On the Simplicity of Ideas Nelson Goodman The Journal of Symbolic Logic, Vol. 8, No. 4. (Dec., 1943), pp. 107-121. Stable URL: http://links.jstor.org/sici?sici=0022-4812%28194312%298%3A4%3C107%3AOTSOI%3E2.0.CO%3B2-Q The Journal of Symbolic Logic is currently published by Association for Symbolic Logic.
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Tag J O U B N ~ 0s L SYMBOLIC LOGIC
Volume 8, Number 4, December, 1943
ON THE SIMPLICITY OF IDEAS' NELSON GOODMAN
1. The problem. 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 perrpit defining A iil 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 a t the outset, making a system both unnecessary and impossible. We may, admittedly, construct a perfectly legitimate system with no defined ideas if in place of definitions we adopt the corresponding equivalence-statements. But such a postulational set-up is not a way of evading the question of economy. Whether our basic correlations are set forth as definitions (with " =dfl') or as postulates (with " = "), what we have in effect are licenses for the use of alternative expressions in stating certain facts. In either case, the virtual primitive terms of the system comprise a least possible set of expressions that will, if we take the fullest advantage of such licenses, do service for the whole set of expressions contained in the system.2 And so long as the scope of the system is not narrowed, its articulation is improved as its minimum vocabulary is reduced-as economy measured by the sparsity of the set of virtual primitives is increased. Just what is meant by the economy or simplicity of a set of primitives is not nearly so clear, however. Little reflection is needed to discover that the mere counting of primitives is no satisfactory measure. If the number of primitives were the sole criterion, then the economy of extralogical bases-which alone concerns us in this paper-would indeed be a trivial matter. For by the purely mechanical application of certain logical devices, we can readily reduce all the Received August 2, 1943. A modified and expanded version of a paper O n the length of primitive ideas, of which a n v,ol. 8 (1943), p. 39. The paper was scheduled for the abstract appeared in this JOURNAL, meeting of the Association for Symbolic Logic t h a t was t o have been held a t Yale University in December 1942. 2 For t h e 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.
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primitives of any system to one: if all the given primitives are of one type, we need merely take instead some sequence of them; if they are of different types, we first apply "L" ("unit-class of") to each of those of the lower types enough times to secure a primitive of the same type as the highest in the set, and then replace the resultant primitives by some sequence of them. Such a condensation quite clearly effects no genuine economy. But we are confronted with the more general problem of prescribing rules that will always enable us to distinguish between genuine and spurious economy. What we apparently need is a way of gauging the relative simplicity and complexity of ideas. The trouble with merely counting them is that it does not reflect the fact that the greater number of primitives in one basis may be balanced by the greater complexity of those in another. Fortunately, however, we are not forced to deal with the notoriously difficult problem of defining simplicity in all its phases. We do not need to be able to determine whether a given idea is simpler than another in every way, but only whether it is simpler in those respects which are relevant to economy. Naturally any criterion of simplicity must, within announced limitations, conform to our stronger and more justifiable intuitions mcerning simplicity in general. But beyond this, the crucial test for our present purposes will be whether the replacement of a given idea by one that is simpler does generally require-and therefore signifythe application of some special knowledge concerning the ideas involved. I t is this that makes an economy significant or genuine. If, through the application of some purely automatic procedure like that by which the number of extralogical primitives in a basis can always be reduced to one, an idea that is less'simple by the criterion in question can always or even usually be replaced by another that is more simple, then that criterion is unsatisfactory for our purposes. Before we look for a satisfactory criterion, however, we must be clear about what we intend it to apply to. As already mentioned, we shall be concerned solely with extralogical bases. The extralogical basis of a system consists of all the primitives that do not belong to the underlying logic. We assume that some logic approximately as strong as that of Principia mathematica is adopted as substructure, and that the further, "extralogical" primitive ideas required are then added. But what, precisely, are primitive ideas? Quite obviously, it is not primitive terms as such that concern us here. We do not conclude that two primitive ideas are equally simple merely because equally simple signs are used for them. Indeed, the simplest possible sign may be used for the most complex primitive idea. Only if the system-builder chooses to subject his terminology to the dictates of some prior categorization--e.g. making "R" and "S" relation- sign^, "a,b" and "c,d,e9' sequence-signs, etc.-can his primitive terms reveal anything of the relative simplicity of his primitive ideas.3 But if we admit only individe 8 Such a prior categorization is tacitly assumed in A. Lindenbaum's Sur la simplicit6 jormelle des notions, Actes du Congrbs International de Philosophie Scientijique, Paris 1935 (pub. 1936), VII Logique, pp. 29-38. In this article, t o which Dr. A. Tarski called my attention, Lindenbaum discusses the problem of simplicity, but proposes no solution.
