From Peirce to Skolem - A Neglected Chapter in the History of Logic

Acknowledgments

This book began ten years ago as a master 's thesis at the University of Chicago, unde r the direction of W. W. Tait and William A. Howard. During the time that I was beginning my research, I learned much about modern mathematical logic from Ted Slaman and Robert Soare, and about category theory fl'om Saunders Mac Lane, at the University of Chicago.

Else M. Barth of the University of Groningen in t roduced me to scholars in Europe who were interested in the history of logic and, most especially, r e c o m m e n d e d me to Dagfinn Follesdal andJens-Erik Fenstad at the University of Oslo. Through the help of these splendid scholars, and the late Burton Dreben of Harvard University, I received a doctoral degree from the University of Oslo for an earlier version of the present

work. Christian Thiel and Volker Peckhaus of the University of Erlangen

deserve much thanks fox their very helpful and thorough answers to my questions about Schr6der and L6wenheim. Nathan Houser, Director of the Peirce Edition Project, always found the time to assist me in my quests for Peirce lnanuscripts that were difficult to locate. I am grateful to Marcus Schaefer for comments on my chapters on Peirce, and to Todd Trimble fox his help recasting Peirce's early algebraic theories. Thanks are due to E W. Lawvere and Sir Michael D u m m e t t for their comments on an earlier draft of this work.

I extend my deepest thanks to Stuart A. Kurtz fox" lively and stimulating

research sessions over the course of many years and for his pat ient help in decoding the source materials for this book. I am also grateful to the Compute r Science Depar tmen t at the University of Chicago for providing me with the scholarly resources for research and writing this book.

1 am most indebted and deeply grateful to Anil Nerode for directing my study of logic and its history, fox" his incisive comment s on my manuscripts in all their various stages, and for his exper t advice on the texts.

vii

viii A C K N O W I . E D G M E N T S

I am also proud to acknowledge the help and guidance I have received through all phases of this project fi-oln Saunders Mac Lane, who initially suggested the topic and whose interest in my work and personal en- couragement have been unrelenting.

I am grateful to the special collections librarian at the Johns Hopkins University for help in locating reference material in the Peirce archives, to the special collections librarians at Harvard University and MIT for access to Norbert Wiener's unpublished doctoral dissertation, to the Peirce Edition Project and the University of Chicago microfilms libra> ians for providing access to facsimiles of Peirce's notes and manuscripts, and to the University of Chicago Press. I am indebted to Elizabeth Huyck for her help in preparing this book for publication, and to Suzanne Kuwatsu, Don Reneau, and, again, Marcus Schaefer for their work on the translations of Schr6der's writings that appear as appendices to this book. I am particularly grateful to John Muenning for his help in type- setting this book.

Finally, I would most especially like to thank my mother for her patience, support, and love.

Geraldine Brady Chicago, June 2000

Introduction

This b o o k is an a c c o u n t o f the i m p o r t a n t i n f l u e n c e on the d e v e l o p m e n t

o f m a t h e m a t i c a l logic o f Char l e s S. Pe i rce and his s t u d e n t O. H. Mitchel l ,

t h r o u g h the work of Erns t Sch r6de r , L e o p o l d L 6 w e n h e i m , a n d T h o r a l f

Sko lem. As far as we know, this b o o k is the first work d e l i n e a t i n g this

l ine o f i n f l u e n c e on m o d e r n m a t h e m a t i c a l logic.

M o d e r n m o d e l t h e o r y b e g a n with the s emina l p a p e r s o f L 6 w e n h e i m

(1915) "On possibi l i t ies in the ca lcu lus of relat ives" a n d S k o l e m (1923)

" S o m e r e m a r k s on a x i o m a t i z e d set theory." T h e y s h o w e d tha t in first-

o r d e r logic, if a s t a t e m e n t has an inf in i te m o d e l , it also has a m o d e l

with c o u n t a b l e d o m a i n . T h e y o b s e r v e d tha t s e c o n d - o r d e r logic fails to

have this p r o p e r t y ; witness the ax ioms for the real n u m b e r field. T h e i r

p a p e r s focused the a t t e n t i o n o f a g rowing n u m b e r o f logic ians , s t a r t ing

with Kur t G 6 d e l a n d J a c q u e s H e r b r a n d , on m o d e l s o f t i r s t -o rde r the-

ories. ~ This b e c a m e the m a i n p r e o c c u p a t i o n o f m o d e l t h e o r y a n d a

la rge c o m p o n e n t o f m a t h e m a t i c a l logic as it d e v e l o p e d over the rest o f

the twen t i e th century . In add i t i on , the work of H e r b r a n d , b a s e d on the

n o t i o n of S k o l e m func t ion , b e c a m e , t h r o u g h J. Alan R o b i n s o n , the m a i n

basis of systems of a u t o m a t e d r ea son ing .

A care fu l e x a m i n a t i o n o f the c o n t r i b u t i o n s o f Pe i rce , Mi tche l l ,

Sch r6de r , a n d L 6 w e n h e i m sheds l igh t on several ques t i ons : H o w did

f i r s t -order logic as we know it deve lop? W h a t are the real c o n t r i b u t i o n s

~We do not discuss here the Frege-Russell-l-Iilbert tradition leading to first-order logic and G6dei, since this development has many excellent treatments in the literature already, such as the beautiful book of the late .lean van Heijenoort, From Frege to GiMeL Van Hei- jenoort's book treats Frege, L6wenheim, and Skolem, but does not cover either Peirce's or Schr6der's work, which led to L6wenheim's paper. This omission is also present in the historical papers of other otherwise very well-read logicians. There are masterful accounts of tile seminal papers of LSwenheim and Skolem in the late Burton Dreben's introduction to G6del's thesis in Collected Works oJKurt (,iidel and in the late Hao Wang's introduction to Skolem's Selected Works in Logic. But Peirce and Schr6dcr get no attention.

2 I N T R O D U C T I O N

of Peirce, Mitchell, and Schr6der, over and above the bet ter known contr ibut ions of Gottlob Frege, Ber t rand Russell, and David Hilbert?

As a result of this investigation we conclude that, absent new historical evidence, L6wenheim's and Skolem's work on what is now known as the downward L6wenheim-Skolem theo rem developed directly f rom Schr6der ' s Algebra der Logik, which was itself an avowed e labora t ion of the work of the American logician Charles S. Peirce and his s tudent O. H. Mitchell. We have been unable to detect any direct inf luence of Frege, Russell, or Hilbert on the deve lopmen t of L6wenhe im and Sko- lem's seminal work, contrary to the commonly held percept ion. This, in spite of the fact that Frege has und ispu ted priority for the discovery and formula t ion of first-order logic.

This raises yet o ther intr iguing questions. Why were the contr ibut ions of Peirce and Schr6der neglected by later authors? Was it because Peirce publ i shed in American journals that were not easily available to Euro- peans? Was it because Schr6der had a verbose and somet imes obscure style as a writer? Was it because the logical notat ions used by Peirce and Schr6der were simply less readable than those of Frege? After reading this book, the reader should be able to form his or her own opinions.

The re is clear evidence that G6del, at the time he wrote his thesis in 1929, in which he prove d the completeness theo rem for the first-order predicate calculus, was directly acquain ted with.at least the special terminology used by L6wenheim. In the open ing paragraph of his thesis, G6del uses the term "Ziihlaussage," in def ining completeness , which he t hough t was L6wenheim's:

The main object of the following investigations is the proof of the completeness of the axiom system for what is called the restricted functional calculus, namely the system given in Principia Mathematica, Part I, Numbers 1 and 10, and, in a similar way, in Hilbert-Ackermann, Grundziige der theoretischen Log~k .... III,w 5. Here "completeness" is to mean that every valid formula expressible in the restricted functional calculus (a valid Ziihlaussage, as L6wenheim would say) can be derived from the axioms by means of a finite sequence of formal inferences. The assertion can easily be seen to be equivalent to the following: Every consistent axiom system consisting of only Ziihlaussagen has a realization. (Here "consistent" means that no contradiction can be derived by means of finitely many formal inferences.) (G6del 1929, pp. 60-61 )2

Ziihlaussage can be translated as "first-order statement." In his 1915 pa- pel, L6wenhe im defines Ziihlausdruck (i.e., "first-order expression") as

In Collected Works of Kurt G6del, vol. 1 (Feferman et al. 1986). Throughout this work, page numbers given are for English translations and modern reprints, where available.

FROM PEIRCE TO SKOLEM

"a relative expression in which every I2 and II ranges over the subscripts, that is, over the individuals of 11 (in other words, none ranges over the relatives)," and which, of course, recurs in the s ta tement of his famous theorem: "If the domain is at least denumerably infinite, it is no longer the case that a first-order fleeing equation is satisfied for arbitrary values of the relative coefficients."

It seems clear that G6del read at least the statements of theorems and definitions in L6wenheim's paper, and in Skolem's 1920 paper as well. In the published version of his thesis (1930), G6del cites Skolem (1920) explicitly:

An analogous procedure was used by Skolem (1920) in proving L6w- enheim's theorem. (G6del 1930, pp. 108-109)

It also is fairly certain that G6del did not know Skolem's later proofs of L6wenheim's theorem, which intriguingly looked just like G6del's completeness proof. In the 1960s, Jean van Hei jenoor t and Hao Wang noticed the similarity of Skolem's 1923 K6nig's lemma-style proof and G6del's 1929 completeness theorem proof and asked G6del about it. 3 Van Hei jenoort apparently asked G6del why he did not cite Skolem (1923) in his thesis, and G6del replied (in 1963) that he was sure he did not know of Skolem's paper when he w r o t e his dissertation; otherwise, he would have quoted it, since, he says, it is much closer to his work than Skolem's 1920 paper, which he did quote. In 1967 G6del wrote, in a letter to Wang:

The completeness theorem, mathematically, is indeed an almost trivial consequence of Skolem 1923. However, the fact is that, at that time, nobody (including Skolem himself) drew thisconclusion (neither from Skolem 1923 nor, as I did, from similar considerations of his own). (Dreben and van Heijenoort 1986, p. 52)

In G6del's 1930 paper proving the completeness theorem, the statement "Every consistent statement has a countable model" replaces the earlier Skolem-L6wenheim formulation "Every s tatement with an infinite model has a countable model." The tree constructions of Skolem- L6wenheim were justified semantically; those of G6del are identical, but are justified syntactically.

Skolem produced several proofs of the L6wenheim theorem, the second not requiring the axiom of choice and close in spirit to L6wen- heim's original proof. Skolem studied at G6tt ingen in the winter of 1915-1916. We do not know whether he first learned of L6wenheim's

"~ This is r eco rded in Dreben and van He i j enoor t ' s i n t r o d u c t i o n to G&del's thesis, in Collected Works of Kurt G6del, vol. 1, pp. 51-52.

4 INTRODUCTION

paper at G6ttingen, or whether he simply read it in Mathematische An- nalen. Skolem's first paper re-proving L6wenheim's theorem (1920) in- t roduced the notion of first-order proposit ion explicitly as a replacement for L6wenheim's first-order equations and dropped the relative sum and product notation that L6wenheim had adopted from Schr6der, and which originated with Peirce. Skolem's 1920 proof was thus a simplified version of L6wenheim's original proof using algebraic notions and the axiom of choice. Skolem did not claim that L6wenheim's original proof was wrong or incomplete; he only said he was giving a simpler and clearer proof.

In 1923, Skolem introduced formal function symbols and used terms in these symbols and associated trees and a K6nig's lemma-style argument to give a second proof of L6wenheim's theorem. This proof avoided the axiom of choice. Skolem's second proof has the same root as Herbrand ' s later theorem and G6del's completeness theorem.

Skolem's 1923 proof also has the same "gap" as L6wenheim's original proof, namely, an application of K6nig's infinity lemma is needed and is absent. Yet another proof Skolem gave of the L6wenheim theorem in a 1929 paper fills this gap. (How to fill it may well have been obvious to all authors concerned, since the proof is about the same as the proof of the Bolzano-Weierstrass theorem, which every rigorous mathematician has known since the time of Weierstrass.) .

. ,

L6wenheim's seminal paper "On possibilities in the calculus of relatives" (1915) proves that if a first-order formula, as expressed in Schr6der 's relational language, has an infinite model, then it has a countable model. L6wenheim's paper was written in the language of the calculus of relatives; its choice of problems and method of solution are natural extensions of material in volume 3 of Ernst Schr6der 's Vor- lesungen iiber die Algebra der Logik (1895). The L6wenheim language of relatives, infinite sequences with subscripts, condensed relatives, and fleeing subscripts can hardly be deciphered without a careful reading of Schr6der 's volume 3, and L6wenheim's proof uses Schr6der 's notation for functions (subscripts with subscripts), used by no one else as far as we can determine.

L6wenheim's theorem was part of his investigation of the expressiveness of the calculus of relatives, the need to ensure that the mathematical system is capable of expressing everything that is involved in a logical argument . L6wenheim proved there was an expressive hierarchy in the calculus of relatives: the first-order fragment of the calculus of relatives can say more than the fragment of the calculus of relatives restricted to the relative operations, and the full calculus of relatives, with quantification over relations, can say still more than the first-order fragment.

Schr6der, on the other hand, used, and largely developed, the calculus of relatives chiefly as a language of and foundation for logic and math-

F R O M P E I R C E T O S K O L E M

ematics. Schr6der gave one of the first expositions of abstract algebraic structures in the form of a very extensive axiomatic deve lopment of lattices, based on both order and algebraic operations, in the first two volumes of his Algebra der Logik. He made substantial investigations into a second-order theory of relatives, which 1.6wenheim, even in 1940, proposed as an alternative to set theory as a foundat ion of mathematics. In Schr6der 's Algebra der Logik one finds for the first time an extensive discussion of the notion of solving [Aufl6sung] a relational equat ion as a generalization of el imination theory in commutat ive algebra. This amounts to in t roducing a relation symbol that acts like a Skolem function and symbolically solves the equation as a function of its parameters . Schr6der then used sequences of prenex universal and existential quantifiers written as algebraic sums and products. Thus, the Skolem function technique itself can be seen as a direct descendent of Schr6der ' s me thod for al ternating quantifiers. His hundreds of individually proved relational identities were also the starting point for Alfred Tarski's theory of relation algebras (1941), in which a few axioms give all these identities. Schr6der 's development of mathematics in the higher order theory of relations is thus the intellectual predecessor of Tarski's logic without variables, which does give an alternate foundat ion for mathematics, as L6wenheim had hoped it would.

Schr6der himself was not a disciple of or seriously inf luenced by Frege, while his work precedes that of both Russell and Hilbert. His research program was explicitly an extension of the calculus of relatives and the theory of quantifiers proposed by the American mathemat ic ian and logician Charles S. Peirce. Peirce came from an algebraic tradition, through the int luence of his father, the great American algebraist, Ben- j amin Peirce, the author of the pioneer ing work Linear Associative Al- gebras (1870). The Peirces' algebra is not the algebra of logic, but the algebra of linear transformations, which is what associative algebras are about. 'i

Charles S. Peirce, building on work of Augustus De Morgan on relatives and of his father on linear associative algebras, was the first to develop a systematic algebra of binary relations based on the Boolean operat ions and the relative operat ions of relative product , relative sum, and converse, to which he added a theory of relations of all arities. In particular, he proposed to develop a calculus of relatives that was an extension of Boole's calculus that would accommodate quantification. In Peirce's earliest version of the calculus of relations (1870), existential and universal quantification are expressed by relational operat ions in the system and not as separate objects; universal quantification is ex-

'~ Benjamin Peirce's work is all abstraction of Arthur Cayley's matrix algebra, possibly earlier than Cayley's first paper on the algebra of matrices, which appeared in 1858.

6 I N T R O D U C T I O N

pressed by the exponential, and existential quantification is expressed by relative product, Peirce was able to represent mixed quantifier expressions in this system by combining terms that included exponentials. All this was done and published nine years before Frege's Begriffsschrift.

Ten years after his initial paper on the calculus of relatives was published in 1870, Peirce developed a system of propositional logic based on implication and negation that essentially anticipates the main features of modern systems of natural deduction and sequent calculus. Within this system, he articulated an early version of introduction and elimination rules. Peirce put a great emphasis on "illation" (deduction) and on implication as an operation arising from illation, as being more basic than identity. He emphasized that a partial order is involved and anticipated Dag Prawitz's view of natural deduction. He thus developed propositional logic as a kind of lattice theory almost twenty years before Dedekind introduced lattices as mathematical objects as such.

Three years later, in 1883, one of Peirce's students, O. H. Mitchell, developed a rudimentary system for quantification, limited to a theory of quantified propositional functions with two prenex quantifiers. In the same year, inspired by Mitchell, Peirce introduced quantifiers as operations on propositional functions over a specific domain and part of the semantics of first-order logic for prenex formulas over this domain. This direction of research culminated two years later in Peirce's system of first-order logic, which is expressively equivalent to our modern-day first-order logic with functions.

There is today a commonplace misconception that since Frege was the first to capture first-order logic, therefore L6wenheim's work must have s temmed from Frege, possibly through Russell or Hilbert. But this is not so. In fact, as we will show, the central ideas of what we now call first-order logic were fully implicit in the works of Schr6der and Peirce from which L6wenheim drew his chief inspiration, al though couched in a now obscure notational form.

Although the most famous foundationalist of the early part of the twentieth century, Bertrand Russell, makes almost no ment ion of the work of Peirce, his impact is clear. Alonzo Church, acknowledged to be the best-read person of his time on the history of logic, judged Peirce to have had a t remendous technical impact on mathematical logic: Church credits him with the introduction of quantifiers, the Sheffer stroke, normal form, prenex form, and equality in second-order logic. Church is, of course, meticulous. Nonetheless, it is still true that in the standard references used today, Peirce is neglected.

What about Schr6der, equally absent from the Principia Mathematica of Whitehead and Russell? One reason that Schr6der's contribution may have been neglected is a distaste for his forests of identities in the


ca lcu lus o f re la t ions , which were n o t r e d u c e d to a smal l set o f funda -

m e n t a l ones , as Tarski d id later.

However , n o t a t i o n a l c o m p l e x i t y a l o n e does n o t necessa r i ly e x p l a i n

his neg lec t . F r e g e ' s c o n c e p t u a l n o t a t i o n a n d his Grundgesetze are o f t en

equa l ly u n r e a d a b l e , as is W h i t e h e a d a n d Russel l ' s Principia Mathematica, especia l ly v o l u m e 3. T h e c u r r e n t n o t a t i o n for f i r s t -o rder logic c o m e s

f rom n o n e of t h e m ; it arrives full b lown in H i l b e r t ' s 1917 lec tu res ,

w i t h o u t any r e f e r e n c e to anyone .

Pe i rce h i m s e l f was e x t r e m e l y cri t ical of S c h r 6 d e r ' s idea o f Aufl6sung. H e said tha t if S c h r 6 d e r ' s n o t i o n of a (Sko lem f u n c t i o n ) so lu t i on is

a c c e p t e d , it w o u l d be like saying for a fifth d e g r e e a lgeb ra i c e q u a t i o n

tha t o n e h a d solved it by i n t r o d u c i n g a f o rma l f u n c t i o n o f the coeffi-

c ients a n d saying its values were the roots . H e d id n o t g rasp tha t the

idea of a fo rma l so lu t ion , o r S k o l e m func t ion , c o u l d be useful .

Russell , on the o t h e r h a n d , a d o p t e d S c h r 6 d e r ' s ideas freely, whi le

r e j ec t ing his m e t h o d o l o g y as o u t m o d e d a n d p h i l o s o p h i c a l l y u n s o u n d . ~

Because of the we igh t o f Principia Mathematica, t h e r e was a t e n d e n c y to

a c c e p t at face va lue the o p i n i o n s e x p r e s s e d by Russel l a b o u t defec ts in

Pe i r ce ' s a n d S c h r 6 d e r ' s t r e a t m e n t of relatives. This d id n o t go u n n o t -

iced. N o r b e r t Wiener , in his d o c t o r a l thesis (1913) , c r i t i c ized the lack

o f c r ed i t tha t Russel l gave to Schr6der . H e too t h o u g h t t ha t Russel l owed

m u c h m o r e to Sch r6de r , a n d h e n c e to Pei rce , t han Russel l was wil l ing

to admi t . Wiener , in the pa r t o f his thesis r e p r o d u c e d h e r e , goes so far

as to sugges t tha t the a l g e b r a of r e l a t ions as c a r r i e d o u t in Principia Mathematica is t aken d i rec t ly f r o m S c h r 6 d e r w i t h o u t c redi t .

P e r h a p s the u l t i m a t e cause for the n e g l e c t of S c h r 6 d e r a n d Pe i rce

can be t r a c e d to the i n f l u e n c e o f Hi lber t . At the t u r n o f the twen t i e th

cen tury , the Russel l p a r a d o x a n d the Bural i p a r a d o x c a u s e d Hi lbe r t ,

who h a d a l r eady r e w o r k e d the f o u n d a t i o n s o f g e o m e t r y , to r e t h i n k how

to set up logic a n d set t h e o r y as a f o u n d a t i o n for the m a t h e m a t i c a l

In his discussion of the calculus of relations in Principles of Mathematics, Russell states:

Peirce and Schr6der have realized the great importance of the subject, but unfortunately their methods, being based, not on Peano, but on the older Symbolic Logic derived (with modifications) from Boole, are so cumbrous and difficult that most of the applications which ought to be made are practically not feasible. In addition to the defects of the old Symbolic Logic, their method suffers technically (whether philosophically or not I do not at present discuss) from the fact that they regard a relation essentially as a class of couples, thus requiring elaborate formulae of summation for dealing with single relations. This view is derived, I think, probably unconsciously, from a philosophical error: it has always been customary to suppose relational propositions less ultimate than class-propositions (or subject-predicate propositions, with which class-propositions are habitually confounded), and this has led to a desire to treat relations as a kind of classes. (Russell 1903, p. 24)

8 INTRODUCTION

paradise of Cantor, and perhaps his interest turned the attention of the mathematical community to Frege and Russell, thus solidifying Russell's account of the development of logic given in Russell's many writings. This work sets out to correct that account.

1. The Early Work of Charles S. Peirce

1.1. Overview of the Mathematical Systems of Charles S. Peirce

Charles S. Peirce, in tile course of his long working life, developed a variety of theories and logical systems. His principal contributions include:

The calculus of relations A lattice-theoretic formulation of Boolean algebra Implicative propositional logic Quantified propositional logic and Boolean algebra Existential graphs An axiomatic arithmetic of the natural numbers. These theories are interwoven, but draw apart from time to time in

Peirce's papers. They are also quite different technically, although they deal with related issues.

1. Ttle calculus ofrelations.~Peirce's major contribution to logic published during his lifetime was his calculus of binary relatives (relations). The calculus of relatives was proposed as an algebra of logic, extending the work of George Boole. Peirce's system combines the linear algebraic methods of his father, Benjamin Peirce, Boole's calculus of propositions and classes, and Augustus De Morgan's relative operations. Since the calculus of binary relatives on a set is a Boolean algebra with additional structure, Peirce is able to lift into the calculus of relations all the laws of Boolean algebra, including the dualities for union, intersection, and complement; he also introduces additional dualities for converse, relative sum, and relative product. Peirce initially viewed the calculus of relatives as accommodating relations of unlimited arity, but soon recognized that binary relations suffice, a result proved by L6wenheim in 1915.

2. Boolean algebra.~Boolean algebra was developed after Boole by John Venn and De Morgan's student W. Stanley Jevons, and by Peirce, who

10 PEIRCE'S EARLY WORK

gave a systematic lattice-theoretic t reatment of Boolean algebra almost twenty years before Richard Dedekind isolated lattices in group theory.

3. Implicative propositional logic.--A third theory first introduced by Peirce is his system of implicative propositional logic. This theory is developed in the first two chapters of Peirce's 1880 paper, "On the algebra of logic," in which he determines the relationship between implication and deduction, deriving implications from deductions and conversely. This is a very close informal predecessor to "natural deduction systems" introduced by Dag Prawitz (1965). Yves Girard has interpreted the Gentzen sequent systems as simply a set of rules for manip- ulating natural deductiosn, and in this sense Peirce's system is a predecessor of the sequent calculus. ~

4. Quantified propositional log~c.--O. H. Mitchell developed a rudimentary system for quantification while attending Peirce's seminars at the Johns Hopkins University (1879-1883). Mitchell's system was limited to a theory of quantified propositional functions with exactly two prenex quantifiers, but it had a great influence on Peirce. After Mitchell's work, Peirce expended great effort on rules for simplifying prenex formulas. He separated the Boolean (propositional) part of a logical statement from its quantifiers, discovering essentially what we now call prenex form. Peirce also gave a number of interesting second-order definitions

. .

of mathematical notions in quantified propositional logic. However, in this period he did not introduce a fixed general notion of a first-order formula. He also certainly did not discover the intricacies of free and bound occurrences of variables.

5. Existential graphs.--Long after Frege's introduction of conceptual notation, Peirce independently developed a formal theory equivalent to first-order logic. However, Peirce's formulas were not linear expressions, but labeled graphs. Peirce's calculus of existential graphs brings together two logical systems Peirce had developed previously, the calculus of relations and his natural deduction system, via his original work in implication. Peirce understood that his rules of inference defined the notion of provable statement by an inductive definition and that the provable statements were true in all domains. But there is no hint that he knew or could formulate what it meant for a system to be complete. Peirce's existential graphs were unpublished and largely unknown until after his death.

6. Theory of arithmetic.~Peirce published an axiomatization of arith-

I The noted category theorist E William Lawvere read an early version of the present manuscript and commented that "Peirce's idea that implication mirrors inference finds a much more rich and explicit formulation in Eilenberg and Kelly's theory of closed and enr iched categories (Eilenberg and Kelly 1965), where the internal or enr iched horn- object represents the actual maps." The role of adjoints in giving common form to all ttle rules of logic is summarized in Lawvere's 1994 paper "Adjoints in and among bicategories."

F R O M P E I R C E T O S K O L E M 11

metic in 1881 in The American Journal of Mathematics. Peirce's paper, "On the logic of number," contained an axiomatization of the natural numbers based on the order relation. This preceded by eight years Peano's axiomatization in Arithmetices principia, which was based on the successor operation, but came twenty years after Herman Grassmann's neglected Lehrbuch der Arithmetik of 1861, which is the first known axiomatic treatment of the natural numbers based on zero and successor, and which is directly referred to by Peano.

1.2. Peirce's Influence on the Development of Logic

Peirce introduced a wide range of logical theories during the course of his professional life. One tends not to see, at least in the post-Hilbert period, logicians spanning quite so wide a range of alternative formal- izations of their discipline as Peirce; not even G6del or Tarski could rival Peirce's broad base in terms of the different representations they worked out.

It is not always clear to what extent Peirce's work influenced later developments in logic and to what extent it simply anticipated them. At least one direct link from Peirce to the rest of history is the influence of Peirce on Ernst Schr6der in his acceptance and systematization of the calculus of relations. Schr6der, in turn, influenced L6wenheim, and through L6wenheim influenced Skolem.

In the initial period of development of the calculus of relatives, Peirce was guided by the work done by his father, Benjamin Peirce, in linear associative algebras. The simplest examples of linear associative algebras were algebras of linear transformations, where product is composition. Peirce saw the analogy between the composition of linear transformations and the relative product of relations. He introduced Boolean matrices and a suitable matrix product to represent the composition of relations.

The calculus of relatives became far more algebraic and less com- putational during the period of Peirce's appointment at the Johns Hop- kins University. Stimulated by and competing with the work on matrix theory ofJ. j. Sylvester and Arthur Cayley, who were with Peirce at the Johns Hopkins for the first six months of 1882, Peirce melded his ideas about the calculus of relatives with the more abstract and algebraic approach of Cayley, and simplified the connections with logic, which Peirce was increasingly becoming aware of. By 1883, the calculus of relatives is laid out with a confidence and neatness and ease of exposition that is not present in Peirce's earlier papers. Schr6der based his development of the calculus of relatives on this later, highly algebraic presentation of the calculus of relatives by Peirce.

12 PEIRCE'S EARI.Y WORK

Schr6der expanded Peirce's calculus of relatives in the third volume of his Algebra der Logik, adding explicit rules for quantification over relations. Schr6der viewed the calculus of relatives as a language for and foundat ion of mathematics, and he successfully formalized in Peirce's system, as an example of a significant and interesting mathematical theory, Dedekind 's chain theory, which includes Dedekind 's work on induction. Schr6der also gave an axiomatic t rea tment of Peirce's lattice-theoretic development of Boolean algebra in the first two volumes of Algebra der Logik.

