SequencesAuthor(s): Nelson GoodmanSource: The Journal of Symbolic Logic, Vol. 6, No. 4 (Dec., 1941), pp. 150-153Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2267107 .

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TDz JOURNAL OF Symiouc LOGIC Volume 6, Number 4, December 1941

SEQUENCES NELSON GOODMAN

1. Definition of sequences. The problem of defining ordered couples on the basis of class theory, and thus reducing relation theory to class theory, was first solved by Wiener.' By Kuratowski's somewhat simpler device,2 the ordered couple x,y is defined as

&X U LXU &y).

The principle involved may be extended for sequences of any finite length. The ordered triple x,y,z would, for example, become:

&&&X U LL(& U ty) U L(L(t U Ly) u HLZ).

By this procedure any sequence of length n would be defined as a class n types higher than the components of the sequence; and there appears to be no alterna- tive extension of Kuratowski's device that would escape this progressive raising of type.' Thus a new method of defining sequences that would construe them as classes only one type higher than their components would effect a considerable simplification in the theory of sequences and relations. My purpose in this paper is to propose and explain such a definition and some of its consequences.4

We may define an ordered couple as the class having as members the unit subclasses of the first component and the two-membered classes which either include the second component or include it if it is not null." This mode of definition can be easily extended to serve for sequences of any length. The ordered triple x,y,z, for example, becomes the class having as members all unit subclasses of x, all two-membered classes that are included in y or that include

Received November 3, 1941. 1 N. Wiener, A simplification of the logic of relations, Proceedings of the Cambridge

Philosophical Society, vol. 17 (1914), pp. 387-390. 2 C. Kuratowski, Sur la notion de l'ordre dans la thtorie des ensembles, Fundamenta

mathematicae, vol. 2 (1921), pp. 161-171. 3That defining the ordered triple x,y,z as stx u t(tx u ty) u t(tx u ty u tz) would fail

to distinguish the sequences x,x,y and x,y,x and x,y,y from each other has been pointed out by W. V. Quine.

4 I hope in a later paper to deal with certain consequences for the measurement of the logical economy of bases for extralogical systems.

6 In Mathematical logic (New York 1940, p. 202, fine type), Quine suggested an alter- native to the definition of ordered couple he used in his system, but later discovered that this alternative definition would fail to distinguish x,y from y,x when xcy. In a later discussion between us he proposed the first satisfactory definition of the ordered couple as a class only one type higher than its components. However his device-which defines x,y as the class of the unit subclasses of x and the complements of unit subclasses of y- is obviously not readily extensible to longer sequences. It has the possible further dis- advantages, as compared with the definition I suggest above, (1) that in an infinite universe an ordered couple of finite components will always have infinite classes among its members, and (2) that the couple A,A will not be identified with the null class.

150

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SEQUENCES 151

y if y is not null, and all three-membered classes that are included in z or that include z if z is not null. In general a sequence x1,x2, -. ,xn becomes

z{(zel: zCxj [. V .xlCz . x1$A]) v (ze2: ZCX2 . V . X2CZ * X2$5A) v

* v (zen : ZCXn v v. Xn Cz xn A)J.

The clause in square brackets is vacuous but included to preserve parallelism throughout. Without the clauses "x2- A", * , "xn# A" conjoined respectively to "x2Cz", *--, "xnCz", all k-membered classes would belong to a sequence of which the kth component were null; but likewise all k-membered classes belong to a sequence of which the kth component is universal. Hence the case of a null component would not be distinct from that of a universal component. The clauses mentioned effect the distinction by making null the class of k-membered members of a sequence of which the kth component is null.

This method of definition is applicable to sequences of any finite length. A sequence is considered to have the null class in all positions where no non-null class appears, and to be of length n if and only if its nth component is the last that is not null. Thus the sequence A,x,A,A,y is of length 5 and identical with A,x,A,A,y,A,A,...; the couple A,A is of length 0 and identical with every other sequence of which all components are null. Sequences of length 0 become iden- tified with the null class.

In order that the proposed method shall operate successfully, two requirements must be fulfilled. The number of elements in the universe that belong to the next lower type than the components of the sequence in question must exceed by one the number of those components. In other words, the method of defini- tion works for any sequence of n components in a universe which contains n+ 1 elements belonging to the type of the members of the components of the sequence. This brings us to the second requirement: that the components of the sequence must be classes-that is, must be of type 1 or a higher type. But for any system of logic which identifies individuals with their unit classes,6 as is quite feasible, this is no restriction at all.

Every sequence becomes not more than one type higher than its components since it is identified with a class of classes which include or are included in, and are thus of the same type as, those components. I have elsewhere7 pointed out that, quite apart from any new definition of sequences, the Wiener-Kuratowski theory of relations could be modified by construing symmetrical relations as classes of unordered classes, so that an n-adic symmetrical relation-as a sub- class of the cardinal number n-would be only two types higher than its relata

6 This is done in Quine's Mathematical logic, but is compatible with systems involving type theory if we merely modify type theory by regarding individuals, and the null class as well, as belonging to all types. Every class of two or more individuals still belongs to type 1 only; every class having as members some classes that belong only to type 1 belongs to type 2 only, and so on. Incidentally, adoption of this point of view makes a sequence of length 1, provided its first component is an individual or class of individuals, identical under our definition with that component. Further, no sequence of individuals is of higher type than its components; it is for this reason that I write "not more than" rather than "only" in the first and last sentences of the paragraph following.

