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The logical simplicity of predicates

Published online by Cambridge University Press:  12 March 2014

Nelson Goodman
Nelson Goodman*
Affiliation:
University of Pennsylvania
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Extract

In an earlier article, I proposed a way of determining the relative simplicity of different sets of extralogical primitives. The calculations assumed a fully platonistic logic, committed to an indefinite hierarchy of classes, with sequences and relations defined as classes. Recently it has been shown that a nominalistic logic, countenancing no entities other than individuals, can be made to serve many of the purposes for which a platonistic logic had been thought necessary. The question naturally arises how we are to determine the simplicity of extralogical bases of systems founded upon a nominalistic logic. In such bases, the only extralogical predicates will be predicates (of one or more places) of individuals. The present paper offers a general method of measuring the simplicity of such bases.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 14 , Issue 1 , 16 May 1949 , pp. 32 - 41
Copyright
Copyright © Association for Symbolic Logic 1949

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References

1 On the simplicity of ideas, this Journal, vol. 8 (1943), pp. 107–121.

2 Steps towards a constructive nominalism by Nelson, Goodman and Quine, W. V., this Journal, vol. 12 (1947), pp. 105–122.Google Scholar

3 Where the complexity-value of a basis is found by adding the values of the predicates comprising the basis.

4 In such contexts in this paper, a capital letter is to be understood as an artificial abbreviation of some actual verbal predicate. The letter with quotes is thus to be taken as the name not of the letter but of the written-out predicate.

5 Here, and in two later cases, a definiens suggested to me by Prof. W. V. Quine has been adopted in place of a longer one I had originally used.

6 Indeed, the difficulties that may result from adopting this principle without all the reservations here outlined are so great that I rejected it entirely in my earlier paper. Yet it does seem to me to embody, however roughly, an intuitive demand that must be satisfied by any finally acceptable assignment of values. For that reason, I have in the present paper made what I hope is a sufficiently guarded use of the principle.

7 If a predicate does not select from among all the possible combinations of occupants chosen in order from certain of its places, it will have fewer than n–1 joints, as we shall see in Section 3. But predicates of this special kind are not in question at the moment, when only applicability (in addition of course to the number of places) is assumed. The degree of simplicity ascribed to a predicate with respect to certain stated characteristics is the maximum simplicity that can be inferred from those characteristics alone (or in other words—cf. Section 4—the complexity ascribed is the maximum consistent with those characteristics). Predicates which also have certain other characteristics may subsequently be shown to be simpler. It must thus be borne in mind that many statements throughout the text are to be understood as qualified by some such phrase as “in the absence of other indications” (cf. Section 1) or “so far as determined solely by the characteristics now in question,” even where no such term as “normally,” “ordinarily” or “in general” is inserted as a remainder.

8 Thoroughly symmetrical predicates I shall hereafter call symmetrical. A symmetrical predicate is one such that if any selection of individuals satisfies the predicate in one order then they also satisfy it in every other order. A partially symmetrical predicate is one which is symmetrical with respect to some but not all of its places; for example, the three-place predicate “Ρ” is partially symmetrical if it is the case that (x) (y) {(∃z) P (x,y,z) ⊃ (∃w) Ρ(y,x,w)} A predicate maybe partially symmetrical with respect to scattered as well as to adjacent places. By a non-symmetrical predicate, I mean one that is not even partially symmetrical.