In his work on the calculus of relatives, Schr6der in t roduced a primitive form of Skolem functions in his notion of a formal solution of a relational equation. Peirce could not see the point of this. L6wenheim, however, did. L6wenheim took the essential idea of a Skolem function from Schr6der and made it the first step of the proof of his celebrated theorem, proving that if a first-order s ta tement has an infinite model , thien it has a countable model. The second part of L6wenheim's proof, giving his model-theoretic construction, has antecedents in Schr6der 's "method of elimination," which forces branchings in a tree of roots by

' + 1 ' = 1 for expanding his fundamenta l representat ion equation, 0 o o ' various values of i and j. (In Schr6der 's and L6wenheim's notation, 1' denotes the identity relation and 0' is its complement . ) L6wenheim explicitly says t h a t h e got this idea from Schr6der.

L6wenheim's theorem was the basis for Skolem's work in the subsequent development of model theory. Conceptually, the nearest analogues to L6wenheim's proof are the early topological compactness-style theorems, such as the Bolzano-Weierstrass theorem, at the point at which logic dissolves into topology. The closest analogue of all is the ari thmetic u l t raproduct construction, which was developed first by Skolem, not Russell or Frege.

Russell expressed disdain for Peirce's and Schr6der 's work on the calculus of relations and never admit ted his dependence on Peirce or Schr6der. But, as Norbert Wiener claims in his doctoral dissertation and presents convincing evidence to show, Russell lifted his t rea tment of binary relations in Principia Mattaematica almost entirely from Schr6der 's Algebra der Logik, with a simple change of notation and without attribution. The calculus of relatives (unions, intersections, relative products, etc.) remains in the mathematical curr iculum without credit to Peirce in the introductory parts of mathematics books to this day.

L6wenheim claimed throughout his life that Schr6der 's relation calculus was as convenient a base for mathematics as set theory. Later, Alfred Tarski, in collaboration with Steven Givant, in t roduced a set theory without variables, which shows that a relational calculus basis for set theory can be fully realized. The abstract theory of allegories of Peter Freyd and Andre Scedrov is another intellectual descendent of Peirce's

F R O M P E I R C E T O S K O L E M x3

and Schr6der 's philosophy that relations, not functions (as in categories) o r sets, can be taken as basic.

Peirce's influence extends even outside the domain of mathematical logic; there is a whole branch of programming, called relational programming, of which the work of James Lipton and Paul Broome is an example, that is based on the calculus of relations of Tarski-Givant, originating in Peirce.

Peirce, and to an extent his student Mitchell, anticipated the devel- opmen t of first-order and higher order predicate logic, but the papers of Peirce do not seem to have influenced Hilbert, Skolem, Herbrand, or G6del directly. The actual historical connect ion between Peirce and the later development of first-order logic runs through 1.6wenheim, via the calculus of relatives, and the link to Peirce results from Skolem extracting 1.6wenheim's theorem flom the calculus of relatives and stating it as a theorem of first-order logic.

First- and second-order predicate logic are fairly explicitly developed in Hilbert and Ackermann's influential text of 1928, but there is no evidence that Hilbert and Ackermann benefited from Peirce's devel- opmen t of it, and the line flom Peirce to Hilbert and Ackermann and "textbook" logic is a link that we can only conjecture about. Indeed, since Russell popularized tile theory of types, both in his Principles of Mathematics and in Whitehead and Russell's Principia Mathematica, and since second-order logic occurs in Russell's work as a separate invention int roduced to make a distinction that would allow him to avoid the paradoxes of Frege's system, it is likely that Hilbert and Ackermann picked out the distinction of first- and higher-order logic from Russell's theol T of types rather than from Peirce. Similarly, the link from Peirce to subsequent natural deduct ion systems looks sequential, but it is not clear that there is an actually historical dependence .

Although Peirce never published any of his writings on existential graphs (with one minor exception), he presented the complete system in his I.owell Lectures, delivered to the philosophy depa r tmen t at Har- vard University in 1903-1904. To what extent Peirce's ideas on modal logic communica ted in those lectures influenced subsequent work in modal logic has not yet been resolved, but C. I. Lewis, for one, had access to Peirce's unpubl ished papers at Harvard after Peirce's death in 1914 (see Lewis's autobiographical essay in Schilpp 1968, pp. 16-17). Peirce's system of existential graphs develops a precursor of the possible worlds semantics for modal logic, a fact apparently well known to Lewis.

Peirce's influence on the development on axiomatic ari thmetic also must remain conjectural. In 1881, in his paper on "On the logic ot number," Peirce int roduced discrete orderings with a first e lement satisfying the principle of induction. He further defined addit ion and multiplication by inductive definitions. Peirce used his recursion equations


for addition and multiplication to prove the standard laws of arithmetic by inductive arguments usually attributed to Peano (1889), and in fact first appearing in the work of Herman Grassmann (1861).2 Peirce did not realize, however, as Dedekind did in his later work (1888), that these definitions of addition and multiplication needed to be justified, viz., that definition by induction is different from proof by induction. It was left to Dedekind (1888) to prove by the method of chains that such functions exist.

In 1888, Dedekind gave an inductive definition of a finite set as the smallest collection of sets containing the null set such that if x is in the collection and y is anything, then x w {y } is in the class. In 1881, obviously independen t of Dedekind, Peirce also set out to capture finite sets, but he did it by characterizing these sets as images under one-to-one maps of initial segments of a discrete order with a first e lement that satisfies the induction axiom. This is equivalent to Dedekind's notion of finite set, that is, a set that cannot be mapped one-to-one into a p roper subset of itself. This may be proved by induction. To show that every Dedekind finite set is finite, however, requires the axiom of choice.

Yet Peirce's influence on the development of logic was not as great as it might have been, considering his substantial contributions to it. This may be due in part to his failure to provide a formal system for logic, in the sense of Frege's. The motivation to create a formal system is lacking in Peirce, as it is for Boole and Schr6der. Boole was not interested in the axiomatic method. Apart from the algebra of logic, Boole's o ther major work was with formal algorithms for solving ordinary differential and difference equations; it is therefore no surprise that his approach to Boolean logic was algorithmic rather than axiomatic.

Peirce and Schr6der similarly first became attached to first and higher order relational algebra and merely used whatever algebraic identities they could discover as they went along to simplify reasoning. They made no early at tempt at an all-encompassing formal system. In this, Peirce and Schr6der were very close in spirit to Peano. Like him, they had a universal language. Like him, they proposed no fixed set of logical axioms and used more or less any logical facts they could identify. Unlike Peano, however, their language was based on relational algebra and relational identities with the pure aim of simplifying reasoning.

1.3. Pe i rce ' s Early Approaches to Logic

Peirce's first publication in mathematical logic was a paper on Boolean algebra, "On an improvement in Boole's calculus of logic," published in 1867. Boole's original algebra (1847) was basically the algebra of

See Shields (1997) for a further discussion of Peirce's axiomatization of alithmetic.

FROM PEIRCE TO SKOLEM 15

"and , " "or," a n d "not . " H o w e v e r , B o o l e ' s "or" was a pa r t i a l o p e r a t i o n

(sum-), a p p l i c a b l e on ly w h e n t he a l t e rna t i ve s w e r e exc lus ive . :~ In his 1867

p a p e r , P e i r c e i n t r o d u c e d the o p e r a t i o n o f log ica l a d d i t i o n ( inc lus ive

"or" ) , w h i c h is always d e f i n e d , in w h a t he t h o u g h t to be an e x t e n d e d

ve r s i on o f B o o l e ' s sys tem. P e i r c e d e f i n e s logica l a d d i t i o n , w h i c h h e de-

n o t e s by "+," ( a n d / o r ) , as follows:

Let the letters of the a lphabet deno te classes whether of things or of occurrences. It is obvious that an event may ei ther be singular, as "this sunrise," or general , as "all sunrises." Let the sign of equality with a comma benea th it express numerical identity. Thus a--, b is to mean that a and b deno te the same class-- the same collection of individuals.

Let a +, b deno te all the individuals conta ined u n d e r a and b together. The opera t ion here pe r fo rmed will differ from ari thmetical addit ion in two respects: first, that it has reference to identity, not to equality; and second, that what is c o m m o n to a and b is not taken into account twice over, as it would be in arithmetic. (Peirce 1867, p. 3)

Pe i r ce r e t a in s in his sys tem B o o l e ' s r e s t r i c t e d "or," w h i c h h e d e n o t e s

wi th a " + " sign.

P e i r c e ' s log ica l m u l t i p l i c a t i o n , a , b, d e n o t e d by a c o m m a a l o n e , is t he

s a m e as B o o l e a n "and" :

Let a, b deno te the individuals conta ined at once u n d e r the classes a and b. (Peirce 1867, p. 4)

Pe i r ce a lso i n t r o d u c e d log ica l s u b t r a c t i o n , a - , b ( a n d - n o t ) , to s u p p l y

an inve r se fo r log ica l a d d i t i o n . Log ica l s u b t r a c t i o n , P e i r c e be l ieves , will

p r o v i d e for n e g a t i o n in his sys tem, a l t h o u g h h e is n o t c o m p l e t e l y c l e a r

a b o u t this. H e d e f i n e s logica l s u b t r a c t i o n as a pa r t i a l i nve r se to log ica l

a d d i t i o n :

Let - , be the sign of logical subtraction; so def ined that

If b +, x =, a x =, a - , b.

Here it will be observed that x is not completely de te rmina te . It may vary from a to a with b taken away. This min imum may be deno t ed by a - b. It is also to be observed that if the sphere of b reaches at all beyond a, the expression a - , b is uninterpre table . (Peirce 1867, p. 5)

'* This probably stems from the origins of Boolean logic in probability theory. Boole restricted "'or" to the circumstance in which the classes being combined were already disjoint, in which case there is no difference between the inclusive and exclusive "'or." This is convenient from the probability point of view because there is no intersection term, so the probability of the sum is the sum of the probabilities whenever the symbol is used in Boole.


Logical subtract ion is only d e t e r m i n a t e if x and b are disjoint, and thus is a partial inverse for exclusive "or," since the lat ter is only de f ined in the disjoint case. Peirce uses a bar over a class term, d, r a the r than logical subtract ion, to d e n o t e the negat ion of a literal (Peirce 1867,

p. 5) and represents the nega t ion of an arbitrary class x by 1 - x, the c o m p l e m e n t of x, where the minus sign is u n d e c o r a t e d with a c o m m a

(Peirce 1867, p. 6). Peirce claims three advantages for his system over Boole's, all due to

his new operat ions:

Boole does not make use of the operations here termed logical addition and subtraction. The advantages obtained by the introduction of them are three, viz., they give unity to the system; they greatly abbreviate tile labor of working with it; and they enable us to express particular propositions. (Peirce 1867, p. 13)

The first advantage is aesthetic: f rom the poin t of view of ma thema t i ca l

duality, Peirce 's system is super io r to Boole 's since Peirce 's ope ra t i on of inclusive "or" is the natural dual of logical mult ipl icat ion, and allows

for the express ion of impor t an t algebraic identities, such as De Morgan ' s

law, whereas Boole 's opera t ions are not mathemat ica l ly dual. This aesthetic gain is offset somewhat by Peirce 's loss of additive inverses, however. The second advantage is a pragmat ic one: by add ing the inclusive "or," Peirce produces a system that is not only m u c h m o r e conven ien t to work in computat ional ly, but also allows for an easy t ranslat ion of n o n m a t h e m a t i c a l logical a rgumen t s into ma themat i ca l r ep resen ta t ions of logical a rguments .

Peirce 's third claim, viz., that his system is m o r e expressive than Boole 's , is not correct . Peirce 's in t roduc t ion of inclusive "or" is an ex- t remely useful cont r ibut ion , but his logical "or" can be expressed using

the ord inary "and" ( comma) , Boole 's restr icted "or" ("+") , and nega-

tion, simply as a +, b = a,/~ + a , b + ,4, b. At this level, Peirce 's system

does not have any more expressiveness than Boole's. Peirce hints that

the re are Aristotelian not ions that are expressible in his system but not in Boole 's calculus. To substant iate his claim, Peirce says only:

Let i be a class only determined to be such that only some one individual of the class a comes under it. Then a - , i, a is the expression for some a. Boole cannot properly express some a. (Peirce 1867, p. ~3)

It is not clear what the expression a - , i , a means. On the one hand,

if a - , i , a means a minus the quanti ty i in tersect a, that class will be a minus a s ingleton, which may be much larger than one individual. On


the other hand, if a - , i , a means a - , i intersect a, that class will be a - , i, which is a with a singleton taken away, ra ther than a restricted to a singleton, as Peirce wants to claim. It is possible that there was a transcription error, and Peirce in tended for the expression to read a - , i ,d . 4

In any case, Peirce's idea fails because his notat ion does not completely capture how we use the word "some" in a mathemat ica l context. Peirce's system does not enable him to say that there exists a class i whose intersection with a is a particular individual and then to determine specific propert ies of that individual. The problem of existentially selecting an e lement from a has been replaced by the problem of existentially selecting a class that contains only a single e l ement of a. But it is not obvious that the second problem is any easier than the first. The second problem is simply one type level higher.

Peirce is struggling toward the solution of the problem of expressing quantification, but the real problem is that the solution is not to be found within the confines of Boolean algebra. All Peirce's at tempts were doomed to fail until it occurred to him how to step out of the framework of purely Boolean operations. We know today that the decision problem for the validity of a s ta tement in propositional logic is decidable; a s ta tement is valid if and only if the last column of its truth table has only "T" values. We also know that the decision problem for the validity of statements in predicate logic is undecidable; this was first proved by Church (1936), after he had given an exact definition of recursive or decidable. It can also be proved by the method of proof used by G6del for his incompleteness theorem. One consequence of these facts is that it is not possible to compile predicate formulas into equivalent propositional formulas, and yet that is what Peirce believed he had done.

In his 1867 paper Peixce is engaged in present ing a cleaned-up version of Boolean algebra in which the "or" operat ion is not restricted to disjoint classes, and is dual to the Boolean "and" that is also not so restricted. But his improvement is algebraic. At the same time, Peirce has muddied the waters somewhat, a l though he did not realize it, by int roducing a minus operat ion that is not, properly speaking, a function.

In order to make clear that his efforts are not simply an improved, pedagogically more useful presentat ion of Boole's work but in fact an extension of Boole's system, Peirce attempts to provide evidence that his system has expressive power that Boole's lacks, namely, that he can express the notion of "some" and so can analyze the Aristotelian syllogisms, where Boole cannot. This we will see as an ongoing theme in Peirce, his a t tempt ing to reconcile Boole with Aristotle and solve the problem of expressing quantification.

1 See l - Ia i lper in (1976) for a full analysis .


In an u n p u b l i s h e d manusc r ip t wri t ten a r o u n d 1896, Peirce descr ibes his early u n d e r s t a n d i n g of Boole as it inspired his own work:

Boole's original algebra is nothing but the calculus of probabilities, as it would be with omniscience. Every probability is necessarily either 0 or 1, and hence every interpretable expression satisfies the quadratic x(1 - x) --0. Although Boole was thinking of probabilities, he reaches the application of his algebra to categorical propositions by some obscure process of thought of which he could give no account. But the true rationale of it is, that each letter is the probability to omniscience that a given individual possesses the character signified by that letter; and that he does not equate two expressions unless their probabilities are the same to whatever individual they apply. Thus, let h be the probability that a given individual is a man. By reason of omniscience, h(1 - h) = 0. Let d be the probability that a given individual dies. Here too, d ( 1 - d)=0 . But all men die. This Boole writes h ( 1 - d) =0, that is the probability that anything, X, is both a man and does not die is 0, no matter what thing X may be. There is the Boolian [sic] algebra in a nutshell. (Peirce 1896, p. 1)

It is well es tabl ished that Boole arrived at Boo lean a lgebra while seeking to give a precise calculus of probabil i t ies. He starts ou t with a proposi t ional calculus and makes probabi l i ty ass ignments to p ropos i t iona l letters. He then discusses the rules for assigning probabi l i t ies to com- p o u n d propos i t ions as a func t ion of their parts. O f course, probabi l i t ies are be tween 0 and 1, and the rule for "or" is P(A or B ) - P ( A ) + P(B) - P(A and B). This rule reflects his implicit class i n t e rp re t a t i on of p ropos i t iona l letters, in which each let ter deno te s a subset o f a fixed (for simplicity of s t a t emen t assumed finite) set. In addi t ion , one is adding up the ( i n d e p e n d e n t ) probabil i t ies of the e l emen t s within the set d e n o t e d by the propos i t iona l letter. We now express this as a finite probabi l i ty space in t e rp re ta t ion of p ropos i t iona l calculus, with each letter d e n o t i n g an event. Boole observed that events with probabi l i ty 1 and 0 (in m o d e r n terms, in finite spaces cer ta in or impossible events, in infini te spaces a lmost cer ta in or a lmost impossible events) have 0, 1 ass ignments associated with them, these be ing ou r m o d e r n t ru th ass ignments .

Having d o n e this, when e x t e n d i n g f rom letters (a tomic p ropos i t ions ) to c o m p o u n d proposi t ions , ins tead of invent ing the Boo lean a lgebra of 0 and 1 as values with opera t ions restr ic ted to them, Boole ins tead copies the fo rma t f rom probability, where P(A or B) is P(A) + P ( B ) - P(A and B), and regards the Boolean ope ra t ions as restr ict ions o f the ar i thmet ica l ones on the real numbers . His successors e l imina t ed this step, i n t roduc ing the ope ra t ions on 0, 1 directly, no t as restr ict ions of ope ra t ions on the real numbers . This is what Peirce is r e fe r r ing to in


the quo ta t ion above. When the ar i thmet ical in t e rp re ta t ion is appl ied to the law of the exc luded middle , it gives the ord inary a lgebra equa t ion x (1 - x) = 0. Peirce 's "characters," i.e., proper t ies , are thus equiva lent to Boole 's sets: each proposi t ional le t ter deno tes a p roper ty of individuals, and the proposi t ional connect ives lead f rom simple to c o m p o u n d proper t ies . Assuming that these are all p roper t ies of e l emen t s of a fixed

set, this is equivalent to Boole. If the proper t ies are no t p roper t i e s of

e l emen t s within a fixed set (e.g., x is a cardinal n u m b e r ) , then Peirce 's

l anguage of proper t ies is be t te r because it is m o r e general . The idea of

equality of proposi t ions u n d e r a probabil i ty in t e rp re t a t ion is clear: both proposi t ions are assigned the same probabil i ty based on the probabil i ty

ass ignment to proposi t ional letters used. W h e n appl ied to 0, 1-valued assignments , this means the proposi t ions are both t rue or both false u n d e r the ass ignment , which is our usual semant ic equivalence.

It is in te res t ing that, like Boole, Peirce saw a potent ia l c o n n e c t i o n between probabil i ty theory and the laws of logic:

Whatever phenomenon is measured by a mathematical quantity, x, is also measured by every function f x of x which has a distinct value for every interpretable value of x. Hence, probabilities, instead of being measured by the ratio of favorable cases to all cases, may be measured by the ratio of favorable cases to unfavorable cases (which simplifies certain problems), or by the logarithm of the ratio of favorable cases to unfavorable cases (which is our psychologically natural way of "bal- ancing probabilities"), or by the negative of the logarithm of unfavorable cases to favorable cases (which represents the modification of Boole's algebra used by me). In short, we may, as I remarked in 1884, take any two determinate numbers, v and f the former signifying true (verum) and the latter false, and representing the principle of excluded middle by the quadratic (v - x ) ( x - f ) = 0, the principle of contradiction being represented by the difference between v and f we have an algebra of logic substantially as good as Boole's. (Peirce 1896, p. 2)

These remarks are ex t raneous to the main lines of the work. Peirce is merely po in t ing out that the f requency in t e rp re t a t ion of the probability of an event as the ratio of cases in which the event holds to all

cases w h e t h e r it holds or not is not the only perfect ly nea t way of in- t e rp re t ing probability. O n e could start by using, ins tead of probability,

the ratio of favorable to unfavorable cases, the ratio of unfavorable to favorable cases, or the log of either. Each gives a d i f fe ren t no t ion of f requency in te rpre ta t ion , each has some intuitive con ten t , and the cal-

culus of probabil i t ies of c o m p o u n d s ta tements could be a l te red to be based on any of them. Any m o d e r n statistics book, for instance, gives

an i m p o r t a n t role to log l ikel ihood est imates, especially for opt imizat ion.

2 0 PEIRCE'S EARLY WORK

Peirce 's motivat ion was similar to that of the m o d e r n statistician. In a way, it amoun t s to saying that our "ruler" of probabil i t ies is to an ex ten t qui te arbitrary.

In the next and final pa rag raph of tile 1896 manuscr ip t , Peirce claims

to have used index nota t ion to d e n o t e individuals in or before 1880,

a l though whe the r he actually did so is quest ionable . T h e r e is a form of

index nota t ion in his 1867 paper, but it is not used to d e n o t e individuals; Peirce employs it ra ther to def ine a funct ion b,, as

Let b,, denote the frequency of b's among the a's. Then considered as a class, if a and b are events, b,, denotes the fact that if a happens b happens.

a b , , = a , b .

(Peirce 1867, p. 9)

The index a in b,, denotes a class, not an individual.

Peirce in t roduced his calculus of relatives in 1870. Like Boole, he was gu ided by an analogy between the laws of proposi t ional logic, the laws of classes, and the laws of ar i thmetic . He was also gu ided by o t h e r

ma themat i ca l analogies. Peirce 's fa ther was the f o u n d e r of abstract linear associative algebra; Peirce edi ted his fa ther 's work and m e n t i o n e d in his own that many laws for the relative calculus opera t ions were ana logues of the laws of l inear associative algebra. For instance, he no ted that an appropr ia te mul t ip l icat ion of matr ices with entr ies 0, 1 a lone and Boolean opera t ions on 0, 1 co r responds to relative product . ~

Peirce saw an analogy between the laws of exponen t i a t i on in arithmet ic and universal quant i f icat ion in logic. For instance, he i n t e rp r e t ed x y+'~ = x y , x ~ in the calculus of relatives as express ing that to be in the relat ion x to every l n e m b e r of y or z is the same as to be in the re la t ion x to every m e m b e r of y and in the relat ion x to every m e m b e r of z v v v .

This can be r ega rded as a primitive insight into the na tu re of quanti-

fication, notat ionally a long the lines of the t r e a t m e n t of quant i f iers in

topos theory (Lawvere 1970; see also J o h n s t o n e 1977; Mac Lane and Moerdi jk 1992).

Peirce was also guided by analogies to a r i thmet ic that were somet imes

conflicting. He tried to establish an ana logue of the b inomial t h e o r e m in the calculus of relatives, and to find m e a n i n g for infinite series and

their identities, a l though these a t tempts are difficult to decipher . In

p r o c e e d i n g by formal manipula t ions , Peirce was following the t radi t ion

An interesting illustration of how knowledge evaporates with the passage of years, this was rediscovered and published in the Journal of Symbolic Logic more than 70 years later by Irving Copi (1948) with no reference to Peirce, whose works had already appeared in the Harvard edition (1933).

FROM P E I R C E T O SKOLEM 21

of Boole, the greates t exper t of his t ime on inverse different ia l o p e r a t o r

formal solut ions of differential equat ions. Like Boole, Pei rce experi-

m e n t e d with ma themat ica l analogies to gain insight into a new subject. For Boole, the new subject was differential equa t ion formal m e t h o d s

and then propos i t ional logic. For Peirce, it was the calculus of relatives. Al though De Morgan in t roduced the not ions of relative p roduc t , con-

verse, involut ion (forward and backward) , and nega t ion in his p a p e r "On the syllogism. IV" (1860), Peirce 's discovery of the calculus of rel-

atives was i n d e p e n d e n t of De Morgan. In an u n p u b l i s h e d lec ture of

t898, Peirce remarks:

But to return to the state of my logical studies in 1867, various facts proved to me beyond a doubt that my scheme of formal logic was still incomplete. For one thing, I found it quite impossible to represent in syllogisms any course of reasoning in geometry or even any reasoning in algebra except in Boole's logical algebra. Moreover, I had found that Boole's algebra required enlargement to enable it to represent the ordinary syllogisms of the third figure; and though I had invented such an enlargement, it was evidently of a makeshift character, and there must be some other method springing out of the idea of the algebra itself. Besides, Boole's algebra suggested strongly its own imperfection. Putting these ideas together I discovered the logic of relatives. I was not the first discoverer; but I thought I was, and had complemented Boole's algebra so far as to render it adequate to all reasoning about dyadic relations, before Professor De Morgan sent me his epoch-making memoir in which he attacked the logic of relatives by another method in harmony with his own logical system. But the immense superiority of the Boolian [sic] method was apparent enough, and I shall never forget all there was of manliness and pathos in De Morgan's face when I pointed it out to him in 1870. I wondered whether when I was in my last days some young man would come and point out to me how much of my work must be superseded, and whether I should be able to take it with the same genuine candor. (Peirce 1898, voi. 4, pp. 8-9)

In the beg inn ing , then, Peirce 's work in logic first set ou t to ex t end

Boole 's t r e a t m e n t so that it covered the syllogisms of Aristotle in a

na tura l and satisfactory way. Peirce self-avowedly acqu i red some of his insights, as well as m u c h of his terminology, such as his "first- intentional

logic of relatives," f rom the scholastic logicians, themselves keen readers of Aristotle. However, r eason ing in geome t ry was in a highly stylized

and f inished state, a n d had been for 2000 years. W h e n Peirce tr ied to

use Boole 's propos i t ional logic with a smat te r ing of syllogistic to represent geomet r i c reasoning, he realized that the Aristotle plus Boole, or roughly what we now call monad ic predica te logic, was no t expressive


enough to represent the reasoning of geometry, which is almost entirely in terms of the two binary relations, incidence and congruence. As soon as these binary relations are introduced, much if not all reasoning in Euclid can approached. We surmise that this was what Peirce mean t by the "various facts" that led him to desire a more perfect logic, and that reasoning in geometry was thus Peirce's route for discovering the necessity of using relations as well as sets, and of using some kind of algebra of relations.

2. Peirce's Calculus of Relatives: 1870

I n t r o d u c t i o n

Peirce published his "Description of a notation for the logic of relatives" in 1870, eight years before the founding of the American Journal of Math- ematics b y j . j . Sylvester (with Peirce's father one of the editors) and the American mathematics research establishment as we know it, and six years before the opening of the Johns Hopkins University, the first graduate research university in the United States (where Peirce served on the faculty from 1879 to 1884). Peirce was not an academic at the time he wrote his 1870 paper; he was engaged in astronomical work at Har- vard Observatory as an employee of the United States Coast Survey, of which his father was superintendent. Peirce's subsequent papers on mathematical logic were published in the American Journal of Mathematics, which circulated in Europe and was available in the libraries of the principal European universities, but his 1870 paper appeared in the Proceedings of the American Academy of Arts and Sciences, which published papers presented at Academy meetings. The European mathematical and scientific community would have had little contemporary access to Peirce's paper except through personally circulated copies, such as the one Peirce delivered to De Morgan in 1870, and would have known of Peirce chiefly via the reputation of his father. The year 1870 also saw the publication of Peirce's father's masterwork, Linear Associative Alge- bras, of which Peirce became editor in 1880 upon his father's death. It is arguable from Peirce's notation and remarks that Peirce's algebra of relatives was a natural by-product in logic of his deep involvement with his father's representation theory for linear associative algebras.

Peirce described his 187.0 paper as ".. .an amplification of the conceptions of Boole's Calculus of Logic." In the Lowell Lectures of 1903, Peirce (not immodestly) evaluated this work as follows: "In 1870 I made a contribution to this subject [logic] which nobody who masters the

23

24 PEIRCE'S CALCULUS OF RELATIVES

subject can deny was the most important excepting Boole's original work that ever has been made" (Peirce 1903).