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uals a t the lowest level, and define sequences and relations as certain classes of . . . classes of these, and if we permit a certain typical ambiguity of terms, then such rules concerning the use of symbols become virtually indefensible. Moreover, even were we presumptuous enough to impose on the system-builder rules to the effect that he must use the most special kind of symbol permissible for what is den0ted-e.g. he must use such a term as "a,b,c" for a triad and not "xu, even though the triad is a class and "x" a legitimate symbol for it-still these very rules are themselves based upon what kinds of things the terms denote. Yet the primitive idea cannot be simply what the primitive term denotes. If we have two diierent terms that in fact-though we do not know or postulate it-have an identical denotation, we have two primitive ideas; for example, in an extralogical system without existential postulates, there is always the possibility that two or more primitive terms alike denote the null class. Furthermore, two primitive ideas may be equally simple even though one of the terms denotes something much more complex than the other; for exsmple, if one primitive term "A" stands for the class of grade-classes of pupils in the Jones School, and the other "B" for the class of grade-classes of pupils in the White School, the two primitive ideas are equally simple even though i t should happen that "A" (through the fact that the Jones School has eight grades, each with a different number of pupils from the others) denotes a sequence4 of eight components, while "B" (through the fact that the White School has exactly the same number of pupils in each of its grades) denotes a sequence of one. By any acceptable standard, a sequence of one component is much simpler-all else being equalthan a sequence of eight components; but the great difference in the simplicity of what "A" and "B" denote does not alter the fact that the two primitive ideas are equally simple. Thus i t is evident that the simplicity of what we call the primitive idea is not directly a function of what the primitive term denotes; but rather of what we know-r, figuratively, of what the interpreted system knows-about what the term denotes. If we like, we may think of the primitive idea itself as the interpretation that is explicitly or implicitly provided for the primitive term; and we may, for convenience, suppose the logical terms in that interpretation to be The definition of sequences presupposed above, and throughout the first three sections of this paper, is t h a t in my paper Sequences, this JOURNAL, v01. 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 t o make three comments. (i) The technical difficulty pointed out by Dr. McKinsey in his review ( V I I 120 (1)) can be met by altering the definiens for a sequence t o l ' i ( y ) :: p l ( z n NC'y) # A z n NCSy f A . p'(z n NCdy) C y v :p l ( z n NCIy) = A v z n NCSy= A : y C s l ( z n NCS1/) 3 . y s 2"; the definiens for the kth component of any sequence z should correspondingly be altered t o read: " i ( z n k # ~ z e p'(znk). v :p'(znk) = A v znk = A z s sl(znk)]." (ii) Finite sequences under this theory will be elements for any logic for which (a) cardinal numbers as defined in Principia mathematics and their subclasses are elements and (b) finite classes of elements are elements. (iii) The class tA is an exception t o the statement that all self-ordered classes are identical with the sequence they establish.
