New Notes on Simplicity Nelson Goodman The Journal of Symbolic Logic, Vol. 17, No. 3. (Sep., 1952), pp. 189-191. Stable URL: http://links.jstor.org/sici?sici=0022-4812%28195209%2917%3A3%3C189%3ANNOS%3E2.0.CO%3B2-G The Journal of Symbolic Logic is currently published by Association for Symbolic Logic.
Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/asl.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.
JSTOR is an independent not-for-profit organization dedicated to and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact
[email protected].
http://www.jstor.org Fri May 18 08:40:53 2007
TEEJOURNAL OP SYMBOLIC LOGIC Volume 17, Number 3, Sept. 1952 Prinled i n U . S . A .
NEW NOTES ON SIMPLICITY
NELSON GOODMAN
The treatment of simplicity in my book and in earlier articles1 needs one important amplification and certain minor amendments. The basic method for computing complexity values nevertheless remains the same. New studies show that so to adjust the actual technique of computation as to take full account of the factor of reflexivity would result in unwieldy rules. Fortunately, however, this is unnecessary; for the method already prescribed works for thoroughly irreflexive predicates2, and every other predicate is replaceable by such predicates. Thus we need only require that before the complexity of a basis is evaluated every primitive be replaced by the simplest adequate set of irreflexive predicates. Consider, for example, two-place predicates that are not irreflexive. Any two-place (or indeed any n-place) predicate that is satisfied only by identity pairs (or identity n-ads) can obviously be replaced by a one-place predicate. Any two-place predicate that is satisfied by some non-identity pairs and that is either exactly reflexive (i.e. satisfied also by just the identity pairs of elements in such non-identity pairs) or totally reflexive (i.e. satisfied also by all identity pairs) can be replaced by a two-place irreflexive predicate. In the former case, for instance, we replace "P" by "Q", so explained that Qx,y if and only if Px,y and x # y ; and then define: Finally, a non-irreflexive two-place predicate "P" that is of none of these three kinds can be replaced by a one-place and an irreflexive two-place predicate as follows:-let Qx if and only if Px,x, and let Rx,y if and only if Px,y and x # y; then define:
Predicates of more t,han two places offer much greater variety since they may exhibit any sort of reflexivity with respect to any selection of places. At the maximum, the set of irreflexive predicates needed to replace a three-place predicate will consist of a three-place predicate, a one-place predicate, and three two-place predicates. As the number of places increases further, the variationsand the maximum complexity of the replacing basis that may be neededincrease very rapidly; but the general procedure of replacement is unchanged. Since all predicates are now to be resolved into irreflexive ones, the earlier Received March 1, 1951. (Cambridge, Mass., 1 The structure of appearance [hereinafter referred t o as "SA"] 1951), pp. 59-75; T h e logical s i m p l i c i t y of predicates, this JOURNAL, vol. 14 (1949)) pp. 3241; and An i m p r o v e m e n t in the theory of s i m p l i c i t y , ibid., pp. 228-229. Some points in the present paper were arrived a t as the result of questions by my students Mr. A. N. Chomsky and Mr. H. C. Bohnert, 2 An n-place predicate is irreflexive if and only if it is satisfied only by sequences consisting of n different components. A one-place predicate is both reflexive and irreflexive. 189
190
NELSON GOODMAN
and minor rule3 requiring that predicates be mechanically consolidated is naturally dropped. Three other matters, not directly related to points discussed above, call for comment: (1) universal predicates, (2) 'kinds' of bases, (3) choice among equally simple bases. (1) Whether a one-place primitive predicate is considered to be universal or not depends upon how the range of the individual-variables of the system happens to be specified; but the complexity value of a predicate should hardly depend upon this. And obviously it must not depend upon the answer to the question whether in fact there are individuals in addition to those admitted as values of the individual-variables. Accordingly, a universal one-place predicate has one segment, like any other one-place predicate, not zero.' Similarly, such places of a many-place predicate as are satisfied by all individuals of the system constitute one segment. Since we are concerned only with applicable predicates, null predicates do not enter into consideration. (2) My remark5 that the 'kinds' of basis are ultimately defined in terms of segments and joints needs correction. What I should have said is that the tentative method of appraising complexity by means of the guiding rule and the adopted notion of kinds leads to and is superseded by the method of segments and joints. The initial relevant kinds of basis are just those describable directly in terms of number of predicates, number of places in each, and status with respect to symmetry and reflexivity. Other kinds, such as those describable in terms of self-completeness, enter only through interreplaceability-relationships with these first kinds. The initial choice of relevant kinds is admittedly arbitrary to a certain extent in that other characteristics also might have been embraced. But it justifies itself by leading to a method- of computing complexity-in terms of selectivity as measured by number of segments and joints -that has independent recommendations as a means for deciding doubtful cases. (3) When we are to choose on formal grounds between bases of equal computed complexity, we make ready decisions conforming to certain obvious rules that nevertheless may well be explicitly set down. In general, the underlying principle is to avoid wasting predicates or places or combinations of places. Thus if A and B are alternative adequate bases of equal computed complexity, we choose A rather than B if: (i) each consists of one predicate, and A has the fewer places; or (ii) A, in number of predicates and of places in each, is exactly like some basis that consists of some but not all the predicates of B; or (iii) after the above two rules have been applied, A and B have the same total number of places but A has the more predicates and thus the lower ratio of places to predicates. To illustrate (iii), two one-place predicates are preferable as a basis to a twoplace predicate that has a value of 2; for the latter wastes either a place (through SA, p. 75.
'SA, p. 70.
SA, pp. 74-75.
NEW NOTES ON SIMPLICITY
191
having but one segment) or the combination of places (through having no joint). (Note added May 17, 1951.) Under the method of computation described in this paper, an n-place predicate that requires resolution into several others may have a complexity value much greater than 2n - 1. Indeed, the maximum value of an n-place predicate becomes
e.g., 4 for 2-place predicates, and 15 for 3-place predicates. I am indebted to Mr. Leon Robbins, Jr. for help in arriving a t this formula. The need for this increased maximum value is further illustrated by a recent discovery by Prof. W. V. Quine. He has found a way of replacing any 2-place predicate by a 3-place symmetrical predicate, thus controverting the statements made in SA, p. 69, lines 14-21. His method can no doubt be extended to predicates of more places. Thus (n 1)-placesymmetrical predicates must have a higher maximum value than do n-place predicates in general. This requirement is met by our present method of computation, which gives 2-place predicates a maximum value of 4 and gives 3-place symmetrical predicates a maximum value of at least 9. I t is interesting to note that Quine's device normally results in a replacing 3-place predicate that is neither irreflexive nor exactly reflexive nor totally reflexive with respect to all three or any two of its places.
+
UNIVERSITY OF PENNSYLVANIA