The opening section of this paper includes a clear statement of Peirce's aim to construct a logical calculus of inference of wide scope:

I think there can be no doubt that a calculus, or art of drawing inferences, based upon the notation I am to describe, would be perfectly possible and even practically useful in some difficult cases, and particularly in the investigation of logic. (Peirce 1870, p. 28)

Peirce emphasized the word "investigation," meaning here research into logic itself, as opposed to applications of logic elsewhere. He expresses the idea that a formal system of rules of inference would be useful for resolving complex questions by logical means. The questions he addresses are of two kinds, examples of which occur in this and later papers. First, there is his use of formal algebraic computat ions to deduce complex logical theorems from simple ones. Second, there is his cod- ification of rules about relations to reveal the mathematical structure of formal logic itself. This was a precursor of metamathematics and proof theory. Both are extensions of Boole's ideas on the algebra of classes to the much more complex and expressive algebra of relations. Both aims were closer in spirit to proof theory and syntax than to model theory as we know it. Peirce's belief that this was the most impor tan t advance since Boole was certainly based on the fact that the algebra of relations is far more expressive than the algebra of propositions, and reflects a great deal more of everyday logical inference than does Boole's theory of sets, since relations, not just sets, are the bread and butter of reasoning.

2.1. Peirce's Algebra of Relations

In Peirce's original logical language, which is no longer in use, the main ingredients of his 1870 version of the calculus of relatives are:

1. Three kinds of logical terms, called "absolute," "simple relative," and "conjugative."

2. A fundamental binary relation, denoted by --<. The symbol "--<" is used by Peirce ambiguously for both inclusion between classes and implication between propositions. This was a convention subsequently followed by his disciple Schr6der and criticized as ambiguous by Frege, a l though in fact, the ambiguity between class and proposit ional inter-

' In the Collected Papers of Charles Sanders Peirce (Hartshorne & Weiss 1933); except where otherwise noted, all page numbers given in this chapter are from Peirce (1870) in this edition.


pretations is a carryover from Boole. In Peirce's 1870 paper, --< is used predominant ly as inclusion.

3. Three binary operat ions on relatives: A. Addition. This includes:

i) Invertible addition (taken from Boole and deno ted by "+") ; ii) Union (denoted by "+,").

B. Multiplication. This includes: i) Relative product; ii) Intersection (taken from Boole and deno ted by a comma) .

C. "Involution," or exponent iat ion. 4. Complementa t ion , treated in two forms:

By subtraction, (1 - a), i.e., as class complement ; By exponent ia t ion, n x, as the relative "not x."

5. Converse (denoted by the opera tor K). In Peirce's original text, he first lays down the condit ions (axioms)

that must hold for operat ions in a domain and the notat ional conventions for those operations, borrowed directly from algebra. Those conventions, essentially, are in tended rules of inference. He then sets forth the interpretat ion he attaches to those operat ions and symbols. Peirce's approach will seem backwards to a modern reader. Normally, when we approach a logical system today, we start with semantics and work back to a system of proof, ra ther than beginning with syntax and a basic set of rules for manipula t ing symbols, as Peirce does. In our discussion, we will preserve Peirce's approach of considering first syntax and then semantics, present ing the notation abstractly, followed by its interpretation in the domain of relatives.

In setting forth his views on the appropria te use of notat ion for the calculus of relatives, Peirce states that one should not use a s tandard mathemat ical symbol for an operat ion or relation unless that operat ion or relation shares certain basic propert ies with the s tandard one:

In extending the use of old symbols to new subjects, we must of course be guided by certain principles of analogy, which, when formulated, become new and wider definitions of these symbols. As we are to employ the usual algebraic signs as far as possible, it is proper to begin by laying down definitions of the various algebraic relations and operations. (p. 28)

Peirce here introduces the by now fully accepted idea of using old symbols for new operat ions when certain commonly accepted laws hold for those new operations, and not using them otherwise. For instance, wherever one uses the addition symbol, the cor responding operat ion should be associative and commutative, that is, should be a commutat ive


s e m i g r o u p in the m o d e r n sense. In this he is, excep t for details of l anguage , a t rue p r e c u r s o r o f m o d e r n abstract a lgebra. H e says tha t

These conditions are to be regarded as imperative. But in addition to them there are certain other characters that relations and operations should possess if the ordinary signs of algebra are to be applied to them. (p. 31)

Thus , he st ipulates that we s h o u l d only use o p e r a t i o n symbols f rom

exis t ing m a t h e m a t i c s in a m o r e genera l c o n t e x t for o t h e r o p e r a t i o n s

w h e n we have a g r e e d by c o n v e n t i o n what set of c o m m o n laws is to be

a s sumed . T h e s e are no t all the laws o b e y e d by the or ig inal system, bu t

only those c o m m o n p rope r t i e s sha r ed with those o t h e r systems we have

d e c i d e d to investigate.

As f u n d a m e n t a l to his logical i n t e r p r e t a t i o n of the a lgebra o f relatives,

Pe i rce i n t roduces th ree classes of logical terms: abso lu te terms, s imple

relative terms, a n d conjugat ive terms. H e descr ibes these th ree classes

o f t e rms a n d the i r d i s t ingu i sh ing character is t ics as follows:

Now logical terms are of three grand classes. The first embraces those whose logical form involves only the conception of quality, and which therefore represent a thing simply as "am." These discriminate objects in the most rudimentary way, which does not involve any consciousness of discrimination. They regard an object as it is in itself as such (qua&); for example, as horse, tree, or man. These are absolute terms. The second class embraces terms whose logical form involves the conception of relation, and which require the addition of another term to complete the denotation. These discriminate objects with a distinct consciousness of discrimination. They regard an object as over against another, that is as relative; as father of, lover of, or servant of. These are simple relative terms. The third class embraces terms whose logical form involves the conception of bringing things into relation, and which require the addition of more than one term to complete the denotation. They discriminate not only with consciousness of discrimination, but with consciousness of its origin. They regard an object as medium or third between two others, that is, as conjugative; as giver of m to ~ , or buyer of m for ~ from ~ . These may be called conjugative terms. (p. 33)

Pei rce makes a typographica l d is t inc t ion be tween these terms, which

we will preserve: absolu te te rms are d e n o t e d by r o m a n let ters (a, b, c,

. . .) ; relative te rms by italic let ters (a, b, c . . . . ); a n d conjuga t ive te rms by

bo ldface letters (a, b, c . . . . ). H e r e we will o f ten give an ex t ens iona l

i n t e r p r e t a t i o n of these te rms and thus u n d e r s t a n d abso lu te t e rms as

s t a n d i n g for classes or una ry relat ions, s imple relative te rms as corre-


s p o n d i n g to binary relations, and conjugative terms as c o r r e s p o n d i n g

to relat ions of arity g rea te r than two; this is mere ly a m o d e r n cru tch to

try to make sense in c o n t e m p o r a r y set- theoret ic terms of Peirce 's ad-

mit tedly in tensional system.

Peirce also in t roduces a bracket ope ra to r that assigns a n u m b e r to each logical term, indicat ing the n u m b e r [x] of individuals in the class

x denotes:

I propose to assign to all logical terms, numbers; to an absolute term, the number of individuals it denotes; to a relative term, the average number of things so related to one individual. (p. 35)

It is not clear that this not ion always makes sense; for instance, what

is [m], the average n u m b e r of individuals re la ted to a mo the r? Is it one?

Is it two? Is its value d e t e r m i n e d by this year 's census? Is it based on a prespecif ied d o m a i n with relations? etc. This is pret ty clearly simply a

half - formed not ion, which we should not try to hold Peirce to.

2.1.1. Inclusion and Equality

In a p p r o a c h i n g a no ta t ion for inclusion and equality t h r o u g h the al-

gebra of binary relations, Peirce first in t roduces the special symbol "--<" and says that whenever it is used, it should d e n o t e a transitive,

ant isymmetr ic , and reflexive relation. Tha t is, it should be cons ide red a partial o rde r ing in the m o d e r n sense. He thus shares with Dedek ind

the h o n o r of invent ing the abstract not ion of a partial order . Peirce 's first discussion o f - -< proceeds as follows:

I use the sign --< in place of _-<. My reasons for not liking the latter sign are that ... it seems to represent the relation it expresses as being compounded of two others which in reality are complications of this. It is universally admitted that a higher conception is logically more simple than a lower one under it .... Now all equality is inclusion in, but the converse is not true; hence inclusion in is a wider concept than equality, and therefore a logically simpler one. (p. 28)

He is po in t ing out that if we are given a partial o r d e r R, the equality

relat ion x =y can be def ined as xRy and yRx, but if we are given an

equality relat ion, we canno t recover the partial o r d e r R f rom it. There-

fore, he regards the not ion of partial o r d e r as m o r e basic than that of

equality. Peirce also views the def ini t ion of a partial o r d e r i n g given above, that

is, as an ant isymmetr ic , transitive, and reflexive relat ion, to be m o r e basic than the def ini t ion of a strict partial order , that is, a transitive


irreflexive relat ion, for similar reasons. O n e c a n n o t def ine equali ty f rom the axioms for a strict partial order , but one can def ine a strict part ial o r d e r f rom a partial order.

H e r e is what Peirce has to say abou t def in ing the strict o r d e r f rom the nons t r ic t order :

Being less than is being as small as with the exclusion of its converse. To say that x< y is to say that x--<y, and that it is not true that y--< x. (p. 29)

Thus , "--<" is jus t what we call a partial order , and de f in ing a strict part ial o r d e r as a derivative no t ion is i n t ended .

This is abou t the same as we teach beg inne r s now. But we go one step fur ther , to a prepar t ia l o rde r ing , a reflexive and transitive re la t ion (not a s sumed to be ant isymmetr ic) . In this case, o n e mus t go to equiv- a lence classes to get a partial order. Thus, the equiva lence class [a] o f a is the set of all b such that aRb and bRa, and [aiR[b] if a nd only if aRb.

As an example , cons ider the def in i t ion of _< for cardinals o f sets. Def ine A _< B to m e a n that the re exists a one- to-one func t ion f f rom A to B. This gives a prepar t ia l o r d e r i n g of the class of all sets; the equiv- a lence classes are the cardinals. If we def ine A ~ B to m e a n the re exists a one- to-one , on to func t ion f f rom A to B, then the Schr6der -Berns te in t h e o r e m says that (A _< B) A (B_< A) ~ A ~ B. Strict inequal i ty of cardinals is simply a case of be ing _< wi thout be ing =. Abou t c o n t e m p o - raneously, such quot ien ts appea r in Georg Can to r ' s d e v e l o p m e n t of real n u m b e r s as equiva lence classes o f Cauchy sequences , and of course in a lgebra as quot ien ts m o d u l o subgroups or ideals. But the q u o t i e n t con- s t ruct ion is no t f ound in Peirce.

In discussing inclusion and equality, we use an ex tens iona l set-theoretic in te rpre ta t ion . T h e equali ty sign is i n t e r p r e t e d as identity, the "less than" sign is i n t e rp re t ed as p r o p e r inclusion, and "--<" is i n t e r p r e t e d as inclusion, p r o p e r or not. Peirce 's first ex a mp le of inclus ion is f--< m, which means "every F r e n c h m a n is a man," wi thout assert ing w h e t h e r

the re are any o t h e r m e n or not. W h e n the two terms are relatives, however, it is not as easy to i n t e rp re t the inclusion. Peirce 's e x a m p l e is m - < l, which he says means ,'every m o t h e r of any th ing is a lover of the same thing." He is thus saying, in m o d e r n pred ica te logic no ta t ion , VxVy[m(x,y) ~ l(x,y)]. His phrase "of the same thing" we normal ly express by using the same variable in two places, but Peirce does no t use variables in his calculus. His in t e rp re ta t ion of m--< l is thus a s imple inclusion of binary relat ions when we speak set-theoretically.

FROM P E I R C E T O SKOLEM

2. I. 2. Addition

29

Peirce dis t inguishes his add i t ion ope ra t i on f rom an invert ible add i t ion

opera t ion . He uses the special symbol "+," ( a n d / o r ) for add i t i on and reserves the plus sign wi thout an affixed c o m m a for inver t ible addi t ion . He states that add i t ion x +, y is a commuta t ive and associative ope ra t i on ,

and invert ible addi t ion x + y is a commuta t ive and associative o p e r a t i o n

that also satisfies the constraint : x + y = x + z implies y = z. So, for ex-

ample , a r i thmet ica l addi t ion , i.e., add i t ion of integers , is invertible. But

if add i t ion is i n t e r p r e t e d to be the union of classes, this is an associative and commuta t ive ope ra t ion , and hence an addi t ion , bu t no t an invert-

ible addi t ion.

Peirce likewise dis t inguishes two subt rac t ion opera t ions . T h e subtrac- t ion x - y is the ope ra t i on inverse to addi t ion , when the add i t ion re-

fe r red to is invertible; he calls it "determinat ive ." Of course , when the

add i t ion is x +, y, the c o r r e s p o n d i n g subt rac t ion x - , y is no t de te rmi-

native; the re can be m a n y y such that x +, y -- z (given that xis n o n e m p t y ) .

Nowadays we use x - y to m e a n set d i f ference, i.e., the set of e l emen t s

in x but no t in y. Unl ike subt rac t ion in the integers , which is de f ined

in terms of addi t ion , this subt rac t ion is no t de f ined in te rms of un ion ,

bu t separately; a semilat t ice of sets closed u n d e r u n i o n is no t necessari ly closed u n d e r subtract ion.

As in his 1867 paper , Peirce in te rpre t s his add i t ion sign +, as inclusive

"or," explicitly d i f fe ren t f rom Boole 's + d e n o t i n g exclusive "or," which is a partial ope ra t ion , de f ined when the two terms of the sum are disjoint,

and not de f ined otherwise:

The sign of addition is taken by Boole, so that

x + y

denotes everything denoted by x, and, besides, everything denoted by y. Thus

m + w

denotes all men, and, besides, all women. This signification for this sign is needed for connecting the notation of logic with that of the theory of probabilities. But if there is anything which is denoted by both the terms of the sum, the latter no longer stands for any logical term on account of its implying that the objects denoted by one term are to be taken besides the objects denoted by the other. For example,

f + u

means all Frenchmen besides all violinists, and, therefore, considered as a logical term, implies that all French violinists are besides themselves. (p. 37)

3 ~ PEIRCE 'S CALCULUS OF RELATIVES

Thus, f + u implies that no F r e n c h m e n are violinists. Peirce prefers to take as the regular addi t ion of logic a non inver t ib le

o p e r a t i o n that is de f ined for all logical terms. He states that

m + , b

"stands for all m e n and all black things, wi thout any impl ica t ion that the black things are to be taken besides the men." Thus , m +, b cor- r e sponds to our inclusive "or." Class addi t ion, then, c o r r e s p o n d s to ar- i thmet ica l addi t ion only for disjoint classes. Thus , if m a nd w s tand for disjoint finite classes, then [m +, w] -- [m] + [w] makes sense and is true, where b racke t signifies cardinality.

Peirce then def ines the zero for his addi t ion:

By a zero I mean a term such that

x + , O : x,

whatever the significance of x. (p. 32)

He does no t give an example of how relative terms are to be added , bu t certainly in set- theoret ic terms he in tends a u n i o n of relat ions.

2 . 1 . 3 . M u l t i p l i c a t i o n

Peirce descr ibes mult ipl icat ion, xy, as an ope ra t i on that is associative and distr ibutes o'eer addi t ion, on bo th sides. It is, in genera l , n o n c o m - mutat ive. This is surely based on his fa ther ' s t r e a t m e n t of l inear associative algebras, that is, on the p roper t i es of matr ix add i t ion and mul- t iplication. Jus t as surely, for that reason Peirce does not requ i re that a mul t ip l ica t ion be commuta t ive . He writes commuta t ive mul t ip l ica t ion with a comma , x, y = y, x.

He calls a mul t ip l icat ion ope ra t i on invert ible if the cance l la t ion law holds and deno tes this ope ra t i on by a do t (per iod) , x . y = z. Thus, if x . y = z a n d x . y ' = z , t h e n y = y ' .

Peirce dis t inguishes be tween left and r ight quot ients , as is necessary w h e n mul t ip l ica t ion is no t a ssumed to be commuta t ive :

Division is the operation inverse to multiplication. Since multiplication is not generally commutative it is necessary to have two signs for division. I shall take:

(x: y)y = x,

y X ~ - -y .

X

(p. 31)


Peirce certainly knew of n o n c o m m u t a t i v e mul t ip l ica t ions f rom his father ' s l inear associative algebras, but by this t ime he also knew that p roduc t s of relatives are no t commuta t ive as well. In case a mul t ip l ica t ion is commuta t ive , he uses a semico lon to d e n o t e the inverse ope ra t ion : (x; y), y = x (p. 32).

Peirce in te rpre t s mul t ip l ica t ion xy as relative p roduc t , or c o m p o s i t i o n of relations. He descr ibes it as follows:

I shall adopt for the conception of multiplication the application of a relation, in such a way that, for example, lw shall denote whatever is lover of a woman. (p. 38)

Thus , lw is the p r o d u c t of a s imple relative and an absolu te term, and itself is an absolute term. Extensional ly (i.e., as a class), we would write lw as

{x:(3y)[l(x,y) A w(y)]},

that is, "x is the lover of y and y is a woman , for some y." We no te that there is an existential quant i f ie r in the ex tens iona l i n t e rp re t a t i on of relative produc t .

T h e nex t e x a m p l e Peirce gives of relative p r o d u c t is s (m +, w), which he says deno t e s "whatever is servant of any th ing of the class c o m p o s e d of m e n and w o m e n taken together ." This is the same l anguage Peirce used in his discussion of addi t ion , where he desc r ibed the o p e r a t i o n +, by saying "the c o n c e p t i o n of taking togetherinvolved in these processes

is s trongly ana logous to that of s u m m a t i o n " (p. 37). Extensionally, (m +, w) is the un ion of the classes m and w, and s (m +, w) can be wri t ten

{x:(3y)(s(x ,y) A [m(y) V w(y)])}.

Because mul t ip l ica t ion distr ibutes over add i t ion on the left, the equali ty

s(m +, w)= sm +, sw

holds, and because mul t ip l ica t ion distr ibutes over add i t ion on the right, Peirce can say that

(l +, s)w will denote whatever is lover or servant to a woman, and

(/-h s)w = lw +, sw.

(p. ~8)

Finally, because mul t ip l ica t ion is associative, Peirce can say that

(sl)w will denote whatever stands to a woman in the relation of servant of a lover, and

32

(p. 38)

PEIRCE'S CALCULUS OF RELATIVES

(s/)w : s(/w).

For the mos t par t , Pe i rce uses the n o t a t i o n for relative p r o d u c t consis-

tent ly in such a way tha t the first t e rm in the p r o d u c t is a relat ive t e r m

a n d the s e c o n d t e rm is an abso lu te te rm. T h e last f o r m u l a is an excep-

t ion. In the p r o d u c t (s/)w, sl is a relative, n o t an abso lu te t e rm. Exten-

sionally, (s/)w c o r r e s p o n d s to the set

Ix: (3z)[(3y)[s(x, y) A l(y, z)] A w(z)]}.

But this is the same as the set

{x" (3y)[s(x, y) A ((3z)[l(y,z) A w(z)])]},

which c o r r e s p o n d s to s(/w), i.e., "whatever s tands to a lover o f a w o m a n in the re la t ion o f a se rvan t o f hers," a n d so Pe i rce ' s th i rd equa l i ty holds .

In la ter pape r s Pe i rce elevates abso lu te t e rms (sets) to b ina ry re la t ions

by iden t i fy ing t h e m with the i r ident i ty re la t ion , a n d gene ra l l y i n t r o d u c e s

a device to r e p e a t an a r g u m e n t in an n-ary re la t ion to get an (n + 1)-

ary re la t ion . By us ing these devices he can t h e n s imply d r o p the dis- t inc t ion b e t w e e n the t h r ee kinds o f t e rms (absolu te , relative, a n d con-

juga t ive ) given here . This is a device used in var iable- f ree ca lculus for

h a n d l i n g the p r o b l e m s tha t in f i rs t -order logic are dea l t with by wr i t ing the s ame var iable at several places in a fo rmula .

Pe i rce specifies a un i t e l e m e n t for relative mul t ip l i ca t ion :

The term "identical with -" is a unity for this multiplication. That is to say, if we denote "identical with -" by I, we have

x l = X,

whatever relative term x may be. (p. 38)

Bo th x a n d I are relative terms.

Pe i rce t h e n descr ibes mu l t i p l i ca t i on by a conjugat ive :

A conjugative term like giver naturally requires two correlates, one denot ing the thing given, the other the recipient of the gift. We must be able to distinguish, in our notation, the giver of A to B from the giver to A of B, and, therefore, I suppose the signification of the letter equivalent to such a relative to distinguish the correlates as first, second, third, etc., so that "giver of m to - -" and "gqver to - - o f - - " will be expressed by different letters. Let g denote the latter of these conjugative terms. Then the correlates or multiplicands of this mul- tiplier cannot all stand directly after it, as is usual in multiplication, but may be ranged after it in regular order, so that


gxy

will denote a giver to x of y. But according to the notation, x here multiplies y, so that if we put for x owner (o), and for y horse (h),

goh

appears to denote the giver of a horse to an owner of a horse. (pp. 38--39)

33

T h e t e r m goh involves two relative mu l t i p l i ca t i ons a n d is diff icul t to i n t e rp re t . T h e p r o d u c t gxy can d e n o t e the giver to x o f y, bu t t h e n x

mul t ip l i es y, a n d so x is a r e l a t ion itself. If we mul t ip ly y by x a n d then xy by g, this has a d i f f e r en t m e a n i n g than if we mul t ip ly x by g a n d t h e n mul t ip ly y by gx. In the first ins tance , goh d e n o t e s the giver to an o w n e r

o f a ho r se o f tha t s ame pe r son ; tha t is, s o m e o n e who gives a p e r s o n to

himself , a n d tha t p e r s o n owns a horse . In the s e c o n d ins tance , goh d e n o t e s the giver o f a ho r se to an o w n e r o f a horse , w h e r e the s am e

ho r se is m e a n t ; h is used twice, syntactically: o n c e as the c o r r e l a t e o f

the relat ive 0 a n d o n c e as the s e c o n d co r r e l a t e o f the con juga t ive g.

This la t te r i n t e r p r e t a t i o n is, in fact, the o n e tha t Pe i rce in tends :

[L]et the individual horses be H, It', H", etc. Then

h = H +, H' +, H" +, etc.

goh =go(H +, H '+ , H"+, etc.) = goH +, goH' +, goH"+, etc.

Now this last member must be interpreted as a giver of a horse to the owner of that horse, and this, therefore, must be the interpretat ion of goh. (p. 39)

Pe i rce wants to c rea te a s ingle b i n d i n g a n d t h e n m u l t i p l e r e f e r e n c e s to tha t b ind ing . His so lu t ion is to r ead goh as t h o u g h it were g(0h)h. But

g(0h)h does n o t m e a n qu i te wha t he i n t e n d s by goh b e c a u s e g(0h)h

m e a n s the giver o f a ho r se to the o w n e r of a possibly different horse . Pe i rce has no easy way to express a giver of a ho r se to the o w n e r o f

the same ho r se us ing the mul t ip l ica t ive n o t a t i o n he prefers . H e says, however , tha t

This is always very important. A term multiplied by two relatives shows that the same individual is in the two relations. (p. 39)

In o t h e r words , the r e a d e r is to understand by this n o t a t i o n tha t the

s ame ob jec t is a p p e a r i n g twice in the two d i f f e r e n t re la t ions .

Pe i rce a t t e m p t s to i m p r o v e the n o t a t i o n by p u t t i n g in n u m e r i c a l indices. H e can t h e n specify w h e r e to f ind the c o r r e c t first a n d s e c o n d co r re l a t e s a n d writes giver of a ho r se to a lover o f a w o m a n in t h r e e


equivalent ways: glzllWh =gll/zhW =g2_~hl~w. The first index specifies how many factors must be c o u n t e d f rom left to r ight to reach the first corre la te , the second index specifies how many m o r e factors mus t be c o u n t e d to reach the second, and so on. A negative n u m b e r indicates that the first corre la te follows the second by the c o r r e s p o n d i n g positive number . A zero makes the te rm itself the correlate:

A subadjacent zero makes the term itself the correlate. Thus,

l0

denotes the lover of that lover or the lover of himself, just as goh denotes that the horse is given to the owner of itself. (p. 40) 2

Peirce says that if the last subadjacent n u m b e r is a 1, it may be omi t ted ,

and so goh and g~Olh are equivalent expressions. Peirce then uses what we recognize as the m o d e r n categories of b inding and o c c u r r e n c e to

i n t e rp re t g~o~h. Thus, each "variable" is b o u n d only once, but it may have mul t ip le occurrences . This is how Peirce in te rpre ts gl~O~h, i.e., he in t roduces a "name" h for the class of all horses, b ind ing an individual horse f rom the class of all horses. He wants that individual horse to be the same horse that occurs in the b inding sites provided by the second

"1" in gll and by the "1;' in 0~, and consequent ly gllOlh is a giver of a horse to the owner of the same horse, which is the i n t e n d e d in terpre- tation.

The express ion g~o~h is an absolute term. To put it in m o d e r n terms, it is a set; we migh t write it as

{x : (3y)(3z)[g(x, y, z) ^ o(z, y) ^ h(y)]}.

Thus, we can view gllOl h as a nota t ion that captures cer tain ideas of quant i f icat ion, inc luding b ind ing and occurrences , in a way that does

not involve the n a m i n g of variables x,y, z to accompl ish that task.

Peirce 's system is s t rong e n o u g h to express the relation: giver of a

horse to the owner of a horse, where the horses are different . Peirce

would formalize this not ion as g~201hh, where the repe t i t ion of the h creates two di f ferent bindings.

Peirce also considers mul t ip l icat ion of absolute terms:

Thus far, we have considered the multiplication of relative terms only. Since our conception of multiplication is the application of a relation, we can only multiply absolute terms by considering them as relatives.

That is, ~ = 1, I; in other words, ~ is I intersected with the identity relation. Our analysis of Peirce's comma operator follows on pp. 35-38. Burch (1997a) gives a detailed analysis of the comma in Peirce.

F R O M P E I R C E T O SKOLEM

Now the absolute term "man" is really exactly the equivalent to the relative term "man that is m," and so with any other. I shall write a comma after any absolute term to show that it is so regarded as a relative term. (p. 41)

35

For Pe i rce , the c o m m a is an o p e r a t o r t ha t inc reases the ari ty o f a t e rm.

In his e x a m p l e , the a b s o l u t e t e r m " m a n " is c o n v e r t e d to the re la t ive

" m a n tha t is - " by the c o m m a o p e r a t o r ; tha t is, "m ," is " m a n tha t is - . "

Pe i rce t h e n app l ies m , to the abso lu t e t e r m b to ge t m , b o r " m a n tha t

is black." T h u s m , b c o r r e s p o n d s e x t e n s i o n a l l y to the i n t e r s e c t i o n o f

the class " m a n " a n d the class "black": m , b - m n b, i.e., all b lack m e n .

It is an abso lu t e t e rm, d e n o t i n g all m e n tha t a re black.

We see h e r e an i n t e r e s t i n g f ea tu r e o f Pe i r ce ' s 1870 paper . Pe i r ce takes

the m u l t i p l i c a t i o n o f re la t ive t e r m s as pr imi t ive . W h e n he s u b s e q u e n t l y

de f ines m u l t i p l i c a t i o n for ab so lu t e te rms, it is as a d e r i v e d c o n c e p t .

Mul t ip l i ca t ion for abso lu t e t e rms is e x p l a i n e d as a m o r e c o m p l e x

s y m b o l n m , b- - - involv ing a c o m m a as well as a s i m p l e j u x t a p o s i t i o n .

Thus , w h e r e a s we w o u l d r ead m , b today as a b i n a r y o p e r a t i o n b e t w e e n

m a n d b, Pe i rce cons ide r s m , b as an o p e r a t o r on m tha t yields a re la t ive

t e r m a p p l i e d to b.