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translated into the language of the underlying logic of the system. Thus the primitive idea is construed as a meaningful expression. I t is not part of the formal system, but i t is what gives a formal system an application. The interpretation must always be supposedly unambiguous, uniquely describing some class,' but i t may be non-committal in any degree about the structure of the class described; for example, i t may-as in the above case of the class of gradeclasses in the Jones School--specify a certain class without disclosing that the class is a sequence, or i t may specify a sequence without revealing the number of components. A method for comparing the simplicity of primitives would thus be useless in the majority of cases if its application required full knowledge of the structure of what the primitive terms denote. No more information can be legitimately demanded than is supplied by the primitive ideas; but a t the same time, the possibility of unrevealed complexities in what is denoted cannot be simply ignored. This is one of the most important and most difficult features of our problem. At the very outset of the search for an adequate definition, the proposal might naturally be made that the kind of simplicity relevant to economy pertains to the defining power of ideas rather than to their structure. A basis might be considered more economical than another only if less can be defined from the former than the latter. If a given idea or basis A were definable solely in terms of B and logic, while B were not definable solely in terms of A and logic, i t would be clear that B is the more powerful idea or basis, A the weaker or simpler. There are, however, two objections against this approach. The first is that the criterion as stated is extremely incomplete; it is never applicable when neither of two ideas is definable in terms of the other and logic. And attempts to generalize and complete the principle involved seem doomed to failure. We might try to establish a rule of this sort: if a basis A is definable solely from logic and some basis that is in a specified sense "of the same kind" as B, while B is not definable solely from logic and some basis that is in that sense "of the same kind" a s A, then A is weaker than B. But to describe "kinds" in the appropriate sense is to decide, quite apart from considerations of defining power, what does and does not constitute equal simplicity. If we can do that, then reference to relative defining power is superfluous in the measurement of simplicity. The second objection is that even the incomplete criterion suggested sometimes gives undeniably counter-intuitive results. From a basis K, consisting of two ideas, A and B, of classes of individuals we can define the three ideas C and D and E as follows:
5. Since sequences and relations are to be construed in terms of a uniform class theory based upon individuals, and individuats are t o be identified with their unit classes (Op. cit., p. 151 and footnote 6), the term "class" here is a comprehensive one.
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From a basis Kt, consisting of C and D and E as primitives, we could notconversely--define A and B. Hence K' is weaker than K. Yet Kt, which consists of three diverse primitives, each of which is an idea of an ordered pair of classes of individuals, is intuitively a much more complex basis than K , which consists of two ideas, each of a simple class of individuals. Thus again we have a strong indication that weakness is not the same as simplicity or economy. Finally, while the association of weakness with economy has an admitted initial plausibility, there is also a certain absurdity in calling a weaker idea a more economical one. The most economical idea, like the most economical engine, is the one that accomplishes most by using least. Simplicity--or fuel consumption-is a different factor from power but has to be taken equally into consideration. And weakness, far from being directly proportionate to economy is inversely proportionate to it: the weaker of two equally simple ideas is the less economical. The economy of a basis may be said to be the ratio of its strength to its simplicity. But superfluous power is also a waste. Adequacy for a given system is the only relevant factor in the power of a basis; and where we are comparing several alternative bases for some one system, as is normally the case, that factor is a constant. Thus in practice the simplest basis is the most economical.
2. Indices of complexity. Accordingly, our attention is returned to the internal structure of bases. The beginning is very easy. If counting fails because any number of primitives may be compounded into a single sequence, then clearly we had better consider the total number of non-null components in a basis rather than the number of separate primitives. The principle is readily extended to cover cases where the components of a sequence are themselves sequences; a pair of pairs of individuals, for example, has two components each of which has two components, and its complexity is 4. In general, the complexity of an idea is the total number of non-null components a t the lowest level. All this seems almost too obvious to mention. The important and interesting problems remain. What we have so far is a method for comparing the simplicity of sequences, sequences of sequences, and so on. But we still cannot compare the simplicity of two ideas if either is an idea of a non-sequence. And we cannot compare two primitive ideas of sequences unless they are fully explicit ideas, revealing exactly the number of components, the number of components of each component, and so on. We have as yet no way of comparing ideas which are-as is normally the case-at all indeterminate on any of these points. Thus we are confronted with two major problems. We must develop a criterion that will enable us to compare the simplicity of any two primitive ideas (1) no matter whether each, or only one, or neither, is an idea of a sequence, and (2) however indeterminate the ideas may be concerning the structure of what the primitive terms denote. Perhaps the first impulse in attacking the former problem is to try to supplement the criterion of sequential length by ad hoc principles covering the comparison of various kinds of non-sequences; for example. of relations to relations, relations to sequences, and primitives that are neither relations nor sequences to
