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On the simplicity of ideas1

Published online by Cambridge University Press:  12 March 2014

Nelson Goodman
Nelson Goodman*
Affiliation:
Camp Pickett, Virginia
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Extract

The motives for seeking economy in the basis of a system are much the same as the motives for constructing the system itself. A given idea A need be left as primitive in a system only so long as we have discovered between A and the other primitives no relationship intimate enough to permit defining A in terms of them; hence the more the set of primitives can be reduced without becoming inadequate, the more comprehensively will the system exhibit the network of interrelationships that comprise its subject-matter. Of course we are often concerned less with an explicit effort to reduce our basis than with particular problems as to how to define certain ideas from others. But such special problems of derivation, such problems of rendering certain ideas eliminable in favor of others, are merely instances of the general problem of economy. Thus it is quite wrong to think of the search for economy as a sort of game, inspired by an abnormal love of superficial neatness. Some economies may be relatively unimportant, but the inevitable result of regarding all economy as trivial would be a willingness to accept all ideas as primitive at the outset, making a system both unnecessary and impossible.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 8 , Issue 4 , December 1943 , pp. 107 - 121
Copyright
Copyright © Association for Symbolic Logic 1944

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Footnotes

1

A modified and expanded version of a paper On the length of primitive ideas, of which an abstract appeared in this Journal, vol. 8 (1943), p. 39. The paper was scheduled for the meeting of the Association for Symbolic Logic that was to have been held at Yale University in December 1942.

References

2 For the rest of this paper I shall write in terms of the more usual and convenient definitional set-up, where the virtual primitives are the actual ones; i.e., where every term other than the virtual primitives is first introduced through a definiendum.

3 Such a prior categorization is tacitly assumed in Lindenbaum's, A.Sur la simplicité formelle des notions, Actes du Congrès International de Philosophie Scientifique, Paris 1935 (pub. 1936), VII Logique, pp. 29–38.Google Scholar In this article, to which Dr. A. Tarski called my attention, Lindenbaum discusses the problem of simplicity, but proposes no solution.

4 The definition of sequences presupposed above, and throughout the first three sections of this paper, is that in my paper Sequences, this Journal, vol. 6 (1941), pp. 150–153. As we shall see later, however, the criterion of simplicity proposed in the present paper is independent of that definition of sequences. Concerning the paper on sequences, I should like here to make three comments, (i) The technical difficulty pointed out by Dr. McKinsey in his review (VII 120 (1)) can be met by altering the definiens for a sequence to “(y) :: p‘ (x ∩ NC‘y) ≠ Λ x ∩ NC‘y ≠ Λ p‘(x ∩ NC‘y) ⊂ y ν: p‘(x ∩ NC‘y) = Λ ν x ∩ NC‘y = Λ : y ⊂ y s‘(x ∩ NC‘y) : ⊃ y ε x”; the difiniens for the (κ th component of any sequence x should correspondingly be altered to read: “{x∩κ ≠ Λ z ε p‘(x∩κ) ν : p‘(x∩κ) = Λ ν x∩κ = Λ z ε s‘(x∩κ)}.” (ii) Finite sequences under this theory will be elements for any logic for which (a) cardinal numbers as defined in Principia mathematica and their subclasses are elements and (b) finite classes of elements are elements, (iii) The class Λ is an exception to the statement that all self-ordered classes are identical with the sequence they establish.

5 Since sequences and relations are to be construed in terms of a uniform class theory based upon individuals, and individuals are to be identified with their unit classes (Op. cit., p. 151 and footnote 6), the term “class” here is a comprehensive one.

6 Op. cit., p. 152.

7 Op. cit., p. 151 and footnote 6.

8 I am indebted to Dr. C. G. Hempel for valuable criticisms of an earlier version of this and the preceding paragraph.

9 While I do not define the class of redundant ideas precisely, it includes not only ideas of redundant sequences, but also other ideas, such as those of redundant relations—e.g., (xRy x = z).

10 I.e. the relationships called for under my earlier cited (footnote 4, above) definition of sequences.

11 For convenience, I often omit the words “ideas of” or “idea of,” but it remains true that I am primarily concerned with the complexity of ideas. Naturally a class cannot literally have a complexity of u, but an idea of a class can. Such a term as “indeterminate class” will be readily understood as short for “idea of a class of which the specific maximum numerical variegation is not determinable from the idea.” By “class” without further qualification, I shall mean a class that is thus indeterminate if not a class of individuals, and such that the logical sums of its cardinal subclasses, the logical sums of their cardinal subclasses in turn, and so on, are indeterminate if not classes of individuals.

12 The zero type is excluded for our present purposes, individuals being construed as their own unit classes and as belonging, along with all other classes of individuals, to type 1.

13 Kuratowski, C., Sur la notion de l'ordre dans la théorie des ensembles, Fundamenta mathematicae, vol. 2 (1921), pp. 161–171.CrossRefGoogle Scholar

14 It should be made clear that I am not concerned in this paper with the relative merits in general of the different theories of sequences and relations considered, but solely with the question which theory enables us to identify any given primitive sequence or relation with the simplest class.

15 Wiener, N., A simplification of the logic of relations, Proceedings of the Cambridge Philosophical Society, vol. 17 (1914), pp. 387–390.Google Scholar

16 Quine, W. V., Mathematical logic (New York 1940), p. 202Google Scholar, small type.

17 The treatment of symmetrical relations here described, but not the material pertaining to reflexiveness, was outlined in my doctoral thesis, A study of qualities (typescript, Harvard Library, 1941), p. 44.