Pe i rce f u r t h e r i n t r o d u c e s the c o m m a as a device t ha t i nc reases ari ty

in s o m e p a r t i c u l a r way. Pe i rce first exp la ins wha t i t .means to have several

co r re la tes :

But not only may any absolute term be thus regarded as a relative term, but any relative term may in the same way be regarded as a relative with one correlate more. It is convenient to take this additional correlate as the first one. Then

l, sw

will denote a lover of a woman that is a servant of that woman. The comma here after the l should not be considered as altering at all the meaning of l, but as only a subjacent sign, serving to alter the a r rangement of the correlates. (p. 41)

In his e x a m p l e , l , sw has exact ly the f o r m of the re la t ive p r o d u c t g iven

by goh, t ha t is, a con juga t ive t imes a re la t ive t imes an a b s o l u t e t e rm;

l , sw w o u l d d e n o t e a c c o r d i n g l y " lover o f a w o m a n who is a s e rvan t o f

tha t w o m a n . "

To see how Pe i rce adds new cor re la tes , we will c o n s i d e r an eas ie r

e x a m p l e , l , mw. We know tha t lw is " lover o f a w o m a n " ; t h e n l , mw will

d e n o t e " lover o f a w o m a n who is a m a n " (i.e., the lover is also a m a n ) .

T h e o t h e r possibi l i ty is for l , mw to d e n o t e " lover o f a m a n who is a

w o m a n , " bu t this i n t e r p r e t a t i o n is n o t c o n s i s t e n t with Pe i r ce ' s conven-

t ion o f always a d d i n g the new c o r r e l a t e to the left of the first co r r e l a t e .

3 6 PEIRCE 'S CALCULUS OF RELATIVES

In his s t andard example goh, "giver of a horse to the owne r of that horse," the second corre la te , "owner o f - , " is to the left of the first

corre la te . For consistency, then, l, mw deno te s the lover of a w o m a n who is a

man. Now, r e tu rn ing to the p r o b l e m of i n t e rp re t i ng l, sw on the ana logy of the relative p r o d u c t of a conjugat ive times a relative t imes an absolu te term, l, sw deno tes the lover of a w o m a n who is a servant of that woman . This in t e rp re t a t ion is exactly in parallel to that of goh, the giver o f a horse to the owner of that h o r s e .

In a passage that is somewha t p rob lemat ic , Peirce states that multi- pl icat ion ind ica ted by a c o m m a is commuta t ive :

It is obvious that multiplication into a multiplicand indicated by a comma is commutative, that is

s,l=l,s.

This multiplication is effectively the same as that of Boole in his logical calculus. (p. 43)

Peirce c a n n o t simply say that mul t ip l ica t ion ind ica ted by a c o m m a is commuta t i ve because he has, in the p r e c e d i n g discussion, given the c o m m a a m e a n i n g de f ined by the mul t ip l ica t ion .of relative terms, which

is no t commuta t ive in general . How can s , l - l , s? First, let us cons ider s , / . Applying the c o m m a to the s imple relative

s results in the conjugat ive te rm s, which deno te s "servant o f - who is -"; l d eno t e s "lover o f - . " To fo rm s, l we identify correlates: using variable no ta t ion , we can write s , as "x is a servant o f y who is z" (where the "who" refers to x) and I as "u is a lover o f v." T h e n in the p r o d u c t s , l the corre la te z is ident i f ied with u, and y and v are ident i f ied, so that s , I co r r e sponds to "x is a servant of y who is a lover o f y"; in o t h e r words, s , l deno te s "servant o f - and lover o f - . " Similarly, l , s corresponds to "x is a lover of y who is a servant o f y"; in o t h e r words, l , s d e n o t e s "lover o f - and servant o f - . " This is evidentially the same as

"lover o f - and servant o f - . " By this in te rp re ta t ion , s, l = l, s, and the c o m m a mul t ip l ica t ion is com-

mutat ive. The ident i f icat ion of the variables is some w ha t artificial, bu t this i n t e rp re t a t ion has similarities with Peirce 's s t andard e x a m p l e of mul t ip l ica t ion by a conjugat ive, goh, in which h is used twice.

Peirce uses the c o m m a as a type coe rc ion device that increases the arity o f his logical terms by increas ing the n u m b e r o f a r g u m e n t s by one . Firs t -order logic uses variables and Cartesian p r o d u c t with the d o m a i n for this purpose . Thus, we can in t e rp re t Peirce 's m , ( "man who is also -"; i.e., a relative clause) in f irst-order logic by taking m ,(x,y) if and only if m(x), i.e., i n d e p e n d e n t of the second parameter . T h e n m , cor-


r e sponds to {x" m(x)} x D = {(x,y) �9 m(x) A y ~ D}. This is the logical un- d e r p i n n i n g of the comma . If we read Peirce in this way, it does no t ma t t e r what the type level of a logical t e rm is.

By applying the c o m m a e n o u g h times on any logical t e rm he likes, Peirce can br ing any con junc t i on of logical terms of various arities up to the same arity. This allows him, wi thout any formal variables, to represent con junc t ions of logical terms in formulas equ iva len t to m o d e r n - day set- theoret ic intersect ions. For instance, let the relative s ("servant of") c o r r e s p o n d to {(x, y) " s(x, y)}, i.e., all pairs such that x is a servant o f y. T h e n the in te rsec t ion of m ,(x, y) in the e x a m p l e given above with s(x,y) yields { ( x , y ) 'm(x ) A s(x,y)}, as desired. This can be d o n e for any finite n u m b e r of con junc t ions of Peirce 's logical terms, to br ing t h e m into a single Cartes ian p r o d u c t in which thei r c o n j u n c t i o n is the i r intersect ion. T h e n using "+," (ord inary m o d e r n d is junct ion) of these, the d is junct ion of such con junc t ions co r r e sponds to a u n i o n of intersections. Thus , Peirce obtains an equiva len t o f any quant i f ier - f ree fo rmu la r e p r e s e n t e d as a un ion of an in tersec t ion of re la t ions within a single Car tes ian power of the domain . He does this with no variables.

Tarski (1956, pp. 195-201) , in def in ing satisfaction, uses a fixed infinite list o f variables, x~, xz . . . . . Ins tead of finite Car tes ian products , he uses the set of all infinite sequences f rom the d o m a i n , wri t ten D ~. He corre la tes the nth t e rm of the s equence with the variable n. T h e n , if R(x l, xs) is a fo rmula and R deno te s a subset of D x D, he identif ies R with R ~ de f ined as the set of all infinite s equences a 0, a~ . . . . f rom D such that R(a l, a~). This is the satisfaction set o f the f o r m u l a R(x l, x~). T h e n all relat ions R are pu l led up to be subsets R ~~ of a single set D ~, and the ope ra t ions of un ion and in tersec t ion c o r r e s p o n d to d is junc t ion and con junc t ion .

Peirce viewed every m-ary relat ion as be ing ex t end ib l e to an n-ary rela t ion in many ways, by dupl ica t ing variables. All o t h e r variables are then left free, as in Tarski. Take, for example , any p red ica t e R(x~). It e m b e d s as S(x l, x 2 . . . . ) if and only R(x 1). But if we dup l ica te the variable as R(xl , ), get t ing a relat ion T(x l, x2) if and only if R(xl ) , a nd e m b e d the dup l i ca t ed R(x~,), that is, e m b e d T(Xl, X2) , the e m b e d d e d fo rm is a d i f fe ren t S, namely, S'(x l, x 2, x~ . . . . ), if and only if R(x l) a nd x I = x 2. This says tha t pu t t ing c o m m a s in to dupl ica te variables gives perfec t ly clear c o r r e s p o n d i n g Tarski semantics. T h e n R(x~) e m b e d s infinitarily as Rl(xl , x2... .) if and only if R(xl ) , while R(xl , ) , o r S(Xl,X2) e m b e d s infinitarily as S~ (x~, x 2 . . . . ) if and only if S(x 1, Xz), that is, if a nd only if x I = x 2 and R(Xl). C o m m a s hand le the no t ion "the same variable occurs in two places" wi thou t variables.

Peirce fu r the r says that

[I]f we are to suppose that absolute terms are multipliers at all (as

3 8 PEIRCE'S CALCULUS OF RELATIVES

mathematical generality demands that we should), we must regard every term as being a relative requiring an infinite number of correlates to its virtual infinite series "that is - and is - and is - etc." (p. 42)

As an example , he writes

l , s w = l , s w , l , l , l , l , l , l , I , etc.,

where 1 is the identity relat ion and the l 's deno te the same individual d e n o t e d by w. Peirce uses ", I, I, ..." to increase (potentially) the arity of any term, so that lower arity relations could be used in mult ipl icat ions.

2.1.4. Peirce's First Quantifiers

From his idea of implicit infinitary relatives, Peirce develops a very limited theory of quantif icat ion, in which Ioo and I0 are quant if iers of a sort:

"Something" may then be expressed by

I~0.

I shall for brevity express this by an antique figure one (l). "Anything" by

I0

I shall often write straight 1 for anything. (p. 43)

Here is an in te rpre ta t ion of I0o and 10 in m o d e r n terms. We can think of 1~ as a unary predica te (i.e., an absolute term). Suppose we have a possibly infinite doma in D. It has a Boolean a lgebra of all subsets of D, which has one -e l emen t subsets I as atoms. (We write I to s tand for a Pei rcean individual.) The ent i re set D is the least u p p e r b o u n d of these infinitely many atoms. Each a tom is a one -e l emen t set, and thus is a unary relat ion (i.e., proper ty) ho ld ing of that e l e m e n t and no other.

T h e whole collection is loo; as Peirce would write it, I~ - I +, I' +, I" +, .... This 1~o is the disjunction of all the unary proper t ies charac ter iz ing single e lements . To say that an (arbitrary) object has p roper ty Ioo is to say it is in the d o m a i n D (existential quant i f ier as dis junct ion) .

T h e second quantifier, 10, is the universal property. In m o d e r n terms, it is the project ion of the diagonal relat ion onto the first a r g u m e n t of the relation. In o the r words, since 1 is the identity relat ion, it includes all pairs (0, 0), (A,A), (B,B) . . . . . where A, B, etc., are all the individuals in the universe. Since the index 0 makes the te rm itself the correlate ,


I0 must be the entire universe, i.e., all individuals 0, A, B . . . . . since it consists of everything that is identical to itself.

Nowadays quantifiers are operators, separate entities that are applied to formulas, but for Peirce the quantifiers Ioo and I 0 are relations themselves. Consider the absolute term "lover of something," i.e., l~o. This is equal to the relative product of the relative "lover of" and the absolute term "something," i.e., lloo. In this formula, a term in the relation, loo, does the quantification: l l= is a lover of something, i.e., " there is something" that the lover loves.

Peirce does not give examples in his 1870 paper using these quantifiers, and he does not explain how they might work with more complex formulas; this is a simple first attempt. Peirce gets far ther than Aristotle did, but he does not yet see quantifiers as separate entities.

2 . 1 . 5 . I n v o l u t i o n

Having already said what axioms should hold for operat ions to be designated as addit ion and multiplication, Peirce states what axioms hold for exponent ia t ion. Following De Morgan, he prefers the term "involution" to "exponentiat ion." Peirce states three principles that involution must satisfy:

The operation of involution obeys the formula

(x~)~ = x~).

Involution follows the "indexical principle"

X y+'z = xY~ X' .

Involution also satisfies the binomial theorem

(x +, y)~ = x ~ +, F,t,x ~-t' , yt, .4, y~.

(pp. 30-31)

The first two principles are present in every later axiomatization of exponent ia t ion.

To read the first principle, r e m e m b e r that x y means the set of things that the binary relation x relates to all e lements ofy. Thus, on the surface x y is a set. But Peirce here and everywhere identifies a set with its identity relation, the set of pairs (u, u) of things u that binary relation x relates to all e lements of y. This then makes x y a binary relation as well, and for a set z we can now form (xY) z. A momen t ' s thought reveals that this is the set of things that x relates to all e lements of y and all e lements of z. But the o ther side of the first equation, x Cyz), is the set of things that x relates to all e lements of the intersection of y and z. These are

4 ~ P E I R C E ' S C A L C U L U S O F R E L A T I V E S

the same set, which is the first axiom. If we i te ra ted fu r the r exponen t s , each t ime we would use the identi ty re la t ion to r e p r e s e n t the set

involved. To read the second pr inciple , p r o c e e d similarly. The left side is the

set of things re la ted by x to every m e m b e r of the un ion of y and z. This

is the same as the set of things re la ted by x to every m e m b e r of y,

in te r sec ted with the set of things re la ted by x to every m e m b e r of z. T h e thi rd pr inciple is Peirce 's set- theoret ic ana logue of the b inomia l

t h e o r e m . The left side is the set of all things re la ted by the u n i o n of x

and y to all e l ements of z. The r ight side is a un ion of the following

sets. The first set is the set of things re la ted by x to every m e m b e r of z. T h e last set is the set of things re la ted by y to every m e m b e r of z. A

typical midd le te rm is then the set of things re la ted by x to every e l e m e n t of z - p and by y to every e l e m e n t of p. The p in the s u m m a t i o n (ex-

t e n d e d un ion) is over all n o n e m p t y p r o p e r subsets p of z. If the sum-

ma t ion were over all subsets of z, the end terms would no t n e e d to be

d e n o t e d separately. If z is infinite, the whole th ing is a un ion of the

infini te col lect ion of all subsets of z, that is, over the power set of z. Peirce thus uses un ions of arbi t rary sets of sets wi thout even m e n t i o n i n g

it, no t jus t finite un ions of sets. If the sets are finite and we take car-

dinalit ies, this is i ndeed the b inomia l t h e o r e m , p roved by set- theoret ic

i n t e rp re t a t i on of cardinal opera t ions .

T h e t radi t ion of simply applying formal m e t h o d s f rom a lgebra or calculus to find a lgebra of logic rules and expans ions was no t o r ig ina ted by Peirce. It was d o n e by Boole. In Boole it s tems pe rhaps f rom the

formal m e t h o d s in his books on the calculus of finite di f ferences , which

are classics still. Peirce was following the Boolean t radi t ion he re in trying the b inomia l t heo rem, etc., wi thout just i f icat ion, as pure ly formal tools

for f inding formulas. Peirce gives his i n t e rp re t a t ion of the involut ion o p e r a t i o n in the cal-

culus of relatives as follows:

I shall take involution in such a sense that x y will denote everything which is an x for every individual of y. Thus l w will be a lover of every woman. Then (st) W will denote whatever stands to every woman in the relation of servant of every lover of hers; and s ~tW) will denote whatever is a servant of everything that is lover of a woman. So that

(sl)w = $(lw).

(p. 45)

H e r e l w deno te s "lover of every woman." In set- theoret ic terms, I w cor-

r e sponds to the set


Ix" (vy)[w(y) ~ t(x,y)]].

We note that there is a universal quant i f ie r in the def ini t ion. Further , s ~ deno tes "servant of every lover of " and co r r e sponds to

the set

{(x, y) �9 (vz)[l(z, y) ~ s(x, z)]}.

Peirce claims that (st) W - s~tW); in o the r words, that the servant of every

lover of every woman is the same as the servant of every lover of a woman. This is not obvious, so we will show that Peirce 's claim is correct .

To show (sl)W=s ~tW)" First, we write (sl) w and s ~tW) as (st)w= {x" (Vy)[w(y) ~ (Vz)( l(y ,z) ~ s(x,z))]} and s ~lw) ={x" (Vz)[(3y)(w(y) ^ l(y, z)) ~ s(x, z)]}. T h e n

(s') * = (Vy)[w(y) ~ (Vz)(l(y,z) ~ s(x,z))]

= (Vz)(Vy)[w(y) =r (l(y,z) =r s(x,z))]

- (u v ",l(y, z) v s(x, z)]

- (Vz)[(~y)(w(y) ^ l(y,z)) ~ s(x, z)],

which is s ~t'). Thus (st)W= s <tw).

Peirce recognizes that the algebraic opera t ions induced by involut ion share some of the same proper t ies as ord inary exponent ia ls . For example , his fo rmula sm+'w = S m , S w says, extensionally,

Yy[(m(y) V w(y)) ~ s(x,y)] r162 Vy[m(y) ~ s(x,y)] ^ Yy[w(y) ~ s(x,y)].

This is a s t a t emen t of a proper ty of universal quant i f ica t ion, which

Peirce 's fo rmula gives in a m o r e succinct form. Paraphrased , it says that a person is a servant of everything in the class of m e n and w o m e n taken t oge the r if and only if that person is a servant of every m a n and is a servant of every woman. This is jus t the s tandard additive law for exponents . Again, we see Peirce p resen t ing logic in a way that makes it r esemble the l inear associative a lgebra of his father.

Peirce also gives o the r laws for man ipu la t ing opera to r s and e x p o n e n t s in formulas express ing universal s ta tements . The m o d e r n coun te rpa r t s of these laws are rules for ext rac t ing Boolean opera to r s ou t of impli-

cations within the scope of universal quantifiers. For example , the law

( s , l ) W = s w, I w

allows one to pull an "and" connect ive out of the impl icat ion Vy[w(y) ~ (s(x,y) ^ l(x,y))]; that is, (s, l) w = s w , 1 w states the tautology

Yy[w(y) ~ (s(x,y) ^ l(x,y))] ~ Yy[w(y) ~ s(x,y)] ^ Yy[w(y) ~ l(x,y)].

This fo rmula exhibits a t echn ique for simplifying express ions by moving quantif iers inward. Many of the e l iminat ion-of-quant i f ier proofs in de-

cidability theory work in this way, namely, by a t tacking the i n n e r m o s t


quant i f i e r in a fo rmula and a t t e m p t i n g to e l imina te it a l t oge the r by push ing it down t h r o u g h the Boolean opera tors . It seems that Peirce was in te res ted in formulas that allowed him to make such reduc t ions

in the e x p o n e n t , which carries the universal quant i f icat ion.

Peirce states the law of con t rad ic t ion as x, nX= 0, where n is a relative s t and ing for "not" (p. 48). The express ion x , n x = 0 says all the x's which

are no t x's are none . Peirce does no t discuss n ~, bu t we will give an

exp lana t ion . If x is an absolute term, n ~ is, in m o d e r n set- theoret ic terms,

{y: (u ~ n(y,z)]}, or {y: (u ~ x)(y ~ z)}. It is every th ing which is

d i f fe ren t f rom every x, which is no t z, where z is an x. This is jus t the

set - theoret ic c o m p l e m e n t . If x is a relat ion, n~= {(y, z) : (Vw)(x(w, z) y ~: w)}, and it mat ters no t what z is. This is the re la t ional nega t ion .

2.1.6. Involut ion and Mixed-Quanti f ier Forms

Peirce also applies involut ion to conjugat ive terms. He says that "bet rayer

to every enemy" should be writ ten b", where b signifies "bet rayer to -

o f - " and a is "enemy o f - , " jus t as "lover of every woman" is wri t ten 1W (p. 46). (The verbal form of b" should, in fact, be "bet rayer to every

e n e m y of - , " since b;' is a [binary] relative term.) Since b has two correlates, the re are six d i f ferent ways, using relative mul t ip l ica t ion and

involut ion, of a t taching corre la tes to b:

ham (ba) m

ba m b am

b=m b =m

be t rayer of a m a n to an e n e m y of h im

be t rayer of every m a n to some e n e m y of h im

be t rayer of each m a n to an e n e m y of every m a n

be t rayer of a m a n to all enemies of all m e n

be t rayer of a m a n to every e n e m y of h im

be t rayer of every m a n to every e n e m y of him.

(p. 46) In the first case, bam deno te s "betrayer of a man to the e n e m y of

that same man." This is comple te ly ana logous to goh, "giver of a horse

to the owner of that same horse." In the second case, the sub fo rmu la

ba d e n o t e s "betrayer to an e n e m y of m ," a relative term. So ( b a ) m deno t e s

"bet rayer of every m a n to an e n e m y of that man." In the th i rd case, the s u b f o r m u l a a m deno tes "enemy of every man," and ba m d e n o t e s "be-

t rayer of every man to an e n e m y of every man."

These six terms c o r r e s p o n d to mixed-quant i f i e r express ions in first-

o r d e r logic,


barn = {x" (3y)(3z)[b(x, y, z) ^ a(z, y) ^ m(y)]}

(ba)m = {x" (u :::* (b(x, y,z) A a(z, y))]l

barn= {x" (3z)(Vy)[m(y) ~ (b(x,y ,z) ^ a(z,y))]} i n

b ~' = {x ' (3y) (m(y) A (Vz)[(Vw)[m(w) = a(z, w)] ~ b(x,y,z)])l

b"m = {x' (3y)(qz)[(a(z, y) ~ b(x, y, z)) A m(y)]}

b "m = Ix" (u y) A m(y)) ~ b(x, y, z)] I.

O f course, Peirce does no t use p red ica te logic to justify his in te rpre- tat ion of these terms�9 Instead, he uses laws for the relative calculus that are of ten s imple ana logues to the laws of l inear associative algebra. For example , in the fifth case above, b" deno te s "bet rayer to every e n e m y o f - . " Peirce says that "any relative x may be conce ived as a sum of relatives X, X', X", etc., such that there is bu t o n e individual to which any th ing is X, but one to which any th ing is X', etc." (p. 50), and thus he can write a = A +, A' +, A' +, "", where there is bu t o n e individual to which any th ing is A, etc. So,

A t A t/ b " = b a +'at +'att + . . . . - - b a b b - - ~ ~ ~ � 9

by the index law for exponenLs. But since the re is only o n e individual to which any th ing is A, etc., the relatives A, A' . . . . each c o r r e s p o n d to

�9 A t - , 4 n

a single pair, and thus b A = hA. :~ T h e r e f o r e , b a , b , b , . . . =

hA" Likewise, an absolute te rm can be wri t ten as a sum of hA , hA' , , .... individuals, and so m - M +, M' +, M" +, ...; thus ham can be wri t ten as

(hA , hA' , hA", . . . )m = (hA , hA' , hA", ...)(M +, M' +, M" +, ...)

= ( h A M , h A ' M , h A " M , ...)

+, ( h A M ' , ba 'M' , bA"M' , ...) +, ...

T h e first s u m m a n d of the result on the r ight is n o n z e r o if the re is a single be t rayer of the individual m a n M to each of his enemies , a n d the r e m a i n i n g s u m m a n d s will be n o n z e r o whene ve r the same cond i t i on is t rue for the individual man d e n o t e d by the M p r i m e d te rm in each s u m m a n d . It is evident , then, that b"m signifies the be t rayer of some m a n to every one of his enemies . 4

In a no te a d d e d at the p r in t ing stage (p. 69), Peirce gives a f o r m u l a

.s In set n o t a t i o n , b A = {(x,y) : u ~ b(x,y,z)]} a n d bA ={(x,y) : 3z[A(z,y) A b(x,y,z)]}, b u t A is a r e l a t i o n t h a t is j u s t t r u e for o n e e l e m e n t , a n d "exists" a n d "for all" a r e t he s a m e ove r a u n i v e r s e with on ly o n e e l e m e n t , t h u s b a a n d bA a r e t h e s a m e .

4 T h e r e has b e e n at least o n e a t t e m p t to e x t e n d this ca lcu lus ; B u r c h (1997a) a t t e m p t s to s h o w t h a t this sys tem has t he p o w e r o f full f i r s t -o rde r logic. (See a lso P u t n a m 1995.)

44 P E I R C E ' S C A L C U L U S OF R E L A T I V E S

showing that involution can be expressed in terms of relative product ,

viz.,

/ ' = 1 - ( 1 - / ) _-2"-

i.e., l ' = / s . Strictly speaking, relative p roduc t is the only ope ra t ion that

he needs to express the not ions of "some" and "all." Exponen t i a t ion , however, is useful in express ing the mixed-quant i f ie r s ta tements in con-

cise form.

2.1.7. Elementary Relatives

We have m e n t i o n e d t h r o u g h o u t our discussion that Peirce did not have

the m o d e r n not ion of o r d e r e d pairs or sets of o r d e r e d pairs. Rather,

he s tar ted f rom three notions:

There are in the logic of relatives three kinds of terms which involve general suppositions of individual cases. The first are individual terms, which denote only individuals; the second are those relatives whose correlatives are individual: I term these infinitesimal relatives', the third are individual infinitesimal relatives, and these I term elementary relatives. (p. 59)

We would unde r s t and individual terms as s ingleton sets; the relatives whose correlatives are individual as functions, i.e., single-valued sets of o r d e r e d pairs; and the e l emen ta ry relatives as sets consist ing of a single

o r d e r e d pair (a, b). However, Peirce 's division does not in fact reflect exactly what he

does. M t h o u g h he will dist inguish individual terms such as, for example , H, H' . . . . for (a possibly infinite n u m b e r of) individual horses, f rom the class te rm h for the collect ion of horses, many times he wants formulas

that allow classes to be man ipu l a t ed in jus t the same way as individuals,

p rovided that the proposi t ions about t hem apply to the whole class (or no m e m b e r of the class). To that end he defines e l emen ta ry relatives

as those that hold between whole classes, such as teachers , pupils, and colleagues, which are collective nouns. He defines these as follows:

By an elementary relative I mean one which signifies a relation which exists only between mutually exclusive pairs (or in the case of a conjugative term, triplets, etc.) of individuals, or else between pairs of classes in such a way that every individual of one class of the pair is in relation to every individual of another. If we suppose that in every school, every teacher teaches every pupil (an assumption which I shall tacitly make whenever in this paper I speak of school), then pupil is an elementary relative. (p. 75)


These classes are subclasses of the original class of individuals. In addition, Peirce insists that the classes be disjoint, so that there must be an absolute distinction between teachers and pupils:

The existence of an elementary relation supposes the existence of mutually exclusive pairs of classes. The first members of those pairs have something in common which discriminates them from the second members, and may therefore be united in one class, while the second members are united into a second class. Thus pupil is not an elementary relative unless there is an absolute distinction between those who teach and those who are taught. (p. 76)

In modern terms, for defining e lementary relatives between classes, there must be an equivalence relation E on the domain given, and these classes must be equivalence classes. Finally, the equivalence relation must preserve the relations involved in the system. That is, for all individuals x, y, all relations R of the relational system or model , R(x ,y)A E(x, x') A E(y, y') ~ R(x', y'). Peirce's scalars (which multiply l inear combinations of the e lementary relatives) are propert ies P of individuals preserved unde r the equivalence relation; that is, P(x) A E(x, y) =, P(y).

In the example Peirce gives (p. 76), A is the class of teachers and B is the class of students, assumed to be disjoint and exhaust ing individuals. Thus

A : A is colleague, the class of pairs of teachers; B : B is schoolmate, the class of pairs of students; A : B is teacher, the class of pairs with first m e m b e r teacher and

second student; B : A is pupil, the class of pairs with first m e m b e r s tudent and second

teacher. The equivalence relation on the domain equates all students and equates all teachers and equates no s tudent with a teacher.

The four relations of colleague, schoolmate, teacher, and pupil are well-defined on the set of two equivalence classes, {set of teachers, set of students}. In Peirce's example, in the case all s tudents and teachers are French and f is the property of being French on the original individuals, he uses the comma relative "f," as the proper ty of being French on the quotient, more or less.

This is an exact definition of a h o m o m o r p h i c image of a relational system, obtained by taking a quot ient with respect to the equivalence relation. The "scalar characters" (e.g., "f,") are the propert ies of the quot ient thought of as propert ies of the original individuals, which are well def ined with respect to the equivalence relation.

In sum, this is a special branch of the calculus of relatives dealing with quotients, accounting for relations between class terms that are


t rue because they hold for all individuals in the class d e n o t e d by the class term. In m o d e r n terms, it is a quo t i en t relat ional system. But Peirce does no t give such an explicit not ion.

2.2. Quantification in the Calculus of Relatives in 1870

Peirce 's 1870 paper culminates with an applicat ion of his theory to an

example that we believe motivated the deve lopmen t of his calculus of relatives in the first place. Peirce claims that by including his own funct ion

0 x, he has e x t e n d e d Boole 's calculus to a c c o m m o d a t e the quant i f icat ion

con ta ined in Aristotelian syllogisms. He states the p rob l em thus:

That which first led me to seek for the present extension of Boole's logical notation was the consideration that as he left his algebra, neither hypothetical propositions nor particular propositions could be properly expressed . . . .