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each other and to sequences and relations. But such a piecemeal process, even if successful, would leave us with a very heterogeneous collection of rules. A better approach is open. By using the concept of the establishment6 of a sequence by a class, we can bring all sequences and non-sequences under a single criterion. Though not every class is a sequence, every class establishes a s e quence; and the definition for the kth component of a sequence gives, when applied to any class, the kth component of the sequence that class establishes. Thus the comparison of classes according to number of components is applicable to sequences and non-sequences alike. I t is a short step to a criterion of complexity that makes no direct reference to sequences, establishment, or number of components. Let us use the term "cardinal subclass" of a class A for the product of A with any cardinal number. Every class has exactly as many non-null cardinal subclasses as there are non-null components in the sequence it establishes. Thus to construe the logical complexity of a class as its numerical variegation-the number of these cardinal subclasses it can be divided into-will give the same numerical result as counting the components. But we now have a criterion independent of my particular definition of sequences and establishment. Later, when we come to the final consideration of the adequacy of our criterion (Section 4))we shall perceive the importance of this independence. For the present we are concerned with seeing how the criterion works in the simplest cases, and then with elaborating it so that it will apply to classes of all types. A class of individuals, since we shall identify individuals with their unitclasses,' is a class consisting of one-membered members; its complexity is 1. I t is, incidentally, a sequence of one non-null component (the logical sum of the class in question) which occupies the first sequential place. Any other class x of which the members are equally-numbered classes of individuals has likewise the complexity 1 ; the established sequence has but one non-null componentthe product or sum of s,as the case may be-which occupies the place corresponding to the cardinal number to which the members of x belong. More generally, any class consisting of classes of k different numbers of individuals has the complexity k. However, ideas of classes of classes of individuals may be indeterminate in varying respects and degrees. Indeterminateness as to which cardinal subclass of a class may be non-null is irrelevant; what counts is how many such subclasses may be non-null. In other words, a class establishing the sequence A,A,a, A,b,A,A,c is as simple for our purposes as a class establishing the sequence a,b,c. But we must allow for ideas which are indeterminate even as to the number of non-null cardinal subclasses. What we have first to recognize is that the complexity of an idea is (1) not a number such that we can infer from the idea, that the class denoted does not have a t least that many non-null cardinal subclasses, and (2) the greatest number satisfying (I), if there is a greatest. The complexity of the idea of the class of all two-membered and all three-membered classes of flyspecks on this page is 2, even though the class may be null. If we Op. cit., p. 152.
Op. cit., p, 151 and footnote 6.
ON THE SIMPLICITYOF IDEAS
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have a primitive idea of a class x having as members eight classes of individuals, but cannot infer from the idea how many members those classes have, 8 is the number satisfying conditions (1) and (2); hence the complexity of the idea is 8. I t is true that the class may be null or that all its members may belong to one cardinal number, but the complexity of the idea, as in the former case, is determined by the number of non-null cardinal subclasses the class may have; it matters not which actually are non-null or whether any are. Briefly, the complexity of an idea is a direct function of the possible numerical variegation of what the primitive term denotes. If the number of non-null cardinal subclasses is not limited in any way by the idea, the class may have an infinite number of them and the complexity-index is " m ". We are most often interested, however, in constructional systems of which the extralogical primitive ideas are all of finite classes, and our discussion in this article will hereafter be confined to such ideas. Since a finite class cannot have an infinite number of non-null cardinal subclasses, such an idea cannot be assigned the index " m ". Yet since, for such an idea, all finite numbers satisfy condition (1) (above), and there is no greatest finite number, condition (2) is here inapplicable. For any finite number n, the class may, compatibly with the idea, have more than n non-null cardinal subclasses. Thus to assign any specific numeral as complexity-index would not be in accord with the essential meaning of complexity as above explained. For the indeterminate complexityindex of such ideas, I shall use the letter "u" to suggest that the number of nonnull cardinal subclasses though finite, is unspecified. To conditions (1) and (2) we add: (3) if there is no greatest among such numbers, as is the case where only the finitude of the class is determinable from the idea, the letter "u" is assigned as complexity-index. That "u" may accordingly be assigned to several ideas such that the denoted classes actually have very different numbers of non-null cardinal subclasses is of course beside the point; for since the simplicity of ideas depends solely upon what we can infer from them concerning the number of such subclasses, obviously two ideas having the index "u" are equally simple. Strictly speaking, "u" does not stand for any number. To say that an idea of a class A has the complexity-index "u", or that A may have u non-null cardinal subclasses, means that A may have (compatibly with that idea) any finite number of them; hence that for any other idea from which we can infer that its denoted class, B, may have some specific finite number of cardinal subclasses but not more, A may have more. But as a result of this purely contextual interpretation, wherever ( 1u 7 9 appears in a set of indices offered for comparison it can be treated arithmetically within that universe of discourse as a constant standing for an unspecified natural number higher than any number specifically named in the set. And with the rigorous contextual interpretation once understood, it will be harmless and often convenient in what follows to speak as if "u" actually stood for such an unspecified number within any such universe of discourse.' That "u" could never be treated as standing for a lower number than pre8 I am indebted to Dr. C. 6. Hempel for valuable criticisms of an earlier version of this and the preceding paragraph.
