual or whole we understand to be whatever is represented in any given discourse by signs belonging to the lowest logical type of which that discourse makes use. What is conceived as an individual and what as a class is thus relative to the discourse within which the conception occurs. One task of applied logic is to determine which entities are to be construed as individuals and which as classes when the purpose is the development of a comprehensive systematic discourse.
  2. The concept of an individual and that of a class may be regarded as different devices for distinguishing one segment of the total universe from all that remains. In both cases, the differentiated segment is potentially divisible, and may even be physically discontinuous. The difference in the concepts lies in this: that to conceive a segment as a whole or individual offers no suggestion as to what these subdivisions, if any, must be, whereas to conceive a segment as a class imposes a definite scheme of subdivision – into subclasses and members.
  3. The relations of segments of the universe are treated in traditional logistic at two places, first in its theorems concerning the identity and diversity of individuals, and second in its calculus of membership and class-inclusion. But further relations of segments and of classes frequently demand consideration. For example, what is the relation of the class of windows to the class of buildings? No member of either class is a member of the other, nor are any of the segments isolated by the one concept identical with segments isolated by the other. Yet the classes themselves have a very definite relation in that each window is a part of some building. We cannot express this fact in the language of a logistic which lacks a part-whole relation between individuals unless, by making use of some special physical theory, we raise the logical type of each window and each building to the level of a class – say a class of atoms – such that any class of atoms that is a window will be included (class-inclusion) in some class that is a building. Such an unfortunate dependence of logical formulation upon the discovery and adoption of a special physical theory, or even upon the presumption that such a suitable theory could in every case be discovered in the course of time, indicates serious deficiencies in the ordinary logistic. Furthermore, a raising of type like that illustrated above is often precluded in a constructional system by other considerations governing the choice of primitive ideas.
  4. The ordinary logistic defines no relations between individuals except identity and diversity. A calculus of individuals that introduces other relations, such as the part-whole relation, would obviously be very convenient; but what chiefly concerns us in this paper is the general applicability of such a calculus to the solution of certain logico-philosophical problems.
  5. The calculus of individuals we shall employ is formally indistinguishable from the general theory of manifolds developed by Lesniewski. Lesniewski’s purpose, quite different from ours, was to establish a general theory of manifolds that would not be subject to Russell's paradox; but since he excludes the notion of a null class, his formal system is virtually the same as that which we interpret as a calculus of individuals. Inasmuch as his system is rather inaccessible, lacks many useful definitions, and is set forth in the language of an unfamiliar logical doctrine and in words rather than symbols, we shall attempt (in Part II) to restate the calculus in more useable form, with additional definitions, a practical notation and a transparent English terminology. In Part III we shall explain how this calculus enables us to describe generally certain important, but often neglected properties of relations, and thereby contributes to the clarification of many philosophical problems.

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