What is wanted, in order to express hypotheticals and particulars analytically, is a relative term which shall denote "case of the existence o f - , " or "what exists only if there is any - " ; or else "case of the nonexistence of-," or "what exists only if there is not- ." When Boole's algebra is extended to relative terms, it is easy to see what these particular relatives must be . . . . Now, 0" is such a function, vanishing when x does not, and not vanishing when x does. Zero, therefore, may be interpreted as denoting "that which exists if, and only if, there is not-". (pp. 90-92)

Immed ia t e ly after this s ta tement , Peirce gives the equa t ions 0 ~ 1 and 0x = 0. Accord ing to his i n t e rp re t a t ion of involut ion, 0 ~ deno t e s all i for which Yj e 0 ~ (i,j) ~ O. Since there are no j in the emp ty class 0, 0 ~ is t rue for all i, and so 0 ~ 1. Similarly, 0x deno te s all (i ,j) for which 3k[(i, k) ~ 0 A (k, j) ~ x]. Since there are no (i, k) ~ 0, 0x is false

for all ( i , j ) , and so 0x = 0. T h e n (p. 93) he says that h , (1 - b) - 0 means

that every horse is black, so 0 h'tl-b) = 0 means that some horse is no t

black; h , b = 0 means that no horse is black, so 0 "'b = 0 means that some

horse is black. Finally, l ( h , b ) = 1 means some horse is black. T h e for-

mu la l ( h , b) is the relative p r o d u c t of the universal re la t ion and the absolu te te rm d e n o t e d by h , b.

Peirce writes 0x= 0. But then he says lx = 1, and neglects to use the

e x p o n e n t i a l in the calculat ion he per forms . O n e can criticize Peirce for

d r o p p i n g the exponen t i a l wi thout exp lana t ion . At this point , he still seems to be in his very ar i thmet ica l theory, in which the func t ion 0 x

serves as nega t ion , and is e x p e r i m e n t i n g with the power of various op-

erat ions . Thus, to say (a , b) = 0 is to say there is n o t h i n g in (a , b); to

say 0 ~"'~) = 0 is to say there is s o m e t h i n g in (a , b).

T h e a r g u m e n t he cons ide r sm"Every horse is black," "Every horse is


an animal," "There are some horses"; therefore "Some animals are b lack"mhe formalizes as h --< b, h --< a, 1 h = 1; therefore 1 (a, b) = 1. His derivation runs as follows. From the premises h--< b , h--< a, he obtains h --< a , b. Hence, by monotonicity, lh --< 1 (a, b). Therefore , if h is nonempty, 0 h = 0 or lh = 1, and 1 --< l ( a , b); therefore, l ( a , b) = 1. This is his alternative to Boole.

What advantage is Peirce gaining over Boole? The only result that Peirce has found that Boole did not is a way of saying "is empty" and "is nonempty." Boole had only terms built up by proposit ional connectives, and these could be empty. However, Peirce's result, 0x= 0 when x is not zero, still applies only to absolute terms. That is, to say that an absolute term is empty is to say that it is 0. To say that it is nonempty is to say that 0 raised to it is 0. Peirce is in fact still far from having a general theory.

Peirce in this early paper is making a claim that he has a theory that can deal with quantification. What one would like to see in such a theory is a way to translate any expression that involves quantification into his particular theory. This he does not do. What he does instead is to give an example that involves quantification and show that this example can be explained in his theory. There is quite a leap of faith involved from saying that we can explain this example in his theory to saying that we can explain all examples in his theory, and he does not extend his analysis to deal with an abstract quantifier in an abstract setting. He does, however, identify some impor tant issues. Most particularly, he presents a recognized version of De Morgan's law for quantifiers: that "exists" is equivalent to "not for all not." This is implicit in his equation 0 ~a'h) = 0, and it is on that observation that he bases his early development of quantification.

If we project ourselves back to the time of Peirce, what is the problem that he is trying to deal with? He is working with two theories, Aristotle's and Boole's, both of which are systems of logic and nei ther of which encompasses the other. Boole had propositional connectives but not quantifiers; Aristotle had quantifiers but not proposit ional connectives. Peirce, in his 1870 paper as whole, seeks to combine Boole's propositional logic with Aristotle's syllogisms. He understands Aristotle's theory and Aristotle's examples; he understands Boole's theory and Boole's examples. What he seeks is a framework in which both kinds of examples can be accommodated. This already makes for a difficult problem. It is made harder by Peirce's insistence that the notational system be based solely on logic operations that imitate such c o m m o n algebraic operations as addition, multiplication, and exponentiat ion.

We can try to reconstruct Peirce's line of reasoning in reconciling Aristotle with Boole. Aristotle in t roduced variables for class objects, with a positional notat ion to indicate where these variables appeared within

4 8 PEIRCE'S CALCULUS OF RELATIVES

the structure of an argument. Peirce augmented Boole's notation: he represented existential quantification by multiplication and universal quantification by exponentiat ion. In both cases the quantification is invisible, without its own symbol. It is something one sees in the form of the formula.

Peirce's s tudent O. H. Mitchell opened Peirce's eyes to the idea of a new notational representation for alternation of quantifiers, that is, for all x there exists a y, or for all y there exists an x. These operat ions have no counterpar t in the common algebra that Boole's algebra imitates, or in Aristotle, whose logic dealt with monadic predicates (classes). Mitchell in t roduced an operator F~y to handle alternating quantifiers as a generalization of the Aristotelian notation for one quantifier in that it has two subscripts that are positional in representing the alternating quantifiers.

Ultimately, these problems were solved by the modern notational system of first-order logic. Why does first-order logic succeed? Because it introduces the notion of predicates over a domain, and variables in predicates that can be quantified over. But as long as we are in Peirce's quasi-arithmetic system without explicit quantifiers over individuals, we cannot get very far.

2.3. S u m m a r y

There are two main points to be made about the approach to the calculus of relatives in Peirce's 1870 paper.

1. The notational system practically fell into Peirce's lap entire by analogy with his father's work in linear algebra. An individual term is like a coordinate, an absolute term is like a vector, and a relative term is like a matrix or linear transformation. This sparked a whole area of logic, matrix logic.

2. The most interesting feature of this approach is the hidden presence of existential quantification in the definition of relative product. We can unders tand something of Peirce's enterprise as a failed a t tempt to get full existential quantification out of relative product. The existential quantifier does already have an algebraic counterpar t in his father 's work in linear associative algebra; existential quantifiers correspond to projections. This is the basis of both Tarski's cylindric algebras (Henkin, Monk, and Tarski 1971) and Halmos's polyadic algebras (1962), two modern algebraic versions of first-order predicate logic. But this seems not to have been clearly recognized at the time.

In this early work on the calculus of relatives, Peirce shows more concern with maintaining the analogies between the notat ion he is setting forth and ordinary algebraic notation than with giving a direct


account of the problem at hand. His notat ion is capable of represent ing some deduct ions naturally, but is in general exceedingly cumber some compared to predicate logic as we know it now and to the way mathematics is traditionally written. His arguments also have a distinct lack of elegance. We encounte r this as soon as we try to work with his exponentials or powers and products and sums. Often, unders tandable formulas written using these operat ions become unreadable and mysterious. In the end, working in the system comes down to identifying and using unfamiliar algebraic patterns. The notat ion is being used to express a collection of algebraic laws and does not enable us to reason more easily or more accurately. The notation has little value for reasoning, however neat the algebra. For arguments we may as well stick to words.

Formal logics, to be useful in reasoning, must express premises in a form in which the proof rules that we need to apply to get to desired conclusions are easy to r e m e m b e r and natural to use, at least after some practice. Formulas express ideas; proofs are computa t ions that produce formulas expressing other ideas. If a formal system has proofs that are easy to work with, but the formulas express few ideas and in a shallow fashion, that system is not likely to be a useful one. If a system is very expressive but the proofs are quite difficult to work with, the system is equally unlikely to be used.

If Peirce was trying to make the case that logical reasoning could best be done by his algebraic formalism and proof rules, he needed to demonstrate that his language was sufficiently expressive to represent a wide variety of concepts and sufficiently convenient for p roof construct ion that people would use it. The calculus of relatives, as he developed it, fails at both. It succeeds as pretty algebra, but fails as a system for everyday or mathematical reasoning because of its lack of expressiveness and the opaqueness of the proof procedures. The lack of expressibility is made clear through Alwin Korselt's example in L6wenheim's 1915 paper, which showed that there are formulas in first-order predicate logic that cannot be expressed in the (quantifier-free f ragment of) the calculus of relatives. In contrast, first-order logic expresses a great variety of ideas, and the formal notions of proof closely imitate those used in everyday reasoning. No wonder it was adopted instead.

3. Peirce on the Algebra of Logic: 1880

In t roduc t ion

Peirce's paper "On the algebra of logic" was completed in April 1880 and appeared in the American Journal of Mathematics in September 1880. Peirce began writing this paper after he was appointed to the faculty of the John Hopkins University in 1879, and he presented its contents in an advanced course in logic in 1880 at the John Hopkins, a course attended by his student Mitchell.' In this paper Peirce reworks some of the material of his 1870 paper, but he also shows evidence of his originality as a mathematician and logician. He gives a lattice-theoretic treatment of Boolean algebra that appears to be the first such approach, and he develops a system of implicative propositional logic that anticipates the main features of modern systems of "natural deduction" by a rule converting deductions (illations) into implications (there called inclusions). This appears to have been completely original with him. Peirce's 1880 paper is long and complex. We first give a summary, with commentary, of its important points; then we discuss in detail some of the ideas that it develops.

3.1. Overview of Peirce's "On the algebra of logic"

Chapter I: Syllogistic

w 1. Derivation oflogic.mDiscusses the mental and experiential sources of habits that give rise to logical symbols and rules of inference.

w 2. Syllogism and dialogism.mAnalyzes implication and negated implication.

i See Nathan Houser ' s in t roduc t ion to volume 4 of Writings of Charles S. Peirce (1986) for this and o the r facts relat ing to Peirce 's career at the Johns Hopkins University.

51

52 PEIRCE'S ALGEBRA OF LOGIC

w 3. Forms of propositions.--Experiments with casting syllogisms in terms of the implication symbol and its negation.

w 4. The algebra of the copula.--Proposes that B can be inferred from A if and only if A --< B, characterizes implication by introduction and elimination rules, and deduces the schema for implication necessary for the implicational propositional calculus.

By section 4 of Chapter I, Peirce has thus given a fairly complete approach to propositional logic based on implication and negation, with inference as the source of the schema, since implication is supposed to mirror inference. This echoes the contemporary point of view on the origin of implication held by many proof theorists, including Prawitz (1965) and Girard (1989). Implication has a special role in logic as the direct expression of the existence of a deduction of the consequent from the antecedent. This gives a natural way to introduce implication in any formal system, classical or otherwise, that has a notion of deduction, and gives implication a syntactical origin in deductions rather than a semantic origin in truth tables.

Chapter II: The Logic of Non-relative Terms

w 1. Internal multiplication and the addition of logic.--Gives the schema that define "+" (disjunction) and " x " (conjunction) in terms of implication. These are the introduction and elimination rules, which, when written out as formal statements in the natural way, contain quantifiers over all propositions. This is the nature of natural deduction systems, of which Peirce's was the first.

In modern algebraic terms, Peirce's implication gives rise to a partial order in a Lindenbaum algebra of propositions. The introduction and elimination rules for disjunction and conjunction assert the existence of least upper (+) and greatest lower (x) bounds in the Lindenbaum algebra, that is, make the Lindenbaum algebra a lattice. In modern propositional logic, there would be a fixed language, and the propositional quantifiers would range over only the formal propositions in that language. Peirce has no fixed language, so these range over all propositions. He then uses these definitions quite carefully to deduce many Boolean identities. Peirce's propositional logic has implication and negation as primitives and gives the rest of the definitions in quantified propositional logic based on these connectives. The whole basis of Peirce's system, in other words, is quantification over propositions in the metalanguage. This is familiar today only to those who have studied natural deduction. But if we think in terms of the Lindenbaum algebra, the exposition here can be seen as developing propositional logic as a


species of lattice theory, where lattices are defined in terms of partial orderings.

Although partial orders are explicitly defined, the not ion of a lattice is not isolated as clearly here as it will be later by Peirce's disciple Ernst Schr6der in volume 1 of Schr6der 's work, Vorlesungen iiber die Algebra der Logik (1890). The origin of the definition of lattice in Peirce and Schr6der, based on the introduct ion and elimination rules for disjunction and conjunction, is quite different from its origin in Dedekind. Dedekind defined lattices based on his experience with lattices of subgroups of a group, or lattices of ideals of a ring. From examples of lattices of subgroups, Dedekind knew that lattices need not be distributive, whereas finding this out was a task for Peirce and Schr6der, who did not start with such examples; the cases naturally arising from the logics they investigated were distributive.

w 2. The resolution of problems in nonrelative logic.~Gives a detailed account of the methods of Boole, Jevons, Schr6der, and Mac Coil for solving inference problems in propositional calculus. These need not be described in detail, except to say that Peirce expresses them in algebraic form. They differ little from those of Boole and his other followers.

Chapter III: The Logic of Relatives

w 1. Individual and simple terms.reintroduces the not ion of individual, in preparat ion for introducing ordered individual relatives.

w 2. Relatives.reintroduces the notion of an individual relative, and makes up binary relations as "sums" of individual relatives (A : B). This makes sense in modern terms if individual relatives are regarded as ordered pairs, and the pairs with coordinates from a set are regarded as atoms generat ing a Boolean algebra, isomorphic to the Boolean algebra of subsets of the Cartesian square of the set. This is an interpretation on the same level as saying that an absolute term can be interpreted as a set.

w 3. Relatives connected by transposition of relate and correlate.mIntroduces the converse of a binary relation, obtained by reversing its pairs, and also examines ternary relations and the transpositions of their arguments. All notation is Boolean algebraic in the sense alluded to above, where least upper bound is sum.

w 4. Classification of relatives.~Classifies pairs and relatives according to the identity of the components , etc. These results are not very important .

w 5. Composition of relatives.mDevelops the algebra of relative product.


w 6. Methods in the algebra of relatives.mThis section tries to imitate

Chap te r II, section 2, on in fe rence me thods in proposi t ional logic for the full a lgebra of relatives, with sum, product , converse, and relative

p roduc t , and lists some rules. w 7. General formulae for relatives.mPresents pages of algebraic identi-

ties, with no indicat ion of how one migh t rationally organize t h e m for

use in reasoning; this is simply a col lect ion of algebraic identi t ies Peirce

had discovered. H e r e Peirce 's pape r ends. He has not just i f ied his initial claim by

indica t ing how to use the a lgebra of relatives for ord inary reasoning.

In a sense, he never just if ied this claim later, either.

3.2. Discussion

3.2. I. The Origins of Logic

In his 1880 paper, Peirce begins with a cons idera t ion of the biological

and psychological basis for logic. He states that

In order to gain a clear understanding of the origin of the various signs used in logical algebra and the reasons of the fundamental formulae, we ought to begin by considering how logic itself arises. (Peirce 1880, p. 104) z

Thinking, Peirce says, is governed by the general laws of nervous action. He discusses the stimulation of a g roup of nerves and connec ted ganglions

as throwing the body into an active state. Stimulation is described as spreading from ganglion to ganglion. He then discusses fatigue and the

subsiding of exc i tement with the withdrawal of the stimulus. The role of repet i t ion is discussed, and also the es tabl ishment of habits, and belief, j u d g m e n t , and inference are def ined in these terms:

A cerebral habit of this highest kind, which will determine what we do in fancy as well as what we do in action, is called a belief. The representation to ourselves that we have a specified habit of this kind is called a judgment. A belief-habit in its development begins by being vague, special, and meagre; it becomes more precise, general, and full, without limit. The process of this development, so far as it takes place in the imagination, is called thought. A judgment is formed; and under the influence of a belief-habit this gives rise to a new judgment, indicating an addition to belief. Such a process is called an inference', the antecedent judgment is called the premzs~ the consequent judg-

Except where otherwise noted, all subsequent page citations in this chapter will refer to "On the algebra of logic" (1880), in the CoUected Papers of Charles Sanders Peirce (Hart- shorne and Weiss 1933).


ment, the conclusion; the habit of thought, which determined the passage from one the other (when formulated as a proposition), the leading principle. (pp. 105-106)

55

Peirce adds, in a f o o t n o t e to the above passage:

Deductive logic, perhaps, does not involve the principle that there is any special character in the peripheral excitation but only that reasoning proceeds by habits that are consistent.

In o t h e r words, logical d e d u c t i o n s are of the same form, i n d e p e n d e n t

o f the empi r ica l cha rac t e r of the p ropos i t i ons a b o u t which they reason.

H e nex t p roposes the idea of logic as evolving f rom habit :

At the same time that this process of inference, or the spontaneous development of belief, is continually going on within us, fresh peripheral excitations are also continually creating new belief-habits. Thus, belief is partly determined by old beliefs and partly by new experience. Is there any law about the mode of the peripheral excitations? The logician maintains that there is, namely, that they are all adapted to an end, that of carrying belief, in the long run, toward certain predestinate conclusions which are the same for all men. This is the faith of the logician. This is the matter of fact, upon which all maxims of reasoning repose. In virtue of this fact, what is to be believed at last is independent of what has been believed hitherto, and therefore has the character of reality. Hence, if a given habit, considered as determining an inference, is of such a sort as to tend toward the final result, it is correct; otherwise not. Thus, inferences become di- visible into the valid and the invalid; and thus logic takes its reason of existence. "~ (p. 106)

In his m u c h later 1903 Lowell Lec tu res Pei rce r e p u d i a t e s these first

two sect ions o f his 1880 p a p e r as be ing no t m a t h e m a t i c a l a n d n o t re-

d u c e d to pr inciples . In his u n p u b l i s h e d m a n u s c r i p t 735 in the Rob in

catalog, en t i t l ed "Exact logic," we f ind the s t a t e m e n t "Logic is the t heo ry

of r e a s o n i n g a n d as such it is no t a b r a n c h of psychology" in the table

of con ten t s . This suggests tha t Peirce came to a view s imilar to tha t of

Frege in The Foundations of Arithmetic, which eschews a psychologica l

basis for the c o n c e p t of n u m b e r .

"~ This passage is cloudy compared to Peirce's 1877 essay "Fixation of Belief," which expresses a similar theory of knowledge in philosophical terms.

5 6 VFIRCF'S ALGEBRA OF I.OGIC

3.2. 2. Syllogism and Illation

A. IUation

Pei rce takes up the t r e a t m e n t o f" i l l a t ion" in sec t ion 2. He first descr ibes

the g e n e r a l type of in fe rence :

The general type of inference is

P .~ C,

where " is the sign of illation. (p. 106)

H e r e P is the premiss , or set of p remises , and C is the conc lus ion ; i l lat ion

is d e d u c t i o n or in fe rence .

Pe i rce t hen i n t roduces the symbol --<, exp l a in ing it as follows"

Thus, the form Pi--< C, implies either, 1, that it is impossible that a premise of the class Pi should be true, or, 2, that every state of things in which P, is true is a state of things in which th____e corresponding C; is true.

The form Pi--< Ci implies both, 1, that a premise of the class Pa is possible, and, 2, that among the possible cases of the truth of a P; there is one in which the corresponding C i is not true. (p. 108)

In m o d e r n terms, P~--<C~ is t ru th - func t iona l impl ica t ion ; Pe i rce ' s "state" is a t ru th valuat ion. Pe i rce ' s de f in i t ion of P~--< C i is thus aston- i sh ing close to the m o d e r n def in i t ion of o n e s t a t e m e n t b e i n g a s eman t i c c o n s e q u e n c e of ano ther . O f course , he does no t have a c o m p l e t e defin i t ion of what he m e a n s by state, which for us is now given by t ru th

valuat ions , p ropos i t i ona l or p red ica te .

B. Rules of Inference

Pei rce c o n t i n u e s to discuss habits , r e f e r r i ng to a "habi t of in fe rence" :

A habit of inference may be formulated in a proposition which shall state that every proposition c, related in a given general way to any true proposition p, is true. Such a proposition is called the leading principle of the class of inferences whose validity it implies. (p. 107)

A " lead ing pr inc ip le" h e r e is cer ta in ly a gene ra l rule of i n fe rence . Today

a rule o f i n f e r e n c e is fo rmal and m e c h a n i c a l in its app l ica t ion , bu t it

F R O M P E I R C E T O SKOLEM 57

is not clear that Peirce 's leading principle necessarily conveys with it the no t ion of being a formal schema.

Peirce also discusses the criticism of reasoning, c la iming that logic supposes inferences not only to be drawn, but also to be subjec ted to criticism (p. 107). In this he follows the Socratic t radi t ion that t ru th is (best) revealed by dialogue, i.e., that a discourse a m o n g reasonab le m e n

will eventually lead to the cor rec t answer. (Peirce does not, however, assume that there is fixed static truth, as Socrates does.) The m o d e r n

view of deduc t ion , not very di f ferent f rom Aristotle 's, is that in a de-

duc t ion one checks that each premise is t rue and that each rule of deduc t ion yields truths f rom truths. To criticize a d e d u c t i o n one mus t

attack e i ther an i m p r o p e r p remise or an i m p r o p e r appl ica t ion of a rule

of deduc t ion . Such an attack is p resumably what Peirce m e a n t by a criticism.

C. Introduction and Elimination of Implication

Peirce re la ted .'. and --< as follows:

[T]herefore we not only require the form P .'. C to express an argument, but also a form, P;--< C;, to express the truth of its leading principle. Here P, denotes any one of the class of premisses, and C i the corresponding conclusion. The symbol -< is the copula, and signifies primarily that every state of things in which a proposition of the class P; is true is a state of things in which the corresponding propositions of the class C; are true. (pp. 107-108)

This is the m o d e r n dist inct ion be tween proving "P implies C" with a mater ia l in te rp re ta t ion of "implies," and showing that the re is a deduction of conclus ion C f rom premise P. In m o d e r n first-order logic, the equivalence be tween these is u n d e r s t o o d as a m e t a t h e o r e m , the deduct ion t h e o r e m plus modus ponens . In some moda l logics, the deduct ion t h e o r e m fails.

Peirce is also po in t ing out he re that one canno t criticize an a r g u m e n t unless the rules of deduc t ion used to cons t ruc t the a r g u m e n t are m a d e

explicit. A "logical principle," in particular, is a special kind of lead ing

principle:

In the form of inference P .'. C the leading principle is not expressed; and the inference might be justified on several separate principles. One of" these, however, P;--< C;, is the formulation of the habit which, in point of fact, has governed the inferences. This principle contains all that is necessary besides the premise P to justify the conclusion. (It will generally assert more than is necessary.) We may, therefore, construct a new argument which shall have for its premisses the two

5 8 PEIRCE'S ALGEBRA OF LOGIC

propositions P and P;--< C; taken together, and for its conclusion, C. (p. 108)

T h e habi t r e fe r red to is what we now call a rule of in fe rence , formal or

not. Peirce cont inues :

This argument, no doubt, has, like every other, its leading principle, because the inference is governed by some habit; but yet the substance of the leading principle must already be contained implicitly in the premisses, because the proposition P;--< C, contains by hypothesis all that is requisite to justify the inference of C from P. Such a leading principle, which contains no fact not implied or observable in the premisses, is termed a logical principle, and the argument it governs is termed a complete, in contradistinction to an incomplete, argument, or enthymeme. (pp. 108-109)

An e n t h y m e m e is a syllogism with an uns ta ted premise . E n t h y m e m e s are used in a r g u m e n t s in which the conclus ion is no t a c o n s e q u e n c e of the s tated hypothesis but r a the r of that hypothesis plus addi t iona l uns ta t ed assumptions , which are of ten prejudices. W h a t Peirce says he re is that "P implies Q" summar izes that there is a logical d e d u c t i o n that takes o n e from P to Q, usually using uns ta ted hypotheses . These then n e e d to be stated explicitly to comple t e the a rgumen t .

Gen tzen ' s theory of sequents for p red ica te logic (1934) bears ou t Peirce 's discussion here. His m i d s e q u e n t t h e o r e m says that to get f rom a hypothes is to a conclus ion, one must first d e c o m p o s e the hypothes is into a tomic parts; to get f rom there to the conclus ion , one simply reas- sembles these atomic parts in a d i f fe rent order. All fu r the r steps are thus appl icat ions of s equen t d e d u c t i o n rules.

Similarly, Peirce says that "a logical pr inc ip le is empty," by which he appears to m e a n that it is a valid pr inciple , ho ld ing in all states o f the universe wi thout any addi t ional assumptions , of which tautologies are examples :

A logical principle is said to be an empty or formal proposition, because it can add nothing to the premisses of the argument it governs, although it is relevant; so that it implies no fact except such as is pre- supposed in all discourse. (p. 109)

Thus , a logical pr inciple does no t tell us any th ing abou t the universe (because it is t rue of all possible states), bu t it shows us s o m e t h i n g abou t the universe, abou t its s t ructure. This view is very close to Wi t tgens te in ' s (see Tractatus, 6.12).


Peirce then talks about immed ia t e inference , which he def ines as "a comple t e a rgumen t , with only one premise" (p. 109).

At the very end of this section, Peirce takes the m a t t e r of logical form

a step further , a rgu ing for a normal form:

Now, the logician does not undertake to enumerate all the ways of expressing facts: he supposes the facts to be already expressed in certain standard or canonical forms. But the equivalence between different ones of his own standard forms is of the highest importance to him. (p. 110)

He goes on to say that some "will not be reciprocal in fe rences or

logical equat ions , but the most impor t an t of t hem will have to have that character ." He is here dis t inguishing be tween rules of in ference , spe-

cifically, be tween rules in which "From P infer Q" is valid but "From Q

infer P" is not a cor rec t rule, and rules in which both are cor rec t rules,

that is, in which P and Q can be i n t e r changed wherever they occur.

What Peirce is seeking is a s tandard form for logical rules of inference . He argues that it is possible to const ruct a l anguage such that we never need to have m o r e than two hypotheses for any given logical rule:

From the doctrine of the leading principle it appears that if we have a valid and complete argument from more than one premise, we may suppress all premisses but one and still have a valid but incomplete argument. This argument is justified by the suppressed premisses; hence, from these premisses alone we may infer that the conclusion would follow from the remaining premisses. In this way, then, the original argument

P Q R S T .'. C

is broken up into two, namely, 1st,

and, 2nd,

P Q R S .'. T--<C

T--<C T

.'. C.

By repeating this process, any argument may be broken up into arguments of two premises each. (pp. 110-111)

This is similar to the a r g u m e n t m a d e in m o d e r n ma thema t i c s that in

analyzing functions, we need only cons ider funct ions of arity 1 because


each funct ion f(x,y) of two variables can be t rea ted as a funct ion g(x) of one variable, the values of which are funct ions of a second variable, y; explicitly, [g(x)](y) =f(x, y). Thus, formally it suffices to have a theory of unary funct ions f rom which all behaviors of funct ions of h ighe r arity

can be explained. This is now called "currying," after Haskell Curry, deve loper of combina to ry logic. He re Peirce is mak ing an a r g u m e n t of

the same form. The Curry-Howard i somorph i sm substant iates that it is

m o r e than an analogy (see Girard 1989). It is also a lmost the same

pr inciple that Hi lber t used to conver t all deduct ive systems into systems

with modus ponens as the only rule of inference.

3.2. 3. Forms of Propositions

A. Implication

In section 3 Peirce lists some connectives that are logically equivalent to the different possible combinat ions of the --< sign and its negation"

In place of the two expressions A --< B and B --< A taken together w___e_e may write A = B; in place of the two expressions A --< B and B --< A taken together we may write__A_A < B or B > A; and in place of the two expressions A - < B and B --< A taken together we may write A ~ B. (p. 111)

Peirce is here def in ing the connect ives "A is s t ronger than B," "A is weaker than B," and "A is i ncomparab le with B," m o r e c o m m o n in the a lgebra of logic than in logic itself.

In a presc ient remark, Peirce upholds the use of restr ic ted universes of discourse, an idea that he at t r ibutes to De Morgan:

De Morgan, in his remarkable memoir with which he opened his discussion of the syllogism (1846, p. 380), has pointed out that we often carry on reasoning under an implied restriction as to what we shall consider as possible, which restriction, applying to the whole of what is said, need not be expressed. The total of all that we consider possible is called the universe of discourse, and may be very limited. (p. 112)

The purpose of this observat ion is to in t roduce the idea of logical

quant i f iers ranging over a restr icted domain , not over the universe of all possible things, as in Frege 's papers. Working in defini te yet variable domains was a characterist ic of the twentieth century d e v e l o p m e n t of f irst-order logic for relat ional systems, of which this is a precursor . Peirce

goes on to say:


The forms A --< B, or A implies B, and A --< B, or A does not imply B, embrace both hypothetical and categorical propositions. (p. 112)

61

Pe i rce ' s use o f the t e r m "implies" h e r e is a little m u d d y b e c a u s e he has

no expl ic i t n o t a t i o n as yet to d i s t ingu ish quan t i f i e r s in h y p o t h e t i c a l a n d ca tegor ica l p ropos i t i ons . Thus , both "If A, t h en B" a n d "all A are B" are

wr i t t en A --< B, m a k i n g it diff icult for the m o d e r n r e a d e r to give an

exac t m o d e r n i n t e r p r e t a t i o n o f any specific usage, e x c e p t by con tex t .