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scribed above follows inevitably from the explanation of what complexityindices mean and of the kind of ideas to which this index is assigned. Observe furthermore that our requirement df non-triviality would be violated if ideas of unspecified classes were rated more economical than ideas of specified classes; for an idea-of a class x-having a specified complexity-index can always be automatically replaced by another-f the complement of x-having the index 11u ), . Moreover, a spurious economy could always be readily achieved in another way: by omitting from the explanation that constitutes the idea all description of the structure of what the primitive term denotes; while with unspecified classes recognized as less economical, increases in economy are significant because they are achieved through securing, and incorporating in the idea, as much information as to structure as possible. Complexity-indices of ideas of classes of classes of individuals thus run from "1" to "u". Zero does not occur; for if it follows from an idea that all cardinal subclasses of the denoted class are null, the idea is of L A and thus of a definable logical constant, not an extralogical primitive. The complexity-index of a basis is the arithmetical sum of the indices of the primitives comprising the basis. Where the basis contains no ideas of classes of higher type than classes of classes of individuals, the index will always be expressible as a case of the formula :
Either of the coefficients k or kl , but not both, may be zero. Before we proceed to consider ideas of higher types, one question may well be settled. We have seen that the complexity of an idea is equal to the number of places that may be occupied in the established sequences by non-null components. I t makes no difference whether these places are occupied by possibly different or by expressly identical components; an idea of a class that establishes the redundant sequence a,b,a is as complex by our criterion as the idea of a class that establishes the non-redundant sequence a,b,c. The criterion could very easily be altered so that the idea of a redundant class would have the same complexity as the idea of the non-redundant class resulting from the elimination of all repetitions. Many words might be expended in debating whether such an alteration is recommended by intuition or not, but actually this question is of little importance since in practice no one would ever take a redundant idea as primitive. As we are concerned with complexity only insofar as it is relevant to the economy of extralogical bases, we may simply exclude redundant ideasg from our universe of discourse on the ground that they are illegitimate as primitives. Convenience alone then decides in favor of leaving our criterion in its present simpler form. The logical type consisting of classes of classes of classes of individuals is especially interesting since relations, and sequences of sequences, first come in a t this level. To begin with, let us consider a primitive idea P of the ordered pair 9 While I do not define the class of redundant ideas precisely, i t includes not only ideas of redundant sequences, but also other ideas, such as those of redundant relations-e.g.,
i.ii(zRy.z=z).