7 A study of qualities (Doctoral thesis, Harvard Library, 1941), p. 44.

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152 NELSON GOODMAN

while an n-adic asymmetrical relation-as a class of sequences of length n- would still be n+1 types higher than its relata. With the above method of defining sequences, we can now treat all relations, symmetrical and asymmetrical and of any degree, as classes not more than two types higher than their relata.

If a component xk of a sequence has k or fewer members, then each of the k-membered classes belonging to the sequence will include xk; for obviously no class can be included in a class of fewer members. Furthermore, since there are at least k+ 1 things, xk will in this case be the only class contained in all k-mem- bered members of the sequence; i.e., Xk will be the product of all such classes.8

On the other hand, if Xk has more than k members, all the k-membered mem- bers of the sequence will be included in k; for obviously no class can include a class having more members. Thus in this case, Xk will be the sum of all such classes. And the product of these classes will be null as is the product of every class of all the n-membered classes included in a class of n+m members (where m ? 1).

In summary, any component Xk of a sequence is the product of the k-membered members of the sequence, or the sum of all such members if the product is null. Symbolically, any component xk is

z(z e p'(knP) . v . p'(knP) = A . z e s'(knP)). The principles underlying our method of defining sequences and their com-

ponents enable us now to replace our schemata by formal definitions. A se- quence is any class of classes x such that for each cardinal number k, if the product of xnk is not null then xnk is the largest of the subclasses of k that have the same product as xnk, or if that product is null then xnk is the largest of the subclasses of k that have the same sum as xnk. Symbolically, Seq =df X(Y) . p'(x n NC'y) $ A . p'(x n NC'y) C y . v .

p'(x n NC'y) = A .y C s'(x n NC'y): yex.

Incidentally this leaves a place open for "infinite sequences" in some sense or other, but we confine our attention here to finite cases.

The kth component of any sequence is then of course defined as follows?: Xk = & 2 { z e p'(xnk) . v . p'(xnk) = A . z e s'(xnk)).

2. Establishment of sequences by classes. Obviously not every class will be a sequence under the definition given. But every class'0 will be related to some

8 Throughout it should be remembered that the components of a sequence may be quite different from the classes belonging to it as members.

9 Since seeing the above definitions, Quine has suggested that sequences might be handled in an alternative way, defining x1,x2,** ,xn where n>2 as yz(yex, . zel . V . YeX2. * V . v * V - . v . yex .zen). A component Xk of a sequence Q would then be defined as Q"k. The simplicity of this method is rather offset by the fact that it rests upon prior definition of the ordered couple, so that two steps are involved. It is questionable whether the formal definitions corresponding to these schemata would be notably simpler than those I give above. Moreover, the interesting consequences concerning the relation of all classes to sequences (dealt with in ?2 of the present paper) would not be forthcoming.

10 Were it not for the identification of individuals with their unit classes, we should have to read "class of classes" here and in similar contexts. However, when this identifica-

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SEQUENCES 153

sequence in the following way: every component Xk of the sequence x will be the product of xnk, or the sum if the product is null. A class is said to "estab- lish" that sequence to which it is so related; such a class need not, as the sequence must, have as members all the k-membered classes which are included in xk, or include Xk if xk is not null. But application of the above definition (?1) of Xk

to any class will give the kth component of the sequence which the class estab- lishes. The many-one relation of establishment is formally defined as follows:

Est =df xy(n) :. n e NC. D p'(xnn) # A . p'(xnn) = p'(ynn) . v .

p'(xnn) =A . s'(xnn) = s'(ynn).

Every class establishes some sequence; and every sequence is established by some class-by itself, indeed, among others. The sequence established by a class is of length n if and only if the largest members of the class have n members. A class might be said to have the length and the components- often quite different from its size and its members, respectively-of the sequence it establishes.

Ordinarily, classes do not establish sequences of their own members. Even classes that are sequences usually have components other than their members. Thus the class x, having as members ty and LZ and all two-membered classes to which w belongs, is the sequence

ty U tz, &W,

of which no component is a member of x. Yet certain classes do establish sequences composed of their own members.

We call a class "self-ordered" if every member of it is a component of the se- quence .it establishes, and every component of the sequence is either null or a member of that class. The formal definition of self-ordered classes is simplified by the fact that a class of classes is self-ordered if and only if it has as a member no more than one member of each cardinal number. For if a class has two or more k-membered members, then the kth member, Xk , of the sequence established by the class is identical with no one of these members but with their product or sum; while if a class has but one k-membered member, then Xk is identical with that member. Accordingly:

SO = df x(Y) (Z) (YeX . ZeX . y sm z. D . y=z).

While not all sequences are self-ordered classes, a self-ordered class not only establishes a certain sequence of its members but is that sequence. For if a class has but one k-membered member no larger class of k-membered classes has the same product as the single k-membered class with itself; and if the class has no k-membered member, no larger class has the same sum.

BOSTON, MASSACHUSETTS

tion is made (as outlined in footnote 6 above), even a class of individuals is a class of classes, i.e., of unit classes.

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