F r o m this iden t i f i ca t ion , Pe i rce gets:

Thus, to say that all men are mortal is the same as to say that if any man possesses any character whatever then a mortal possesses that character. To say "if A, then B" is obviously the same as to say that from A, B follows, logically or extralogically. (pp. 112-113)

T h a t is, to say tha t all m e n are m o r t a l is to say tha t for any p r o p e r t y

p, if p is possessed by a ma n , t h e n p is possessed by a mor ta l . N o t e tha t

this s t a t e m e n t has a universa l quan t i f i e r over p r o p e r t i e s p, a n d so this

is a s e c o n d - o r d e r de f in i t i on o f impl ica t ion . In o t h e r words , it is a k ind

o f g e n e r a l i z a t i o n to imp l i ca t ion (i.e., to Pe i rce ' s --< ) o f Le ibn iz ' s pr in-

ciple o f the iden t i ty o f ind iscern ib les , which says tha t objec ts are the

s ame if they have the s ame p rope r t i e s . Pe i rce t h e n c o n t i n u e s with p e r h a p s the first f o r m u l a t i o n o f a na tu ra l

d e d u c t i o n rule:

By thus identifying the relation expressed by the copula with that of illation, we identify the proposition with the inference, and the term with the proposition. This identification, by means of which all that is found true of term, proposition, or inference is at once known to be true of all three, is a most important engine of reasoning, which we have gained by beginning with a consideration of the genesis of logic. (p. 113)

Thus , we can infe r "if A, t h e n B" f ro m a d e d u c t i o n tha t infers B f r o m

A. This is the co re idea of a na tu ra l d e d u c t i o n system, wha t in c a t e g o r y

t h e o r y wou ld be r e f e r r e d to as an ad jo in tnes s c o n d i t i o n (see Mac L a n e

1971 a n d Lawvere 1966). T h e passage q u o t e d above, b e g i n n i n g "Thus ,

to say tha t all m e n are mo r t a l is the s ame as to say .. ." exp res ses the

very s ame idea. Pe i rce uses the t e rms A a n d B in such a way tha t "all A

is B" a n d "if A, then B" are r e g a r d e d as the same. 4 I n f e r e n c e (A ." B,

i l lat ion) is thus a j u d g m e n t , the resul t o f a logical d e d u c t i o n . T h e m o d -

e rn s eman t i c equ i v a l en t is a s e c o n d - o r d e r de f in i t i on (i.e., for all m o d e l s

�9 I Peirce's use of "term" is from the Aristotelian tradition, which is also adopted by Boole. Roughly, a term denotes a predicate.


in which A holds , B ho lds as well). W h e n Pe i rce first no te s this for A

--< B he is a l r eady e x p r e s s i n g the e q u i v a l e n c e o f j u d g m e n t a n d

imp l i ca t ion .

B. Boole and Quantification

In this sect ion, Peirce criticizes Boole ' s t r e a t m e n t of quan t i f i ca t ion , re-

i t e ra t ing his ob jec t ion to Boole ' s use of v to express an i n d e t e r m i n a t e "

m

Of the two forms A --< B and A - < B, no doubt the former is the more primitive, in the sense that it is involved in the idea of reasoning, while the latter is only required in the criticism of reasoning. The two kinds of proposition are essentially different, and every at tempt to reduce the latter to a special case of the former must fail. Boole attempts to express 'some men are not mortal ' in the form 'whatever men have a certain unknown character v are not mortal. ' But the propositions are not identical, for the latter does not imply that some men have that character v; and, accordingly, from Boole's proposition we may legitimately infer that 'whatever mortals have the unknown character v are not men'; yet we cannot reason from 'some men are not mortal ' to 'some mortals are not men. ' (p. 113)

B o o l e t r i ed to de f ine ex is ten t ia l a n d un iversa l q u a n t i f i c a t i o n by ex-

t e n d i n g w h a t e v e r B o o l e a n a l g e b r a o f sets is b e i n g d i scussed by a B o o l e a n

i n d e t e r m i n a t e v. Thus , Boo le was in fact w o r k i n g in a f ree e x t e n s i o n o f

the B o o l e a n a lgeb ra g e n e r a t e d by o n e a d d i t i o n a l e l e m e n t ( H a i l p e r i n

1976). To express "Some A is no t B," Boo le can say ins t ead tha t in this

e x t e n s i o n ring, v n A :g 0 a n d v n A 5g B. Pe i rce says, in effect, a n d n o t

with accuracy, tha t Boole expresses "Some A is n o t B" as v n A ~ B,

which will n o t work in the case v n A = 0. A l t h o u g h he was a b r i l l i an t

a lgebra i s t a n d t h o r o u g h l y fami l ia r with his f a the r ' s work on associat ive

a lgebras , Pe i rce d id n o t grasp tha t Boo le was w o r k i n g in a B o o l e a n

p o l y n o m i a l ex t ens ion . But t hen , unt i l H a i l p e r i n , no o n e f i gu red o u t

wha t Boo le was do ing . This is n o t to say t ha t B o o l e ' s t r e a t m e n t was

sat isfactory; it was not . Even if c o r r e c t e d , it wou ld dea l on ly with m o n a d i c

p r e d i c a t e logic with a s ingle variable .

In any case, hav ing d i smissed Boo le ' s i n d e t e r m i n a t e s , Pe i rce in t ro-

d u c e s a p r imi t ive n o t a t i o n for the ex is ten t ia l q u a n t i f i e r in imp l i ca t i ons

b e t w e e n terms, the cup o p e r a t o r :

On the o ther ha___nd, we can rise to a more general form unde r which A__.__~ B and A - < B are both included. For this_purpose we write A - < B in the form A - < B , where A is some-A and B is not-B. This more general form is equivocal in so far as it is left unde t e rmined whether the proposit ion would be true if the subject were impossible. When


the subject is general this is the case, but when the subject is particular (i.e., is subject to the modification some) it is not. The general form supposes merely inclusion of the subject under the predicate. The short curved mark over the letter in the subject shows that some part of the term denoted by that letter is the subject, and that it is asserted to be in possible existence. (pp. 113-114)

63

Peirce wants to express the idea that A is no t c o n t a i n e d in B as some A

is not B, and we are back to Aristotle 's te rm calculus. T h e cup o p e r a t o r expresses quant i f ica t ion wi thout using a variable, bu t un fo r tuna t e ly B mus t be known before the cup o p e r a t o r on A has m e a n i n g . Peirce follows this m e t h o d only t h r o u g h his t r e a t m e n t of the a lgebra of nonrelative terms in this paper , and the rea f t e r reserves the cup symbol to d e n o t e the converse ope ra t i on on relatives.

C. Aristotle and Quantification

Peirce a t t empts to deal with Aristotle 's syllogistic us ing impl ica t ion, negat ion, and the cup opera tor . He translates the Aris totel ian proposi- t ional forms as follows:

The modification of the subject by the curved mark and of the predicate by the straight mark gives the old set of propositional forms, viz.:

A. a - < b Every a is b. Universal affirmative.

E. a -< / ; No a is b. Universal negative.

I. d--< b Some a is b. Particular affirmative.

O. d--</; Some a is not b. Particular negative.

(p. 114)

Peirce gives these p ropos i t ions an in t e rp re t a t i on that differs f rom thei r t radi t ional mean ing . Accord ing to the t radi t ional i n t e rp re t a t ion , affirmative p ropos i t ions imply the exis tence of the i r subjects a n d negative p ropos i t ions do not. Thus "every a is b" is valid only when the re exists some object that is a. Accord ing to Peirce 's i n t e rp re t a t ion , however, par t icu lar p ropos i t ions imply the exis tence of the i r subjects, while universal p ropos i t ions do not. Thus , for Peirce the t ru th of a- -< b or a--</~ does no t imply the exis tence of a.

Hi lbe r t and A c k e r m a n n (1928), in fitting Aris totel ian syllogisms into the f irst-order p red ica te calculus, i n t e rp re t the Aris tote l ian proposi- t ional forms as Peirce does. The issue is that in the Aris tote l ian concept ion , the re were immed ia t e in ferences f rom A to I and f rom E to O; i.e., f rom "every A is B," one would n e e d to i n f e r that "some A is


B." Neither Peirce nor Hilbert and Ackermann agree with this reasoning. They observe that to define a class does not of itself imply that the class is nonempty. Thus, to assert "every A is B" does not enable us to infer that "some A is B," because the possibility remains that nothing is A. But is this a criticism, or does it simply point out that it is a convention whether or not the empty class is allowed? In modern logic based on nonempty domains, the universal quantifier implies the existential because the domain is assumed to be nonempty. It is just a small twist to write out an alternate predicate logic that uses exactly those rules that work in all domains, including the empty domain.

Peirce's cup operator is also a poor notation for indicating the scope of the quantification. For instance, in order to express "some A is B," Peirce would attach the cup operator to the A, which seems linguistically natural. However, this gives rise to opaque rules for handling quantifiers, as we will see in the next section.

Perhaps Peirce was misled by the fact that negation binds as a unary operator; he might have thought that quantification binds in the same way as negation, that is, that in "no A is B," the "no" scopes inside the "is." Peirce had the scope of negation correct when he expressed "no A is B" as A --< B, since "no" binds more tightly than "is," but "all" does not, nor does "some"; this is the basis of the trouble with his approach here.

Propositional logic and Boolean algebra by themselves can provide a way to talk about containment , but despite Peirce's best attempts here, they provide no Way to talk about existential quantification. Later, in Peirce's work following O. H. Mitchell, a formal theory of quantification grows out of an at tempt to explain Aristotelian syllogisms using propositional logic and an algebraic notation for quantifiers as sums and products.

3.2. 4. The Algebra of the Copula

A. Deduction and Implication

In his "algebra of the copula," Peirce develops an informal system of natural deduct ion in which the binary connective - < is in t roduced and el iminated by introduction and elimination rules, which are basic to his system.

The algebra of the copula begins with the general assertion that "from the identity of the relation expressed by the copula with that of illation, springs an algebra" (p. 116). Peirce is here referring back to his identification of deduction and implication on page 118. The notion ".'. is equivalent to --<," in other words, that every deduct ion proves an im-


plication and every implication arises f rom a deduct ion , is Peirce 's starting point. (However, we formalize, as he did not.)

We assume as given variables x, y, z . . . . . Formulas are made up of variables and the binary connective --<. A deduc t ion is an express ion x, y . . . . . ". z, where x, y, z, ... are formulas. Peirce allows lists of formulas on e i ther side of the .'. sign, retroactively in the Gentzen tradition.

Peirce says that the identification of the relation of the copula with that of illation gives us, in the first place,

x---~ x, (1)

an identity axiom, and, in the second place, the equivalence of the two inferences

X x

y and . . . y - < z ~

�9 �9 Z

(2)

This is his version of implication in t roduct ion and e l iminat ion rules. Tha t is, suppose we have a deduct ion end ing

x

Y �9 �9 Z ,

where the vertical dots stand for the previous lines of the deduct ion , then the rule of implication in t roduct ion says that we also have the deduc t ion

x

.'. y - < z ,

where the vertical dots stand for the same previous lines of the deduction. Conversely, the rule of implication el iminat ion says that if we have the deduc t ion

then we also have the deduc t ion

x

y - < z ,

Y �9 �9 Z ~


These are the rules Peirce employs to derive a series of formulas that compr ise his algebra of the copula. Peirce 's theorems, with proofs in the Peirce style, are as follows:

{x -< (y - < z)} = {y - < (x - < z)}. (3)

Proof of (3): x - -< ( y - < z ) ~ x .'. y - -< z r {x,y} .'. z r y .'. ( x - < z ) r162

y --< (x --< z), using (2) repeatedly. We assume that, for Peirce, if a and b are two well-formed expressions

in his algebra, then a = b means a r b, i.e., a .'. b and b .'. a, which by (2) we can also express as a- -< b and b--< a.

x - < y , x (4) �9 �9 y o

Proof of (4)" (x--< y) ." (x--< y); hence , x--< y, x ." y.

x - < y , y - -< z (5) . . X - - - ~ Z .

Proof of (5)" {x --< y, x, y --< z} ." {y, y --< z}, f rom (4). ". z, f rom (4). F rom {x--< y, y - -< z, x} ." z, it follows from (2) that {x--< y, y - -< z} ." ( x - < z ) .

S - - < P (6) .'. ( x - < S ) --< ( x - < P ) .

P roof of (6)" From (5) IS--< P, x--< S} �9 x--< P, it follows that S--< P �9 (x--< S) --< ( x - < P), by (2).

S - < P (7) .'. (P---< x) - < ( S - < x ) .

Proof of (7)" From (5) {S--< E P --< x} " S --< x, it follows that S--< P ." (P- -<x) --< (S--<x) , by (2).

M - - < P (S--< P) --< x

.'. (S - -<M) --<x.

(8)

Proof of (8)" Equivalent to {S --< M, M --< P, (S --< P) --< x} ". x, by (2). But { S --< M , M --< E ( S --< P ) --< x} " { S --< E ( S --< P ) --< x} " x by (5)

and (4).

S - - < M (S--< P) --< x

.'. (M --<P) --<x. (9)

Proof of (9)" Same as for (8).

F R O M P E I R C E T O S K O L E M 6 7

M--<P x--< ( S - < M) (10)

.'. x--< (S --< P).

Proof of (10)" Equivalent to {M--<E x--< (S--<M),x} ". S- -<P, by (2). But {M--<Ex- -< (S--<M),x} ." {M --< E S--< M} ." S--<P, by (4) and (5).

S--<M x - < (M--< P) (11)

.'. x--< (S--<P).

Proof of (11)" Same as for (10).

M--<P

S - - < M (12) ~ m

.'. S--< P.

Proof of (12)" Following Peirce, we define the negat ion opera tor A as (A--< x). We want to prove {M --< P, (S--< (M --< x)) --< y} �9 (S --< (P --< x)) --< y. But {M --< P, S --< (P --< x), (S --< (M --< x)) --< y} .'. {S--< (M --< x), (S--< (M --< x)) --< y} " y, by (8) followed by (4), which gives the assertion, by (2).

This definition of negation is incompletely stated. Specifically, there is an unstated quantification on x, in order for it to make sense. It is much better to first introduce the constant 0, as Peirce does in section 1 of Chapter II ("On the Logic of Non-relative Terms").

For example, since the quantified x in (A--< x) becomes a dummy variable, one negation, (A--< x), is as good as another, (A--<y). This was used in proving (12), but it can be confusing to say " A - < x = A - < y . "

S--<M

M - - < P (13) m ~

�9 S - -<P .

Proof of (13): We want to prove {S--<M, ((M --< x) --< P) --< y} .'. ((S--< x) --<P) --<y. But {S--< M, (S--< x) --<P, ((M--< x) --<P) --<y} .'. {(M--< x) - < P, ((M--< x) --< P) --< y} .'. y, by a transposition of the literals in (9) followed by (4). This gives the assertion, by (2).

M - < P

S - - ~ P (14)

.'. S--< M.

68

S - < M

S - < P

.'. M - < P.

PEIRCE'S ALGEBRA OF LOGIC

(15)

S - < P

.'. P--< S. (16)

Formulas (14), (15), and (16) are proved by various applications of formulas (5) and (2); in particular, (14) follows from (8), (15) follows from (9), and (16) follows from (7).

x - < x , (17)

x--< x. (18)

Formula (17) can be proved as follows: from (4) and (2), x " ( x - < y) --< y; hence ." x - < ((x-< y) - < y). Since y is a dummy variable, put y = 0 (al though Peirce has not at this stage introduced 0). However, (18) cannot be proved from the algebra so far. Of course, (17) and (18) together yield x = x.

S - < P

.'. P - < S . (19)

Proof of (19)" {S-< ( P - < x)} = { P - < ( S - < x)} by (3).

S - < P

.'. P --< S. (20)

Proof of (20)" This follows from (16), (18), and (5).

S--<P

.'. P --<S. (21)

Proof of (21)" This follows from (16), (17), (18), and two applications of (5).

In formulas (22) and (23), there are two levels of inference taking place"

If (P ". C) is valid, then (C ". P) is valid. (22)

Proof of (22)" (P " C--< x) ~ ({P, C} " x) ~ (C ." P--< x), by applying (2) twice.

If (P �9 C) is valid, then (C " P) is valid. (23)


P r o o f of (23)" Since P = P and C = ~ , (23) follows f rom (22).

69

B. The Cup Quantifier

Peirce p roceeds to prove two u n n u m b e r e d formulas:

( S - < P ) - < { ( S - < x) - < (P - < x)}. (U1)

(p. 122) P r o o f o f (U1): S - - < P .'. (S--< x) --< (P --<x) by apply ing (15) and (2).

By a n o t h e r appl ica t ion of (2), .'. (S--< P) --< [ ( S - < x) --< (P - < x)]. ~ m

(S - < P) --< {(S - < x-) - < (P - < x)}. (U2)

(p. 122) P r o o f of (U2)" This follows f rom (U1) and (3) and A - ~ (set t ing

x = P, whence P = ~). Peirce then in t roduces the cup o p e r a t o r above the literals. This is

again a very p o o r no ta t ion , pr imari ly because he a t taches a cup o p e r a t o r to each letter, a l t hough the cup ope ra t ion d e p e n d s on what both letters are, and his no ta t ion suppresses that d e p e n d e n c e . He offers the following explana t ion :

Denoting__this by a short curve over the subject, we may write S--<P for S--< P. We see then that while for A we may write A - < x, where x is anything whatever, so for ,~ we may write A--< ;. If we attach a similar modification to the predicate also, we have S - < P or (S--< ;) --< (P -< ;) , which is the same as to say that you can find an S which is any P you please. We thus have

(S-< P) -< (P-< S), (24)

a formula for contraposition, similar to (16). (p. 122)

It is no t clear what he means by fo rmula (24). F o r m u l a (16) is contraposition"

S - - < P

.'. P- -<S.

He then gives a der ivat ion in which some of the fo rmulas do no t seem to make sense. We discuss these in g rea te r detail below; for now, we simply quote:

It is obvious that

(S-< P) -< (P-< S)" (25)

7o P E I R C E ' S A L G E B R A O F L O G I C

for, negating both propositions, this becomes, by (16), m

(P-<S) -< (S-< P).

From (25) we infer

x-<x, (26)

which may be called the principle of particularity. This is obviously true, bec___ause the modification of particularity only consists in changing (A-< x) to (A--<,~), which is the same as negating the copula and the predicate, and a repetition of this will evidently give the first expression again. For the same reason we have

x--< ~, (27)

which may be called the principle of individuality. (p. 123)

Peirce states that this gives

(S - < f~) --< (P --< S), (28)

and formulas (26) and (27) together give

(S --< P) --< (P--< S). (29)

He goes on to say after (29) that it is doubtful whether the proposi t ion S--<P should be interpreted as signifying "that S and P are one sole individual, or that there is something besides S and E" Finally, he ends this section perspicaciously by remarking that he is leaving this branch of the subject in an unfinished state.

3.2.5. The Logic of Nonrelative Terms

Peirce next extends his algebra of the copula to take in nonrelative operations. He begins a second set of numbered formulas. The first three, (1), (2), (3), give the schema that define the binary operat ions + and x , and the nullary operations 0 and o0. He first defines the nullary operations, again taking the equivalence of deduct ion and implication as his starting point:

We have seen that the inference

x and y �9 ~ ~.

is of the same validity with the inference

X

.'. Either 3~ or z


and the inference

with the inference

In like manner,

is equivalent to

and to

x

.'. Either y or z

x and 37 �9 �9149 ~ .

x - < y

(The possible)--<Either )~ or y,

x which is )~ --< (The impossible).

To express this algebraically, we need, in the first place, symbols for the two terms of second intention, the possible and the impossible. Let o0 and 0 be the terms; then we have the definitions:

x--~ ~, 0--< x, (1)

whatever x may be. (p. 125)

71

H e t h e n de f ines the b ina ry o p e r a t i o n s , by i n t r o d u c t i o n a n d el imi- n a t i o n rules:

We need also two operations which may be called non-relative addition and multiplication. They are defined as follows:

If a--< x and b--< x, then a + b--< x;

and conversely

if a + b--<x, then a ---< x and b --< x.

(pp. 125-126)

If x--< a and x--< b, then x--< a x b; (2)

if x---< a x b, then x--< a and x--< b. (3)

H e r e the words "and" a n d "or," as in (x a n d y) .'. z, a re p a r t o f the

m e t a l a n g u a g e , w h e r e a s x a n d + b e l o n g to the a l g e b r a o r f o rma l lan-

guage . E q u a t i o n (2) in the first set o f e q u a t i o n s at the b e g i n n i n g o f

Pe i r ce ' s a l g e b r a o f the c o p u l a m i g h t thus read: x .'. y - -< z ~ x a n d

y . . z.

We now e x a m i n e Pe i rce ' s first g r o u p of p r o p o s i t i o n s a n d p roo f s for

t h e m . All t oge the r , t hen , Pe i rce has 36 p r o p o s i t i o n s in his logic o f n o n -

re la t ive terms.


F r o m (3), se t t ing x = a + b a n d x = a x b, respectively, we ge t (4):

a - < a + b, a • b--< a, b - < a + b, a • b - -<b. (4)

Se t t ing a a n d b equa l to x in (2), x + x - < x. C o m b i n i n g with (4),

x = x + x (5)

follows. T h e case for t imes is dual .

a + b = b + a, a x b = b • a. (6)

P r o o f o f (6): F r o m b - - < a + b a n d a - < a + b a n d (2), b + a - - < a + b . Similarly, a + b--< b + a, so a + b = b + a; a n d dually.

( a + b ) + c = a + ( b + c ) , a x (b x c) = ( a x b) x c. (7)

P r o o f o f (7): ( a + b ) + c - - < a + ( b + c) ~ [ a + b - - < a + ( b + c ) a n d

c--< a + (b + c)] ~ [ a - < a + (b + c) a n d b--< a + (b + c) a n d c - < a +

( b + c ) ] ~ [ a - < a + ( b + c ) a n d b + c - - < a + ( b + c ) ] ~ a + ( b + c ) - - <

a + (b + c), which is t rue; a n d dually. A ce r ta in a m o u n t o f p r e l i m i n a r y ca lcu la t ion is n e e d e d for (8), the

d is t r ibut ive law, which is m o r e soph i s t i ca t ed a n d will no t be p r o v e d here :

( a + b ) x c = ( a x c) + (b x c) (a x b) + c = ( a + c ) x ( b + c ) . (8)

( a + b) + c = ( a + c ) + ( b + c) (a x b) x c = ( a x c) x (b x c). (9)

P r o o f o f (9): This follows f rom (5), (6), (7). At this po in t , we r e m a r k tha t x a n d + r e spec t --<, i.e.,

L e m m a : If a --< b, t h e n a + c --< b + c.

P r o o f o f l emma: It suffices to show tha t a + c--< b a n d a + c - < c. T h e first follows f rom a + c--< a - -< b (transitivity of --< follows f r o m tran-

sitivity o f .'. a n d (2) o f Pe i rce ' s impl icat ive logic [i.e., the i n t r o d u c t i o n

a n d e l i m i n a t i o n ru les] ) ; the s e c o n d is au toma t i c . (The p r o o f for x is

dual . )

It follows f rom this l e m m a tha t + a n d x are well d e f i n e d (i.e., if

a = b [ m e a n i n g a - - < b a n d b - < a ] , t h e n a + c = b + c a n d a x c = b x c).

a + (a x b) = a , a x ( a + b) = a . (10)

P r o o f of (10): First, we p rove a - -< b ~ b = a + b. Given a - -< b, we have

a + b--< b + b - b, whereas b--< a + b is au tomat i c . H e n c e , b = a + b. In

the o t h e r d i rec t ion , if b = a + b, t h e n a - -< a + b = b. Now (10) follows

easily, s ince a x b--< a. (The s e c o n d e q u a t i o n is dual . )

(a + b--< a) - ( b - < a x b). (11)


P r o o f of (11): We show x--< (a + b--< a) r x--< (b--< a x b). T h e first

ho lds if a n d only if x x (a + b) --< a, i.e. (x x a) + (x x b) - < a; the sec-

o n d ho lds if a n d only if x x b--< a x b. But b o t h c o n d i t i o n s h o l d if a n d only if x x b- -<a : b o t h imply x x b - -<a , by x x b - - < ( x x a) + ( x x b ) - - < a a n d X x b - - < a x b - -<a ; bo th are imp l i ed by x x b - < a , by (x x a) + ( x x b ) - - < ( x x a) + a = a , by (10), a n d x x b - - < a x bca

(x x b - - < a a n d x x b- -<b) .

3.3. Conclusion

In his 1880 p a p e r Peirce s t rugg led to work o u t the sense in which

universal a n d exis tent ia l quan t i f i ca t ion were dua l via n e g a t i o n , in an

a t t e m p t to b r ing the Aris tote l ian syllogisms within his system of impli-

cative logic. But his quan t i f i ed variables were no t explici t , n o r d id he

have a g o o d p rope r ly s coped subst i tute . His na tu ra l d e d u c t i o n system

for p ropos i t i ona l logic, however, is qui te beaut i ful .

In Pei rce ' s u n p u b l i s h e d m a n u s c r i p t 520 (Robin ca ta log) , in com-

m e n t i n g on the d e v e l o p m e n t of quan t i f i e r theory, Pei rce claims tha t he

e m p l o y e d indices to d e n o t e individuals as early as 1880:

This simple device of indices was used by me in 1880, or earlier; but I never appreciated its importance until I saw the use make of it in 1883 by Prof. O. C. [sic] Mitchell, then my student. Nor did he see its powers, until I showed them. The credit of the notation must be divided between us. Not only does this simple device remove the difficulty with regard to particulars but it also furnishes, at once, by attaching two or three or more indices to a letter (especially if we use as quantifiers, not merely II and I~, but II' and ~' where the multiplication or summation is to omit some one individual), it furnishes at once the best possible general algebra of relatives. (Robin ms. 520, pp. 2-3)

T h e only passage in the 1880 p a p e r that con ta ins a n y t h i n g r e m o t e l y

suggest ive of this claim is in sect ion 6 of C h a p t e r III, on m e t h o d s in

the a lgebra o f relatives. In the re levant passage, Pei rce says:

[L]et us investigate the relations of tb and l h to /b when l and b are totally unlimited relatives. Write

l= g,(L; : M;), b = I]j(Bj : Cj).

Then . . . . by the second and third propositions above, u

tb--< (L;" M;)b + L;, l b--</(B i �9 Cj) + kBj.

But by the first rule of the last section

74

hence,

PEIRCE'S ALGEBRA OF LOGIC

(L,: M,)b-< lb, /(B/: Cj) -< lb;

n

tb--< lb + L i, I h --< lb + k Bj.

There will be propositions like these for all the different values of i and j. Multiplying together all those of the several sets, we have

m

lb---< lb + I I i L i, l I' --< lb + I I j kB j .

But

n,L,= 2L,, IIjkBj= E, ikB j,

and since the relatives are unl imited,

Z,L,= ~, CikBj =~, I]~,= O, E, jkBj=O.

(p. 152)

We can see in latter equations that he indeed is using the notat ion for quantification over individuals at this early date. Still, he can be given no credit for unders tanding quantifier logic at this time, because he did nothing with it. After all, sums and products over indices occur all over mathematics in infinite series and products; the challenge is the al ternat ion of quantifiers. The best that can be said for this early work is that Peirce did begin to write some infinite sums and products over individuals in logic, perhaps for the first time. Mitchell at least was interested in working out the rules of logic for two-quantifier statements, and this is where the real future of the work lay.