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(a,b),(c,d,e), where the five sub-components of the two major components are classes of individuals. The pair (a,b),(c,d,e)is then a class that may have two non-null cardinal subclasses: H , the class of one-membered classes bearing the requisite1' inclusion-relationships to a,b; and Ii. the class of two-membered classes bearing such relationships to c,d,e. P thus has what we may call a primary complexity of 2, though obviously the total complexity of P is much greater. Now neither H nor Ii can have more than one non-null cardinal subclass, since each is included in a cardinal number. Nevertheless, the classes belonging to the several one-membered members of H may some of them have one member and some have two; for the members of H are the unit-classes of the one-membered classes bearing certain inclusion relationships to a, and the two-membered classes bearing such relationships to b. Thus the logical sum of H may have two non-null cardinal subclasses; we express this by saying that the secondary complexity of H is 2. Since the logical surn of each of the cardinal subclasses of H is identical with that subclass, there is no further complexity a t lower levels. By similar reasoning, we find that the secondary complexity of K is 3. The total complexity of P , found by adding the secondary complexities of its possibly non-null cardinal subclasses, is 5 . The procedure illustrated may be more generally described as follows: to determine the complexity of any idea of a class C of classes of classes of individuals, first find how many non-null cardinal subclasses C may have; then find, and add together, the numbers of non-null cardinal subclasses the logical sums of these possibly non-null cardinal subclasses of C may have. This arithmetical sum is the total complexity of the idea. The index "u" may appear a t either or both levels. (i) If we have an idea of a dyadic asymmetrical relation of individuals, R, we commonly have a class of which the numerical variegation is unspecified by the idea; that is, the number of members in the pairs belonging to R are indeterminable from the idea. The primary complexity-index of R is thus "u". Since, however, we know that the members of R are pairs of individuals, we know that the logical sum of each of the u non-null cardinal subclasses that R may have consists solely of one-membered and two-membered classes. Thus R may have u non-null cardinal subclasses each of which has a secondary complexity of 2. The total index of the idea of R is thus "2u". (ii) In an idea of an ordered pair P of classes of classes of individuals, "u" may appear a t the lower level. P has a primary complexity of 2; but since we presume nothing is specified concerning the cardinal numbers of the members of the members of P , the logical sum of each of the two possible cardinal subclasses of P has the complexity-index "u". Hence the total complexity-index of the idea of P is "2u". Observe that an n-ad of indeterminate classes has the same value as an indeterminate class of n-ads. (iii) Finally, if we have an idea of an indeterminate class Q of indeterminate classes of classes of individuals, then Q may have u non-null cardinal subclasses, the logical sum of each of which likewise may have u such subclasses; thus the total complexity-index of the idea of Q is "u2". 1 0 1.e. the relationships called for under my earlier cited (footnote 4, above) definition of sequences.
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The complexity-index of any basis consisting of primitives of no higher type than classes of classes of classes of individuals will be an instance of the formula
Any two of the coefficients may be zero. The procedure described above can obviously be so extended that it will be applicable to ideas of classes of any type. We first find how many non-null cardinal subclasses the class may have, then find how many the logical sums of those subclasses may have in turn, and so on; the total number a t the level a t which the logical sum of each subclass is identical with that subclass, is the total complexity of the primitive. The complexity-index of any basis consisting of ideas of specified types will be an instance of the formula
So long as a t least one coefficient is greater than zero, any number of others may equal zero. If we want to encompass ideas which do not disclose even the logical type of the class denoted-as we may if we adopt an underlying logic that places no type restriction upon meaningfulness-we need merely extend this formula by introducing the uth power of u. The completely general form of index for any bmis then runs:
Since we have explained how to determine the value of each coefficient in any given case, we thus have a complete method for Jinding the complexity-index of any basis.
3. The arithmetic of complexity. Since, however, "u" stands for no specified number, finding the complexity-indices for two bases may not always make it immediately evident which basis is the more complex. Some cases present little difficulty if we remember that "u" always in effect stands for some natural number and that certain simple theorems in the arithmetic of natural numbers will therefore apply; e.g. if x < y, then x un < y unand xun < yun. It is easy to see that, whatever natural number "u" may stand for in any context, (i) two indices express equal complexity if occurrences of "u" having identical exponents in the two indices also have identical coefficients, and (ii) an index A expresses a greater degree of complexity than an index B if no coefficient of A is less, while a t least one is greater, than its corresponding coefficient in B. The more difficult cases are those where some coefficient of one index is greater, while some other is less, than the corresponding coefficient in a second index. In such cases, which index expresses greater complexity is not a matter entirely independent of the number "u" stands for. Eor example, 3 2u is greater than 1 3u if u is 1, but less if u is 3 or more; and 3u is less than u2 if u is 2, but greater if u is 4 or more. Nevertheless, all such cases can be decided without any further information than we already have. We have seen that "u" may be treated as standing in any set of indices for a number higher than
