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The calculus of individuals and its uses1

Published online by Cambridge University Press:  12 March 2014

Henry S. Leonard  and
Nelson Goodman
Henry S. Leonard
Affiliation:
Durham, North Carolina
Nelson Goodman
Affiliation:
Auburndale, Massachusetts
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Extract

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.

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.

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 5 , Issue 2 , June 1940 , pp. 45 - 55
Copyright
Copyright © Association for Symbolic Logic 1940

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Footnotes

1

A somewhat elaborated version of a paper read in Cambridge, Mass., before a joint meeting of the Association for Symbolic Logic and the American Philosophical Association, Eastern Division, on December 28, 1936.

References

2 The relation is somewhat analogous to the more familiar one between classial and serial concepts, dealing as they do with the same material, but in a manner that makes the latter more highly specialized.

3 Since this paper was presented, the convenience of such a calculus of individuals has been well illustrated by Dr.Woodger's, J. H.Axiomatic method in biology (1937)Google Scholar.

4 In O podstawach matematyki (in Polish), Przegląd filozoficzny, vols. 30–34 (1927–1931)Google Scholar.

5 Leśniewski employs discreteness as his primitive relation in the final version of his system. See Chapter X of his above-mentioned paper.

6 Identity is not defined nor taken as primitive since it is already defined in our logistic vehicle, Principia mathematica. Had it been desirable to develop this calculus in isolation from other treatments of logistic, identity could have been defined as mutual part-whole, to which it is equivalent by theorem I.315:

This isolation, however, seemed undesirable because of the uses, as in Part III below, that we intended to make of the calculus.

7 The present form of this definition has been suggested by the corresponding definition in Dr. A. Tarski's appendix to Woodger, op. cit.

8 Leśniewski employs only two postulates. One is identical with our I.13 expanded in terms of the primitive relation; the other asserts both the existence and uniqueness of the fusion of any (non-null) class. Our postulate I.1 is weaker, since it asserts only the existence of some such individual but not its uniqueness. Accordingly we require also postulate I.12.

9 Berlin, 1928. See especially sections 67–93, pp. 108–120. This work is subsequently referred to as the Aufbau.

10 The following discussion of the problem here illustrated is that referred to by Quine, W. V. in Relations and reason, Technology review, vol. 41 (1939), pp. 325, 327nGoogle Scholar.

11 A unigrade relation is a relation of any one degree; a multigrade relation is one having at least two different degrees.

12 The two definitions just proposed are equivalent only for cases in which the cardinality of the argument, α, is greater than or equal to two. The first definition makes “S′(α)” true when the cardinality of a is less than two, while the second makes it false under the same circumstances. This difference, however, rests upon trivial cases that may be decided by considerations of convenience in dealing with the particular subject-matter and problem in hand; and either definition may be easily adjusted to accord with whatever decision is made.