In section 7 of Chapter III, on general formulas for relatives, Peirce uses both sums and products over relative letters p to give distributive laws for relative product and exponentiat ion. Schr6der later uses these same formulas for rules of quantifier manipulat ion (Schr6der 1895, p. 491). For example, a typical Peirce formula is

(a x b)c= IIt,{a(c x p) + b(c • fi)},

a(b x c): IIt,{(a x p)b + (a x fi)c},

where 1-It, is a quantifier over all p. These are quantif ier rules for I-It,. But Peirce did not regard quantifiers as basic building blocks at this time and did not perceive this as a definition in terms of quantifiers. We will see in the next chapter the extent to which Mitchell worked out the rules of transformation in the two-quantifier case and brought Peirce's inklings to bloom.

4. Mitchell on a New Algebra of Logic: 1883

I n t r oduc t i on

O. H. Mitchell's paper, "On a new algebra of logic," was published in 1883 in Studies in Logic, a collection of papers written by C. S. Peirce's students at the Johns Hopkins University and edited by Peirce. In his paper, Mitchell develops a recognizable notion of and notation for existential and universal quantifiers, but does not have the general concept of a formula and therefore of bound or free occurrences of variables.

Mitchell studied logic with Peirce at Johns Hopkins in the early 1880s and was enrolled, with all the other eventual contributors to Studies in Logic, in Peirce's course in advanced logic in the fall of 1880. Along with Peirce's other advanced students, Mitchell was a member of "The Metaphysical Club," founded and moderated by Peirce, which met monthly for the reading and discussion of a research paper presented by one of its members. Mitchell presented a preview of his paper for Studies in Logic to the Metaphysical Club in November 1882 and published an account of it, prior to his presentation, in the first volume of The Johns Hopkins University Circulars in May 1882. This account, entitled "On the algebra of logic," is a concise summary of the quantifier theory that appears in Mitchell's paper for Studies in Logic and ends with the sentence: "For further development of the subject, reference is made to Mr. Peirce's forthcoming volume of contributions to logic." Thus we can say with certainty that by May of 1882 at the latest Mitchell had worked out his quantifier theory in the form in which it appears in our discussion here.

4.1o Mitchell's Rule of Inference

Mitchell's 1883 paper begins with standard Boolean algebra, augmented by a special rule of inference:

75

76 MITCHELL'S ALGEBRA OF LOGIC

The algebra of logic which I wish to propose may be briefly characterized as follows: All propositions---categorical, hypothetical, or disjunctive--are expressed as logical polynomials, and the rule of inference from a set of premisses is: Take the logical product of the premisses and erase the terms to be eliminated. No set of terms can be eliminated whose erasure would destroy an aggregant term. (Mitchell 1883, p. 72) 1

Note that this is a single rule of inference, adequate as a proof or deduct ion procedure for propositional logic. In that regard it is like Robinson's resolution rule, a single rule that is also sufficient for propositional logic. The idea is the same, but Mitchell obviously was not thinking in terms of machine theorem proving, and so did not come up with an efficient form. Mitchell says that all propositions are Boolean polynomials. In an argument, one Boolean polynomial can be deduced from several others as premises using as a rule of inference: conjoin the premises and then "erase the terms to be eliminated." What does this last phrase mean? We might not call this a single rule of inference. In modern language, it is equivalent to first writing the premises in disjunctive normal form, as a disjunction of conjunctions of atomic statements a n d their negations. Then take the conjunct ion of these proposit ions and use the distributive law to write the conjunct ion of all premises in disjunctive normal form. Some of the conjunctions will contain a propositional letter and its negation (a literal and its opposite). These are self-contradictory and can be eliminated (this is the elimination he refers to). But Mitchell does not go on to explain how to get to the conclusion or show that one cannot; he only refers to the premises.

Here is one modern way of formalizing Mitchell's process. Write the disjunctive normal form above using all the propositional letters occurring in both the premises and conclusion. This means that for all p occurring only in the desired conclusion, one is to conjoin p v--,p to the conjunc t ion of the premises before forming the disjunctive normal form above. Doing the same for the conclusion, namely, writing it conjoined with all p v--,p with p occurring in some premise but not in the conclusion, one gets a disjunction of conjunctions, none containing a propositional letter and its negation. Then the premises imply the conclusion if and only if each conjunction in the representat ion of the conclusion is, up to its order, a conjunction in the representat ion of the premises. In terms of the finite Boolean algebra generated by the proposit ional letters occurring in premises and conclusions, this is just the assertion

'Except where otherwise noted, all subsequent page citations in this chapter refer to O. H. Mitchell's "On a new algebra of logic," in Studies in Logic (1883), in the reprint edition by John Benjamins (1983).


that a < b if and only if every atom <a is also <b. This is the same as a logically implying b.

Such is Mitchell 's rule of inference for Boolean polynomials. He says clearly that he is using + for inclusive disjunction, unlike Boole. He uses concatenat ion for conjunction, and a bar over a letter to indicate negation.

Turning to the subject of propositions, Mitchell says that

Logic has principally to do with the relations of objects of thought. A proposition is a statement of such a relation. (p. 73)

He goes on to say that objects of thought may themselves be class terms, or propositions. Such propositions about proposit ions were called secondary propositions by Boole. Mitchell's class terms seem to be ei ther names of classes or variables ranging over classes, and the operat ions are operat ions on classes. Similarly, propositional terms seem to be propositions or variables ranging over propositions, and operat ions on them are proposit ional logic operations.

This dual interpretat ion of Boolean expressions as ranging over ei ther classes or propositions is traceable to Boole himself, and we find it in Hunt ing ton ' s Boolean algebra axioms in 1904. It is also present in Peirce and Schr6der. Peirce seems to have had these two models and their commonali t ies continually in mind when developing his implicative propositional logic and the lattice-theoretic version of the algebra of logic in his 1880 paper "On the algebra of logic" (see Parts I and II of Peirce's 1880 paper, especially note 1, p. 118).

As we now unders tand it, there are two sources of Boolean algebras: sets and propositions. The propositional interpretat ion leads to the construction of free Boolean algebras as L indenbaum algebras of propositional logics. In mathematical terms, every Boolean algebra is isomorphic to a quot ient of a free Boolean algebra. That is, all Boolean algebras are isomorphic to Boolean algebras that are L i n d e n b a u m algebras of theories in propositional logic, which in turn are obta ined by equat ing propositions, if their equivalence is a consequence of the theory. The set point of view, on the other hand, leads to a considerat ion of power sets as Boolean algebras. Stone's representa t ion theorem (1936) says that every Boolean algebra is isomorphic to a subalgebra of a power set algebra. Thus, from an abstract algebraic point of view, there is little or no difference between Boole's two interpretat ions of his notation, set and propositional, but this was not recognized at the time. These same remarks do not apply to relations, for which the subject is much more complex.

Turning to relations, Mitchell makes interesting use of the term "uni-

7 8 M I T C H E L L ' S ALGEBRA OF LOGIC

verse of re la t ion" a n d the symbol 0o as the set of all possible states o f the universe. He says:

MI. Peirce uses oo indifferently as a symbol for the universe of class terms, or for the universe of relation, but in the method of this paper it seems most convenient to have separate symbols. We can speak of "all of" or "some of" U, but hardly, it seems to me, of "all of ' or "some of ' the universe of relation; that is, the state of things. For this reason o0 seems an especially appropriate symbol for the universe of relation. (p. 73)

O n e assumes that w h e n Mitchell refers to the "state of things," he m e a n s

no t only tha t the d o m a i n U that is given, but also tha t an i n t e r p r e t a t i o n

of the re la t ion symbols on that d o m a i n is specified.

In the d o m a i n of classes, U is the universe of class terms. In the d o m a i n of p ropos i t ions , o0 is the universal ly t rue p ropos i t i on (i.e., t rue in all

states of things) . In the d o m a i n of relat ions, 00 is the universal re la t ion.

We can also th ink of it as the p ropos i t iona l func t ion tha t is always va lued

1. T h e symbol ~ seems to be rese rved for the later par t o f Mitchel l ' s

paper , whe re the state (what is t rue, what t ru th va lua t ion is be ing used)

is a func t i on of time. So, the "universe of re la t ion" is he r e because which re la t ions hold , that is, what the state of th ings is, d e p e n d s on what t ime

it is. T h a t is, t he re is an "all of" a n d a "some of" at o n e t ime, or over an interval of t ime, or over all time:

The relation implied by a proposition may be conceived as concerning "all of" or "some of" the universe of class terms. In the first case the proposition is called universal; in the second, particular. The relation may be conceived as permanent or as temporary; that is, as lasting during the whole of a given quantity of time, limited or unlimited,rathe Universe of Time,- -or as lasting for only a (definite or indefinite) portion of it. A proposition may then be said to be universal or particular in time. The universe of relation is thus two-dimensional, so to speak; that is, a relation exists among the objects in the universe of class terms during the universe of time. (pp. 73-74)

This is, in o t h e r words, a pr imit ive t e m p o r a l logic.

4 .2 . S ing l e -Va r i ab l e M o n a d i c L o g i c

4.2.1. Single-Variable Monadic Propositions

Mitchel l begins the first pa r t of his system with a discussion of what he

calls "o rd ina ry propos i t ions ," i.e., those wi thou t t ime d e p e n d e n c e . He


says that all Boo lean polynomials F are dis junct ions of c on junc t i ons of class t e rms and their negat ions:

Let F be any logical polynomial involving class terms and their negatives, that is, any sum of products (aggregants) of such terms. (p. 74)

O n e example of a logical po lynomia l is F = ab + db, whe re a a nd b are class terms. Note that the free variables implicit in class te rms are no t explicitly m e n t i o n e d . If we put t hem in, F c a n be c o n s t r u e d as a m o n a d i c quant i f ier - f ree formula . Tha t is, we would now write "(x is a a nd x is b) or (x is no t a and x is b)" for F. Mitchell would write ab + 6h, where it is u n d e r s t o o d that he is re fe r r ing to the same m e m b e r x o f bo th classes.

Mitchell then in t roduces his quant i f ie r forms:

The following are respectively the forms of the universal and the particular propositions:

All U is F, here denoted by Fl,

Some U is F, here denoted by F,.

(p. 74)

This formalizes Aristotle but is poor ly sui ted to express the quantif iers . Ins tead of saying (3x)(U(x) A F(x)), as we might , "some U is F" is viewed as a re la t ion be tween U and b, d e n o t e d by F,,. Likewise, "all U is F ' is viewed as a re la t ion be tween U and F, d e n o t e d by F I.

This is a highly asymmetr ic nota t ion . T h e existential o p e r a t o r associated with U applies to a Boolean po lynomia l b, c o n s i d e r e d as a predicate o f one variable. T h e result is " there exists an x in U such that x is also in F." T h e r e f o r e the existential quant i f ie r " there exists an x such that U(x)" binds to F(x) to form " there exists an x such that U(x) and F(x)," d e n o t e d F,,, and the subscript takes care of the b ind ing . However , U and F seem to be only monad ic , that is, buil t up f rom class terms ( cons ide red as unary p red ica te letters) by Boolean opera t ions . Similarly for hi, which is again d e p e n d e n t on F and U and says that if we b ind "for all x in U" to F(x), we get "for all x in U, F(x)." These are i n d e e d

quan t i f i ed p ropos i t ions in the m o d e r n sense, since we have quan t i f i ed the single variable occu r r ing in U and F. But they are a l imi ted class. In o t h e r words, this system seems to be monad ic , and m o n a d i c p red ica te logic seems to be a natura l m o d e l for his logic o f class terms.

Mitchell points ou t that

FI +F, ,=~, m

F~F,,=O"


that is, "F l or (not F)u" is t rue and "F 1 and (not F) ," is false. Using this pure ly single-variable monad i c nota t ion , he also gets

FIF 1 =0 ,

Fu + F,, = 00;

that is, "F 1 and (not F)l" is false (i.e., F l and F L a r e cont rar ies of each o the r ) and "F, or (not F),," is t rue (i.e., F u and F,, are subcont ra r ies ) .

T h e no ta t ion for nega t ing a quant i f ied form is also poor. Mitchell 's no ta t ion for nega t ion is the usual one , i.e., a bar over a t e rm negates that term. However, his no ta t ion for quant i f ica t ion, via the subscripts, does no t makes it qui te clear what the over l in ing refers to. He gets a r o u n d this p r o b l e m by assuming that nega t ion binds m o r e s trongly than quant i f ica t ion , but pa ren these s must then be a d d e d to make the scope of nega t ion clear:

The line over the Fdoes not indicate the negative of the proposition, only the n_egative of the predicate, F. The negative of the proposition F I is not F l, but (Fi), which, according to the above, = F u. (p. 74)

4.2. 2. Disjunctive Normal Form

Mitchell writes out the Aristotel ian propos i t ions E, I, A, O in his no ta t ion . They read

E (d+/~)~ = All of Uis d + / ~ = No a i s b,

I (ab),, = Some of Uis ab = Some a i s b,

A ( d + b ) ~ = All of Uis d + b = All a i s b,

O (a/;),, = Some of Uis a/~ = Some a i s no t b.

He applies his two forms F~ and F u to all possible sums of the 2 2

di f fe ren t min imal propos i t ions ab, db, ab, and d/; and obta ins a com- p rehens ive list of 16 equivalences. These are ob t a ine d by pu t t ing the universal quant i f ie r "1" in f ront of each of the 16 dist inct disjunctive n o r m a l forms r e p r e s e n t e d by the sum of the four min imal proposi t ions , in which each of the min imal propos i t ions e i the r occurs or does no t occur, and then nega t ing each express ion. In o t h e r words, in one possible sum of the four min imal proposi t ions , all four min imal proposi- t ions cou ld occur. This would be the sum ab + db + ab + dD. In a n o t h e r possible sum, ab would not occur, but the o t h e r th ree min imal prop- _ osi t ions would appea r in the sum. This would be the sum db + ab + d/~. T h e r e are 2 4 d i f ferent possible sums, since there are four d i f fe ren t min imal proposi t ions . Quant i fy ing these sums with "1" or "'u" gives all

F R O M P E I R C E T O S K O L E M 8x

the assertions that can be made about a and b. Mitchell presents this informat ion in a table:

(ab + af~ + fib + ff-~)l . . . (0),,

(af~ + fib + df~)~ .. . (ab),,

(fib + f {~ + a b ) , . . . (of), ,

( d f~ + a b + aft) 1 .. . (fib),,

(ab + af~ + fib), . . . ( dl~),,

(ab + a[~)~ .. . (fib + fl~),,

(ab + fib), . . . (a[~ + d/~),,

(ab + f{~), . . . (af~ + fib),,

(af~ + fb) 1 ... ( 6J~ + ab)u

(af~ + diS) 1 .. . (fib + ab),,

(fib + fD)! . . . (al~ + ab),,

(ab) l . . . (af~ + f b + df~),,

(alS)~ . . . ( f b + dl~ + ab),,

(86)1 . . . ( 6J~ + a b + of) , ,

( ff~), . . . (ab + ab + fb), ,

(0)1 .. . (ab + al~ + f b + ff~),,.

(p. 75) If we take any Boolean polynomial in two variables a and b, then,

using De Morgan's laws, each is equivalent to a disjunctive normal form, i.e., a disjunction of conjunctions of atomic a and b and their negations. The four basic propositions are then: ab, ab, fb, and f/~. These are exclusive (disjoint), and every Boolean polynomial in a and b is a disjunc t ion of none, some, or all of them. Therefore , any subset of this four-element set, 16 in number, gives exactly one disjunctive normal form, with no two equivalent.

In modern language, these 16 terms represent the e lements of the free Boolean algebra on two generators. For Mitchell, this table dem- onstrates that he can take any monadic predicate built up from two basic propert ies of objects, such as man (x is a man) and mortal (x is mortal) , look at every Boolean combinat ion, and tell us how to write its universal quantification equivalently as an existential s ta tement applied to ano ther (dual) disjunctive normal form. In o ther words, this is the familiar rule for negating a universal quantif ier applied to a prenex (monadic) formula by replacing the "for all" by "there exists" and replacing the disjunctive normal form present by the disjunctive normal form of its negation, that is, by the disjunction of the four terms above that were not present.

Mitchell unders tands that if three distinct terms a, b, and c are treated

82 M I T C H E L L ' S A L G E B R A O F L O G I C

in a similar way, then instead of having 22= 4 atoms, we would have 2 ~ - 8 a toms and eight dif ferent "minimal" proposi t ions, where the minimal proposi t ions would be the p roduc t of three literals, for instance, abg. The n u m b e r of different disjunctive normal forms that we obtain

2 3 for the quantif ier-free par t is then 2 - 256. For n distinct terms, there would be 22'' disjunctive normal forms:

2 3 If three terms be treated in a similar way we get 2.2 ,= 512, different propositions. With n terms the total number is 2.22". (p. 76)

(Mitchell 's totals are mul t ip l ied by 2 because he counts the nega t ion of each normal form.)

4.2.3. Rules of Inference for Single-Variable Logic

Perhaps the most interest ing features of Mitchell 's system are his inference rules. He gives an algebra of quantif iers for his single-variable logic, which he later modifies slightly for two-variable forms. In o the r words, he actually extends algebraic rules to quantifiers. Even today we do not th ink of quantif iers this way. We have come to th ink of formulas as syntactical objects to be hand led by algebraic values for plus and times, but this idea is not general ly e x t e n d e d to quantifiers. Mitchell 's laws for u + 1, 1 + 1, u x 1, etc., however, take Boole's original idea to an ex- t reme. In spirit, this is very m u c h like Peirce 's symbolism for universal quant i f ica t ion as exponent ia t ion .

Mitchell, going back to Aristotle, builds on the assumpt ion that "all" is not asserted unless some th ing exists:

Since the universe of class terms is supposed greater than zero, the dictum de omni gives

F, - < F,,;

that is, "all U is F' implies "some U is F." (p. 77)

In this regard, he is not following Peirce (see our discussion in w 3.2.3). He does, however, use Peirce 's symbol - < for implicat ion and inclusion.

To develop his rules, Mitchell considers Boolean combina t ions of F~ and F,, s ta tements (e.g., F 1 + Fu). First of all, he can put any such state- m e n t in disjunctive normal form, where now the atoms are of the form F~ and F u. He already knows that the negat ion of an F l takes the form G,,, and conversely, because of the work he has done previously in this paper. His concern now, therefore , is the disjunctive normal form of Boolean combina t ions of s ta tements F, and F~.


Mitchel l conce ives of the quan t i fy ing suffix u as r a n g i n g over the in terval f r om 0 to 1"

m

To say "no U is F' is evidently the same as to say "all U is F"" that is, F 0 =F~, and since a proposition whose suffix is 0 is thus expressible in a form with the suffix equal to 1, each suffix will be supposed greater than zero. The suffix u in F, is taken to be a fraction or part of U less than the whole; that is, "some of" U. In the proposition "some U is F ' it is not denied that all U may be F, but the assertion is made of only a part of U. Thus u is taken as greater than zero or less than 1, or U. (p. 77)

T h e suffix u (no t e tha t this u is lowercase) is thus a f rac t ion , o r pa r t o f

U. H e r e he says tha t it mu s t be a p r o p e r part , a n d thus less t h a n 1. O n

o t h e r occas ions this seems to be incons i s ten t ; he s o m e t i m e s appea r s ,

for e x a m p l e , to have t h o u g h t o f u as s o m e t h i n g tha t c o u l d be instan-

t ia ted if the f o r m u l a is actual ly i n t e r p r e t e d . Most impor t an t ly , however ,

he u n d e r s t a n d s tha t the u quan t i f i e r s in d i f f e r en t p r o p o s i t i o n s c a n n o t

be c o m b i n e d , s ince it is no t c lear w h e t h e r t he r e is a c o m m o n u:

When u is written as a suffix of different propositions in the same argument, it is not meant that the same part of U is concerned in each case. (p. 77)

T h e a lgeb ra for his quan t i f i e r s is given by the g e n e r a l ru le

F,C,, - < (FC),,,;

dual ly

F, + 6;,,--< ( F + G),+,,.

B r o k e n d o w n in to cases, this g e n e r a l ru le says:

all U i s F a n d all U i s G

all U is F a n d s o m e U is G

s o m e U is F a n d all U is G

s o m e U i s F a n d s o m e U is G

impl ies

impl ies

impl ies

impl ies

all U is ( F a n d G),

s o m e U i s ( F a n d G),

s o m e U is ( F a n d G),

s o m e U is ( F a n d G).

T h e last case is a pu re ly a lgebra ic result , which he rejects explicitly, s ince

clearly ( 3 x ) ( x e U A x e F) A ( 3 x ) ( x e U A x e G) does not imply

(3x)(x e UA (x e FA x e G), or, in his own terms,

There can be no inference when nothing is known about the relation of the two suffices; that is, F,, G,,--< o0. (p. 78)

Dually, we have


all U is F o r all U is G implies

all U is F o r some U is G implies

some U is F or all U is G implies

some U is F o r some U is G implies

all U i s ( F o r G),

some Uis ( F o r G),

some U is (F or G),

some U is ( F o r G).

These formulas appea r in a table that follows:

(1) F 1G 1 = (FG)l ,

(2) F~ G,,--< (FG),,, (3) F,,G.---< oo.

F, + G, = ( F + G),, (1')

F,, + G,--< ( F + G)u, (2')

F 1 + G , - < ( F + G)I. (3')

(p. 78) T h e quant i f i e r a lgebra is 1 x 1 = 1, 1 x u = u x 1 = u, a nd u x u ' =

u n d e f i n e d ; u + u = u , u + l = l + u - u , and 1 + 1 - 1 . It is no t clear f rom the discussion why u + 1 = u.

Mitchell then says that by cons ide r ing (1) and (1'), we can see that:

The most general proposition under the given conditions is of the form

n(~, + ~G,), or ~:(FlrIG.),

where F and G are any logical polynomials of class terms, II denotes a product, and I~ denotes a sum. (p. 79)

T h e second of these two forms is most impor tan t . It says that every Boo lean c o m b i n a t i o n of F~, G,, p ropos i t ions is a d is junct ion of con- j unc t ions , each of which is a con junc t ion of a single F~ and a c on junc t i on o f mul t ip le G,,'s.

Now any con junc t of a disjunctive no rma l form of Fl'S and Gu's is cer tainly a con junc t ion of some F~'s and some Gu's, since it consists of some universal formulas and some existential formulas . At this point , Mitchell is on track again, no twi ths tand ing some ambigui ty above, since a con junc t i on of universals i~ a universal, i n d e p e n d e n t o f what variable is used, and a con junc t ion of existentials c a n n o t be simplif ied, since the individuals r e fe r red to n e e d no t be the same. This discussion therefore correct ly de t e rmines the form of Boolean combina t i ons of F l and F,, s ta tements .

Mitchell relates his two genera l forms to De Morgan ' s p ropos i t ions in this way:

If F, G, etc. be logical functions of any number of class terms, a, b, c, etc., the general proposition

ri(~. + EGz), or I;(F~IIG.),


may be reduced to a function of the eight propositions of De Morgan of the form

II~tz.

(pp. 79-80)

85

The eight propositions of De Morgan are (d +/~)~, (d + b)~, (a +/~)~, (a + b)l, and their negations. Mitchell then gives the el iminat ion rule above for a universally or existentially quantif ied disjunctive normal form of a quantifier-free statement.

Mitchell 's theory th roughout the first part of his paper (pp. 72-87) can be viewed as constituting the algebra of single quantifiers for formulas of a single fixed monadic variable. He then proceeds (pp. 81-87) to give applications of the proof procedure he has developed, verifying Aristotle's syllogisms and solving problems presented in Boole (1854).

The only class of statements Mitchell considers in the proof procedure is the class of Boolean combinat ions of F~, G u statements. The F~, G,, s tatements are single quantifiers applied to a Boolean combinat ion of monadic predicates in a single variable. An example of this is (ab + dc)~. The most general statements Mitchell considers are Boolean combinations of these. He gives enough rules of inference to obtain a decision me thod for this class of propositions. Implications between such propositions are then reduced to the same form by c~--</3 is & +/3, where o~ and /3 are again propositions. The canonical forms II(F u + EGI) or E(F~IIG,,) (p. 79) are for statements of this form.

Mitchell 's material here is literally a f ragment of monadic first-order logic. Mitchell has all the ideas for the decision me thod for monadic predicate logic, which essentially reduces to questions about finite Bool- ean algebras. 2 In addition, Mitchell gives an exact s ta tement of his, as contrasted to Boole's, me thod for drawing consequences:

As already stated, this algebra is the negative of Boole's as modified by Schr6der, so far as universal premises are concerned. Thus Boole multiplied propositions by addition, and eliminated by multiplying coefficients. The method here employed multiplies propositions by multiplication, and eliminates by adding coefficients. When many eliminations are demanded in a problem, the advantage in point of brevity of this method over Boole's is of course greatly increased. (p. 81)

Boole used the dual conjunctive normal form. Mitchell thinks his me thod is faster, but he is probably wrong. Automatic theorem proving

~The decision method for the latter is in Hilbert and Ackermann (1928), from L6w- enheim (1915).


is always based on the conjunct ive no rma l fo rm because the r educ t ion to conjunct ive no rma l form is in polynomia l time, while the r educ t ion to disjunctive normal form is no t known to be.

4.3. T w o - V a r i a b l e Monadic Logic

4.3.1. Mitchell's Dimension Theory

T h e first par t of Mitchell 's paper , p r e c e d i n g the discussion of p ropo- sitions that are d e p e n d e n t on time, is, in m o d e r n terms, a t r e a t m e n t o f m o n a d i c p red ica te logic with one variable. The second par t of his paper , dea l ing with propos i t ions that are d e p e n d e n t on time, is a fo rm of m o n a d i c t empora l logic, with jus t one a l te rna t ion of quantif iers . Mitch- ell i n t roduces an odd no ta t ion for the no t ion o f " t h e r e exists an x, there exists a t ime t such that R(x, t)." Peirce simplif ied this, as we shall see in the nex t chapter , but for now, let us concen t r a t e on Mitchell 's theory.

Mitchell presents a system of all possible forms, assuming that we only allow two d imens ions , choos ing t ime as the second d imens ion :

Let U stand for the universe of class terms, as before, and let V represent the universe of time. Let F be a polynomial function of class terms, a, b, etc. Then let us consider the following system of six propositions:

F,,~, meaning some part of U, during some part of V is F,

F,, l, meaning some part of U, during every part of V, is F,

F1, ,, meaning every part of U, during some part of V, is F,

F,, l, meaning the same part of U, during every part of V, is /~;

F1,/, meaning every part of U, during the same part of V, is E

Fll, meaning every part of U, during every part of V, is E

(p. 87)

For example , let Fis the p red ica te "is ill" and let U b e the te rm "Brown." T h e n F~ says all Browns are ill. Thus, F is a descr ip t ion of all Browns, o r a descr ip t ion of all parts of U, and "is a descr ip t ion of" is a b inary re la t ion be tween predica tes F and class terms U. We do not dis t inguish he re be tween predicates and class terms; they are bo th m o n a d i c predicates for us. Similarly, F~ says "is ill" is a descr ip t ion of every par t o f Brown du r ing every par t of t ime (every t i n V). In o t h e r words, F~ is a descr ip t ion of every par t of U d u r i n g V. This is a te rnary rela t ion be tween F, U, and V; that is, a re lat ion be tween a pred ica te and two class terms. Again, for us these are all m o n a d i c predicates .

Mitchell explains how these six p ropos i t ions are related, a nd remarks in a f oo tno t e that this system is essentially due to Peirce:


The dictum de omni gives the following relations a m o n g these six propos i t ions:m

Fll--<F1,,,F~,~F~,,F~1F,,,,, and Fl, + F~,,, + F~,, + FI,, + F~, - < F,,,,;

and since same is included unde r some, we have

FI,,-<FI, ,, and Fuq--<F,, l .