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any specific number represented in that set; thus when we are comparing indices of a given set, we know that "u" stands for a number higher than any coefficient appearing in that set. This principle alone is enough, together with the arithmetic of natural numbers, to enable us to compare any two indices. The following theorems of that arithmetic are relevant: (i) If x < y, then xy < y2. xy < Y2. (ii) If z < y and x < y, then z ty2 < (t l)y2. (iii) If x < y and z < y and t < y, then z xy And more generally: (iv) If each of the coefficients k . . . km is less than t and less than x, and . . kmxn< txn. t < x, then k klx k2x2 From the principle that "u" always stands for a number which, though still unspecified, is higher than any specified coefficient in the sphere of comparison, arithmetic alone thus leads us to the following general theorem that enables us to determine which, if either, of any two indices expresses greater complexity than the other: Of two indices, that one expresses greater complexity which has the higher coeficient for the highest power of u for which the two indices have unequal coefiients. Naturally, if every pair of corresponding coefficients is an equality, the two indices are equal. To illustrate the rule in application: we can now determine that 1 3u is greater than 3 2u, and that u2is greater than 3u. Again, we know immediately when confronted with the complicated set of indices: (a) 7582u7 (b) 17 8u 23u2 9u3 u4 30u6 2u8 (c) 4 3~ 1 5 ~ ' u3 2u4 3ou5 2 2 (d) 1 2~~ 3~~ (el u9, that a < b < c < d < e. We need only find in each case which is the highest power of u for which the two indices have unequal coefficients; the higher index is the one with the higher coefficient for that power of u. Thus we have a complete method for comparing the complexity of any two bases; and as earlier noted, the least complex among alternative bases for a system is the most economical.
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+ +
4. Consequences. The classes" which my earlier cited definition identifies with n-adic sequences of m-type12 classes will have the complexity n.um-', and those which it identifies with n-adic relations of m-type classes, the value n.um. l1 For convenience, I often omit the words "ideas of" or "idea of," but i t 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 t h a t 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. l2 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, t o type 1.
118
NELSON GOODMAN
The question arises what the complexity of sequences and relations as defined under other theories will prove to be. n'aturally we shall not expect that the several classes which divergent theories identify with a given sequence, for example, will all prove to be equally complex. Certainly there are more and ess complex ways of defining a sequence, since virtually the only requirement is that the components be unambiguously recoverable from the definition. But it is worthwhile examining how our criterion operates in application to some other theories, with the purpose of testing its intuitive adequacy further and a t the same time finding what classes, according to the criterion, are the simplest that we may identify with any given sequence or relation. Let us begin with the well-known definition of sequences devised by Kuratowski.13 Observe especially that his definition also depends on numerical variegation, which is the very principle upon which our criterion of complexity is based; for under his definition of the ordered pair x,y as LLX
u r(1x u LY),
if x is not identical with y, then y is distinguished as the second component by the fact that it belongs only to the two-membered class belonging to the couple. Furthermore, a couple of individuals so defined has a complexity of 2; for (i) since it has not more than two members it can have not more than two nonnull cardinal subclasses, (ii) the logical sum of each of these is a class of individuals and hence identical with its own logical sum, (iii) hence the total complexity is 2 times 1. Kuratowski did not extend his theory to provide direct definitions for longer sequences, but a seemingly natural method of extension was suggested in my article on sequences; this would, for example, define the ordered triple z,y,z a s
However, if the components of the triple are individuals, this class proves to have a complexity, by our criterion, not of 3 but of 4. Confidence in our criterion would lead us to suspect that the suggested definition is unnecessarily complex; and, indeed, further consideration reveals that the member L ( U u cy) is superfluous. The ordered triple is adequately defined by the class i r a u L(L(LX u ry) u LLZ),
which, where the components are individuals, has a complexity of 3. T h e ordered quadruple x,y,z,w becomes
which gives a quadruple of individuals a complexity of 4; and so on. Thus an n-ad of individuals under Kuratowski's definition and such extensions will have the same complexity as under my definition-namely, n. If, however, the components of an ordered couple are classes, the Kuratowski 1 3 C . Kuratowski, Sur l a notion de l'ordre dans l a thdorie des ensembles, Fundamenfa mafhematicae, vol. 2 (1921), pp. 161-171.
ON THE SIMPLICITY
119
OF IDEAS
definition identifies the couple with a more complex class than mine,14 a class having a conlplexity of 3 ; and in general a ICuratowski n-ad of m-type l).um-' as against n.um-' under my theory. classes has the complexity (n I