The following pairs of proposit ions,

Fu,, and Fll, F,l and FI,,,, Fu,l and Fl,,,

satisfy the two equations

ot + / 3 = ~ ,

~ = O,

and the members of each pair are therefore the negatives or contra- dictories of each other. (p. 88)

87

His f o o t n o t e e x p l a i n s t h a t he i n t r o d u c e d the p r i m e n o t a t i o n to t r e a t

t he two cases, F~v, a n d F,/~, t h a t he h a d ini t ia l ly o m i t t e d :

The natural first thought is that F~l, F,, l, F1, ,, F,,, form a system of proposit ions by themselves, but it is seen that F1~, and F,, l must be added to the system, in o rder to contradict F,1 and F1, ,. Mr. Peirce pointed out to me that these proposit ions are really triple relatives, and are therefore six in number. F~, for instance, means "Fis a descript ion of U during V." See Johns Hopkins University Circular, August, 1882, p. 204. 2 (p. 88)

T h e logic invo lved h e r e is c lear ly a t e m p o r a l logic. T h e r e is a u n i v e r s e

V o f t i m e m o m e n t s . P r o p o s i t i o n s can h o l d at s o m e t i m e in V a n d at all

t imes in V. Ex i s t en t i a l p r o p o s i t i o n s m e n t i o n wi tnesses . T h e wi tness m a y

be the s a m e at d i f f e r e n t t imes o r n o t spec i f i ed to be t h e s ame . W h e n

t he i n d i v i d u a l is t he s ame , a n o t a t i o n is n e e d e d to i n d i c a t e this. M i t c h e l l

uses a p r i m e for this p u r p o s e : s u b s c r i p t u ' i n d i c a t e s t h a t t h e s a m e wi tness

is u s e d at d i f f e r e n t t imes. So,

u v 3x3t

u l Yti lx

l v Yx3t

u ' l 3xYt

l v ' 3 tYx

11 YxYt.

:4 The page number that Mitchell intended is p. 208 from the May 1882 issue of 7"he .Johns ttopkins University Circub~rs.


Thus, Mitchell's system does present alternations of two quantifiers, but only for a very special set of propositions, namely, Boolean combinat ions of the forms "(all, some) elements of U are in F during (all, some) times," where a notation is in t roduced to distinguish when the same elements are referred to at different times. He cannot accommodate nested quantification beyond one alternation. This is definitely a limi- tation of his system, and one that comes automatically with his way of thinking of dimensions as separate.

The algebra of these propositions is not complicated. Mitchell goes on to obtain the rules for handling the alternating quantifiers AE and EA. They are the same as those for single-variable logic, discussed above.

Mitchell indicates that this system can be generalized, but again his theory is so closely tied up in unary predicates that he does not arrive at a general predicate logic. Rather, he has a monadic temporal logic, and only two quantifiers at that: one for the domain, and one for time, awkwardly put. If we allow any number of dimensions (he indicates three on p. 95, but does not give examples, which would show the shortcomings of his notation), we can express any first-order logical formula. Yet it is not at all certain that he thought this far.

In sum, Mitchell's theory is possibly the first occurrence of a systematic notat ion for one-quantifier monadic statements and one-quantifier monadic statements in a temporal logic, the latter giving at least the rules for handl ing a pair of quantifiers. Systematic proof procedures are given based on disjunctive normal form and elimination and quantifier rules for Boolean combinations of prenex monadic statements, classical or temporal. Mitchell got no further, but his system can be seen as a definite precursor of full predicate logic, based on the detail given in both proposit ional and quantifier proof rules. However, he did not arrive at the general notion of a formula, even for the monadic case, probably because he was tied so closely to the Aristotelian class te rm-predica te formulat ion of propositions.

4.3. 2. Contrast to Peirce

It is not clear to what extent Mitchell actually built on Peirce's work. Mitchell's dimension theory, in which propositions depend on time, is clearly the source of his t reatment of alternation of quantifiers. He does not, however, derive it from Peirce's binary relations, with which one can certainly write, as Peirce does, using summation and product notation, IIx]2y %, i.e., "for all x, there exists a y such that R(x, y)." Instead, Mitchell says, in so many words, that the truth or falsity of a monadic predicate U(x) depends on what time it is-- that is, on the current state of a world that is changing. In modern terms, he is considering a truth


valuation that is changing as a funct ion of time. Thus, "some par t of U, dur ing some part of V, is F' means "there exists an x in U and there exists a time t in the set of possible times Vsuch that F(x) holds at time t . ~

Now, that F(x) holds at time t can be though t of as a binary relation F(x, t). Therefore , time-valued truth or falsity raises us f rom monad ic predicate logic to two-sorted first-order logic: one sort being the doma in in tended , and the o ther sort being time, with the variables ranging separately over these sorts. In this system, the above s ta tement as an opera to r on U, V, and F is written F,,~. The subscripts and their o rde r indicate binding.

If we look at the s ta tement "some part of U, dur ing every part of V, is F," or, equivalently, using Peirce's binary relations, " there exists an x in U for all y in the set of times V for which F(x, y)," where F(x, y) is the assertion that F(x) holds at time y, this is definitely a binary two-sorted relation in m o d e r n terms. Mitchell writes it as F,,~, using the subscripts and their o rde r to indicate binding. However oddly it is written, this is monadic temporal logic.

4.4. Th ree -Var i ab l e M o n a d i c Logic

Mitchell 's proposi t ions of three d imensions (p. 95) are simply those that are built up from Boolean operat ions applied to unary predicates and allow three consecutive quantifiers associated with predicates as above. Hence , they cor respond to certain three-variable formulas. The quantifiers are ( there exists an x in U), (for all x in U); ( there exists a y in V), (for all y in V); and ( there exists a z in W), and (for all z in W), used to bind in order. Mitchell apparent ly could not formula te any in terpre ta t ion of these, since no examples are given in this section of his paper (pp. 95-96) . But if we think of proposi t ions true at a time and place (space point) , we might get this in his notation: ( there exists an x in U)(for all y in V)(for all z in W)(F(x) is true at t ime y at point in space z). 4

In sum, Mitchell unde r s tood monadic predicate logic with two quantifiers and Boolean operat ions very well, and also formally unde r s tood three quantifiers. His calculations would today be calculations in monadic predicate logic, covered by the decision p rocedure for such statements in Hilber t and Ackermann (1928). Yet he did not get general quantifiers for binary relations, as did Peirce.

4 We note that b~(x,y, z) is ternary. However, Mitchell took monadic predicates as basic, as did Boole (although not Peirce), so he works out reduced forms based on monadic predicates, not binary or ternary ones.

9 ~ MITCHELL'S ALGEBRA OF LOGIC

In the last section of his paper, Mitchell explains Peirce 's no ta t ion for quant i f ica t ion (from Peirce 1880) and why his is better:

The propositions A and O in Mr. Peirce's notation are, respectively,

X-<Y,

x - < Y.

Mr. McColl expresses them in a similar way, using a different symbol for the copula. Both Mr. McColl and Mr. Peirce have given algebraic methods in logic, in which the terms of these propositions are allowed to remain on both sides of the copula.

In the method ofw 1 (of which w 2 is an extension), the propositions A and O are expressed as follows:m

(X + Y) l, equivalent to 00--< X + Y,

(XY),, equivalent to XY --< 0;

that is, all the terms of the universal proposition are transposed to the fight-hand side of the copula, while those of the particular proposition are transposed to the left-hand side. (p. 96)

T h e ch ie f advantage of Mitchell 's m e t h o d is that e l imina t ion can be p e r f o r m e d by mult iplying the quant i t ies to be e l iminated .

Mitchell is thus the crucial p r ecu r so r of Peirce 's discovery of full quan t i f i e r logic, but his work is l imited to the m o n a d i c p red ica te logic case with up to th ree quantif iers. It is not a c o m p l e t e t r e a t m e n t of m o n a d i c quant i f iers in general , but could easily be so ex t ended . It was i n t e n d e d to make sense of Boole ' s a t t emp t to hand l e m o n a d i c quantification. Al though t ime d e p e n d e n c e brings binary relat ions into Mitch- ell 's system, they are no t exploi ted . Mitchell is a disciple o f Boole in this paper , and the binary relat ions po in t of view is practically absent . T h e r e is no h in t of genera l p red ica te logic quant i f iers on relat ions; rather, he employs quant i f iers restr ic ted to m o n a d i c predicates , as appea r in Boole.

4.5. Pe irce on Mitchell

At various points in his writings, Peirce c o m m e n t e d on Mitchell 's work. In his Preface to Studies in Logic, da ted D e c e m b e r 12, 1882, Peirce says that Mitchell 's p a p e r presents original work, a d d i n g new no ta t ion to e x t e n d the expressibility of Boolean algebra:

These papers, the work of my students, have been so instructive to me, that I have asked and obtained permission to publish them in one volume.


Two of them, the contributions of Miss Ladd (now Mrs. Fabian Franklin) and of Mr. Mitchell, present new developments of the logical algebra of Boole. Miss Ladd's article may serve, for those who are unacquainted with Boole's "Laws of Thought," as an introduct ion to the most influential and fecund discovery of modern logic. The followers of Boole have altered their master's notation mainly in three respects.

i) A series of writers,--Jevons, in 1864; Peirce, in 1867; Grassmann, in 1872; Schr6der, in 1877; and McColl, in 1877,msuccessively and independent ly declared in favor of using the sign of addition to unite different terms into one aggregate, whether they be mutually exclusive or not . . . . The two new authors both side with the majority in this respect.

ii) Mr. McColl and I find it to be absolutely necessary to add some new sign to express existence; for Boole's notation is only capable of representing that some description of things does not exist. Besides that, the sign of equality, used by Boole in the desire to assimilate the algebra of logic to that of number, really expresses, as De Morgan showed forty years ago, a complex relation . . . . For these reasons, Mr. McColl and I make use of signs of inclusion and non-inclusion. Thus I write

Griffin --< breathing fire

to mean that every griffin (if there be such a creature) breathes fire; that is, no griffin not breathing fire exists; and I write

A n i m a l - < Aquatic

to mean that some animals are not aquatic, or that a non-aquatic animal does not exist. Mr. McColl's notation is not essentially different.

Miss Ladd and Mr. Mitchell also use two signs expressive of simple relations involving existence and non-existence; but in their choice of these relations they diverge both from McColl and me, and from one another. In fact, of the eight simple relations of terms signalized by De Morgan, Mr. McColl and I have chosen two, Miss Ladd two others, Mr. Mitchell a fifth and sixth. [Ladd's two relations are A v

m

B for "A is in part B" and A v B for "A is excluded from B."] iii) The third important modification of Boole's original notation

consists in the introduction of new signs, so as to adapt it to the expression of relative terms. This branch of logic which has been studied by Leslie Ellis, De Morgan, Joseph John Murphy, Alexander MacFarlane, and myself, presents a rich and new field for investigation. A part of Mr. Mitchell's paper touches this subject in an exceedingly interesting way. (Peirce 1883a, pp. iii-iv)

9 x

Mi tche l l ' s p a p e r effectively i n t r o d u c e s q u a n t i f i c a t i o n to B o o l e a n al-

gebra , r e p l a c i n g Boo le ' s unsuccess fu l n o t a t i o n v with Mi tche l l ' s two-

q u a n t i f i e r forms. A l t h o u g h Aris to t le in the Organon, Pe i rce in his 1870

92 M I T C H E L L ' S A L G E B R A O F L O G I C

paper on the calculus of relatives, and Frege in his Begriffsschrifi all had included devices for expressing "all," "some," and "none," Peirce's students were working from Boole's foundation, which was purely propositional, and, until Mitchell's work, had no good formalism for quantificational notions. Mitchell's paper introduces formalism for quantification, adjoining it to Peirce's version of Boolean algebra (where "+" means inclusive "or" or union) , rather than Boole's. This is Mitchell 's advance over Boole. Mitchell 's theory is limited, as we have seen; he can say "some" and "for all," but he is not able to combine these notions in interesting ways.

In Peirce's own Studies in Logic paper ("Note B"), Peirce appears to unders tand Mitchell 's advance completely and to generalize it immediately. Peirce first translates Mitchell 's six two-quantifier forms into the calculus of relatives, demonst ra t ing that Peirce's system is capable of expressing everything that can be said using Mitchell 's formalism:

Suppose that fand gare general relatives signif},ing relations of things to times. Then, Dr. Mitchell's six forms of two dimensional propositions appear thus:

F,, = 0 t f t 0

F,,, = 0 t f ~

r . , = o ~ f t 0

F,,,, = (0 tf)~0

F,,,1 = ~ ( / t 0)

F~IJ ~ (x)foo.

(Peirce 1883c, pp. 205-206)

The dagger denotes the dual of relative product, i.e., a t b = d b, which Peirce calls "relative sum," o0 is the universal relation, and 0 is the empty relation. Then, immediately following the passage jus t quoted, Peirce gives his first published example of II and I] as quantifiers: 1-I~ISil O (Peirce 1883c, p. 207). He introduces a notation to represent individuals, which is a key idea but is missing in Mitchell 's theory. Peirce's notat ion for the quantifiers supports our modern interpretat ion of them. We would write II~lSjl, i differently, but all of the syntactic elements are present in Peirce's 1883 discussion.

By 1885, Peirce is credit ing Mitchell for discoveries that we would not, on the basis of Mitchell 's paper, attribute to him. Peirce, in his 1885 paper "On the philosophy of notation," gives Mitchell credit for the introduct ion of indices, i.e., individual variables, to logic, saying


The introduction of indices into the algebra of logic is tile greatest merit of Mr. Mitchell's system. (Peirce 1885, p. 312)

93

Peirce is h e r e c red i t ing Mitchell with the i n t r o d u c t i o n of variables to

logic, 5 even t h o u g h Peirce uses the word "indices" in his s t a t emen t . A l t h o u g h Mitchel l does use subscripts, they do not , strictly speaking , play the role o f indiv idual variables in Mitchel l ' s theory, as Pe i rce ' s i

a n d j do in Peirce 's quan t i f i e r logic. In Mitchel l ' s two-quant i f ie r fo rm

Fu, ,, for e x a m p l e , u a n d v do no t d e n o t e the free variables of F; ra ther ,

the free variables of F over which o n e m i g h t quant i fy are d e n o t e d by

the posit ions tha t u a n d v occupy. T h e first pos i t ion, o c c u p i e d by u, if

for e l e m e n t s of the universe , a n d the s econd posi t ions, o c c u p i e d by v,

is for t ime. Peirce h imse l f used indices qu i te freely in his 1870 paper ,

bu t no t as indiv idual variables. He cal led t h e m " subad j acen t n u m b e r s "

(see Peirce 1870, pp. 40-43 a n d ou r discussion in w 2.1.3).

Peirce also gives Mitchell c red i t for i n t r o d u c i n g quant i f i e r s a n d prenex form:

We now come to the distinction of some and all ... All attempts to introduce this distinction into the Boolian [sic] algebra were more or less complete failures until Mr. Mitchell showed how it was to be effected. His method really consists in making the whole expression of the proposition consist of two parts, a pure Boolian expression referring to an individual and a quantifying part saying what individual this is. (Peirce 1885, p. 226)

Peirce ' s c o m m e n t on Mitchel l ' s m e t h o d is precisely the s t a t e m e n t tha t Mitchel l writes fo rmulas in p r e n e x form, namely, with a quant i f ie r -

free par t plus a pref ix tel l ing what the variables are a n d how they are quan t i f i ed . This is a g e n e r o u s r e a d i n g of Mitchell .

Pei rce also claims that Mitchell had the idea of e x t e n d i n g his single- quan t i f i e r forms to the logic of relatives:

Mr. Mitchell has also a very interesting and instructive extension of his notation for some and all, to a two-dimensional universe, that is, to the logic of relatives. (Peirce 1885, p. 228)

As we have seen, Mitchel l e n c o d e d two quant i f iers , o n e for each uni-

~ Even now the subscript notation for variables survives. When we write a sequence as {a,}, we often write, "for all i, a, has the property that ..."; this antique notation must be the origin of that used by Peirce and Mitchell. Also, the double sequence {a;i} had been in use for a long time. In much of Peirce's early work, he attempts to adapt a current mathematical notation, such as the exponential, to see if this is suggestive by analogy of" the right rules for his new operations. It thus appears that Peirce, and Mitchell following Peirce, took subscripts as the closest notation available for individual variables.


verse over which the quantifier ranges. But where Mitchell saw a mul- t idimensional parameter ized structure, with quantification of any given formula being over distinct axes, or universes, in that structure, Peirce saw a more flexible quantificational system that would permit multiple quantifications over a single structure. This is what Peirce recognizes as more general.

Peirce's extravagant praise of Mitchell suggests that Peirce may have unders tood Mitchell as having made a greater advance that he actually did. It might have been the case that Peirce developed Mitchell 's theory to include something more substantial than Mitchell 's original presentation, all the while believing Mitchell 's limited work to be the mature form of quantifier logic. On the other hand, Peirce's I-Ii~,jlij notat ion of 1883 may support s more modern interpretat ion than Peirce was actually making at the time. He may still have thought of E and II as sums and products, and not yet have had the logical ideas, i ndependen t of algebraic ideas, that he was attributing to Mitchell in 1885.

Conversly, how original was Mitchell? How strong was Peirce's influence on Mitchell 's work? Was Peirce praising ideas that he suggested for his s tudent to work on? We know, from Mitchell 's footnote in which he says that Peirce told him he was missing two of his quantif ier forms (p. 88), that Peirce read Mitchell 's manuscr ipt before publication in complete detail. (That is how one locates missing cases!) In Peirce's

n l

1870 paper, Peirce presented a list of formulas, ham, (ha) m, ba m, b" , b"m, b ..... , which express existential and universal quantification in various combinations. How is Peirce's list of six forms related to the six two-quantifier forms proposed by Mitchell? Did Peirce have some sort of general conception of quantifiers thirteen years before Mitchell 's paper, and did he communica te the idea to Mitchell? These questions now seem imponderable . Peirce gave credit to Mitchell for discovering quantifiers, but perhaps this credit is really for the rules governing al- ternat ing quantifiers, which Peirce did not have in 1870. He had an exponent ia l notation, as we examined in chapter 2, which expresses single quantifiers. Later he would take the two basic ideas of quantification for class terms and, identifying terms by his "lines of identity," give a rigorous pictorial (iconic) way of depicting quantifiers. Mitchell had a notat ion (subscript 1, subscript u) that identified variables by position of a rgument (therefore dimension) , a clumsy notation. Mitchell therefore might have been the first to use a separate sign for quantifiers, but his formal rules of el imination were not obviously generalizable to full quantif ier logic.

5. Peirce on the Algebra of Relatives: 1883

Introduction

Peirce's next paper on the calculus of relatives was completed in 1882 and published in 1883, as "Note B" to Studies in Logic. Its closing pages present some formulas containing quantifiers and some of the semantics of first-order logic. This was the first use of quantifiers II and E in Peirce's work.

The presentation of the calculus of relatives in Peirce's 1883 paper is more abstract and algebraic in form than in his 1870 and 1880 papers, showing the influence on Peirce of J. J. Sylvester and, most especially, Arthur Cayley, who was visiting at the Johns Hopkins University in 1882. Peirce introduced relative sum as the dual of relative product, in place of exponentiation, to obtain a system that is totally symmetr ic--addi t ion is symmetric with multiplication and the Boolean operators are symmetric with the relational opera tors - -and easier to work with computationally. Schr6der's (1895) expansion and systematization of the calculus of relatives is based on Peirce's exposition of the calculus in this paper, and in the preface to their book formalizing set theory in the language of relation algebras, Tarski and Givant (1987) remark that the framework of the calculus of relations that Tarski employed is presented here in its final form.

5.1. Background in Linear Associative Algebras

In 1881, Peirce published a paper on "Associative algebras" ill The Amer- ican Journal of Mathematics as an addendum to his father's Linear Asso- ciative Algebras, which was published posthumously in the same issue. In this addendum, Peirce proved that there are only three linear associative division algebras, and reproved a theorem that he had first published in 1875, viz., that any associative algebra can be put into relative form

95

9 6 PEIRCE'S ALGEBRA OF RELATIVES

(i.e., has a matrix representat ion). His explanation of this theorem relating associative algebras and the algebra of relatives provides a useful background to his presentation of the calculus of relatives in 1883.

Given an associative algebra, Peirce shows how to construct its relative form in a series of steps (Peirce 1881b, pp. 171-172). First, given an associative algebra with letters i, .1, k, etc., and a multiplication table

i 2 = all i + bll j + cllk + ...

ij = al2i + bl2 j + c l 2 k + . . .

j i = a.21i + b21 j + c21k + ...

he introduces a number of new units A,/ , J, K, .... each corresponding to a letter of the original algebra, except for A. The new units can be multiplied by scalars and can be added, but they cannot be multiplied together. They are basis elements for the new algebra. Peirce calls them nonrelative units (Peirce 1881b, p. 171).

Next, Peirce introduces a number of new relative units, which he calls "operations," each formed by bracketing together two nonrelative units separated by a colon. A typical such operat ion is (I: j ) . He does not define what he means by an "operation"; in his 1870 paper he called A: B, B :A, A : A , and B : B "elementary relatives," where A and B are individuals in a given domain.

Peirce arranges these new operations in a matrix as follows:

(A:A) ( A : I ) (A: J ) ( A : K ) (I: A) (I: I) (I: j ) (I: K) (J: A) ( j : I) ( j : j ) ( j : K)

(Peirce 1881b, p. 171 ) Peirce remarks that the number of operations is equal to the square

of the number of nonrelative units, where the number of nonrelative units is assumed to be finite.

Peirce first gives a rule for multiplying the (I: j ) operations and nonrelative units:

Any one of these operations performed on a polynomial in nonrelative units, of which one term is a numerical multiple of the letter following the colon, gives the same multiple of the letter preceding the colon. Thus, (I:J)(aI+ bJ+ cK) =bI. (Peirce 1881b, p. 172)

Thus (I:J)J= I is relative multiplication. The multiplication of ( I : J ) ' s with each other is defined as:

These operations are also taken to be susceptible of associative combination. Hence (I:J)(J: K) = (I: K), for (J: K)K =J and (I:J)J= I,


so that (I:J)(J: K)K=I. And (I:J)(K: L) =0, for (K: L)L=K and ( I : j ) K = ( I : j ) ( o . j + K ) = 0 . I = 0 . (Peirce 1881b, p. 172)

97

Peirce's multiplication rule is

(I : J ) ( K : L) = ( I :L) i f J = K ,

0 otherwise,

i.e., relational composit ion, equivalently matrix multiplication. Peirce observes that the (I: j ) operations distribute over addition:

We further assume the application of the distributive principle to these operations; so that, for example, {(I :J) + (K :J) + (K : L)}(aJ+ bL) = aJ+ (a + b)K. (Peirce 1881b, p. 172)

After these observations, Peirce introduces the operat ions i', j ' , k', etc., cor responding to the letters of the original algebra and de te rmined by the multiplication table of the original algebra as

i '= (I: A) + al l ( l : I) + bll(J: I) + ...

+ a12(I: J ) + bl2(J: J ) + ...

j ' = ( j : A) + a2~(I: I ) + bzl(J: I) + ...

+ ~ 2 ( I : J ) + bz2(J:J) + ...

and then shows that the multiplication tables of the two algebras, i.e., that of the original associative algebra of i, j, k, etc., and that of the derived relative algebra of i', j ' , k', etc., are the same. Thus, given an algebra defined on n letters, we can find a subalgebra in n x n space that gives a representat ion of it. This proves Peirce's theorem, namely, that any linear associative algebra has a matrix representa t ion .

In modern terms, (I: j ) is an ordered pair of individuals from a domain. Forming a simple finite sum of them gives us an associative semigroup. Every element , collecting terms, is of the form: integer times (I: j ) plus ano ther integer times (I' : J ' ) , etc.; i.e., a l inear combina t ion of scalars times basis elements. When all the coefficients are 1 or 0, the terms that have coefficients 1 define the members of the relation. The same notat ion can be used for infinite linear combinations. We then get terms with all coefficients 1 or 0, represent ing arbitrary relations.

Peirce represents individuals I as pairs (I: I) , with a resulting confusion of notation. Product is relative product , so ( I : J ) J is really (I: j ) relative product with (J : j ) , which is (I: j ) . Since there is a distributive law, one obtains the identities Peirce gave. In summary, Peirce uses the associative infinite semigroup genera ted by individual o rdered pairs (I: j ) ; those that have nonzero coefficients all 1 represent relations. One can, in fact, allow the coefficients to be positive or negative

9 8 PEIRCE'S ALGEBRA OF RELATIVES

integers, p roduc ing freely g e n e r a t e d Abelian groups in the case of finite sums, and some th ing we do not usually name, infinite integral combinat ions of free gene ra t ing e l ement s (I: j ) , in the m o r e genera l case.

Peirce could go even fur ther and allow coefficients f rom any field; the

mult ipl icat ion, which is really relative mult ipl icat ion, then gives us l inear associative algebras. Coefficients that are ne i the r 0 no r 1, out of integers , c o m e up if such coefficients are allowed.

5.2. T h e Algebra of Rela t i ve s

5. 2.1. Types of Relatives

Peirce begins his p resen ta t ion of the a lgebra of relatives in his 1883

p a p e r by descr ibing the types of relatives, which are e i the r binary ("dual") or individual. He explains dual relatives in terms of o r d e r e d pairs:

A dual relative term, such as "lover, .... benefactor, .... servant," is a common name signifying a pair of objects. Of the two members of the pair, the determinate one is generally the first, and the other the second; so that if the order is reversed, the pair is not considered as remaining the same. (Peirce 1883c, p. 195) ~

The individual relatives are the pairs consisting of two individual objects, separa ted by a colon, taken f rom all the individuals A, B, C, D ..... in the universe. Peirce arranges all such individual relatives in a matrix:

(A:A) ( A : B ) ( A : C ) ... (B :A) (B :B) (B: C) ... (C:A) (C:B) (C: C) ...

(p. 195)

He then states what he means by a "general relative":

A general relative may be conceived as a logical aggregate of a number of such individual relatives. (p. 195)

The ques t ion is what is m e a n t by "aggregate." The p rob l em in t ranslat ing

to m o d e r n nota t ion is that we now dist inguish between an o r d e r e d pair

and a set the only e l e m e n t of which is an o r d e r e d pair. This dis t inct ion was not p resen t in Peirce 's writing.

i Except where otherwise noted, all subsequent page citations in this chapter will refer to "Note B" (1883), in the CoUected Papers of Charles Sanders Peirce (Hartshorne and Weiss 1933).


As an e x a m p l e of a genera l relative, Peirce takes l as d e n o t i n g the

dual relative te rm "lover"' and then gives the e q u a t i o n

l = E,E i (/)o (I: J )

as the def in i t ion of I. This equa t i on is critical to the system Peirce is ar t icula t ing in this p a p e r and requires some exp lana t ion . T h e ques t ion ,

in wri t ing this as a formal expl ica t ion of the te rm "aggregate" above, is that now the E sign needs to be expla ined , as well as the m e a n i n g of

the subscripts.

In his exp lana t ion of this equa t i on Peirce says:

(l)q is a numerical coefficient, whose value is 1 in case I is a lover of J, and 0 in the opposite case, and ... the sums are to be taken for all individuals in the universe. (p. 195)

In o t h e r words, l;i is 1 or 0 as the ijth pair (I: j ) of individuals is in the

re la t ion 1 or not. F rom our analysis of Peirce 's 1881 work on associative

algebras, we can say that Peirce is he re using the infini te associative

s e m i g r o u p g e n e r a t e d by individual o r d e r e d pairs (I : J ) . Those that have n o n z e r o coefficients (all 1) r ep re sen t relations. Thus , coefficients with subscripts ij are scalars, mul t ip lying basis e l emen t s of the algebra. H e r e the coefficients are in the Boolean a lgebra of two e lements .

Why does Peirce write i and j ins tead of I and J when they clearly c o r r e s p o n d perfectly? F rom many o t h e r contex ts in Peirce as well as

this one , the na tura l i n t e rp re t a t ion is as follows. All the individuals of

the d o m a i n c o m e listed in a def ini te o r d e r in a sequence . Thus, I is the

ith individual and J is the j th individual in the mas te r sequence . Why is i d i s t ingu ished f rom j? Simply because we mus t have a way of distin-

gu ish ing the first f rom the second m e m b e r of an o r d e r e d pair. T h e n we get a mat r ix with i as the row index and j as the c o l u m n index. T h e

ijth ent ry of the matr ix is the o r d e r e d pair (I: J ) consis t ing of the ith m e m b e r of the s equence (of the d o m a i n ) as the first m e m b e r and the j th m e m b e r of the s equence (of the d o m a i n ) as the s econd m e m b e r . 2

T h e equa t i on l = E;E i(/)O (I: J ) can be u n d e r s t o o d as de f in ing the

re la t ion l by a matr ix l o that tells which pairs (I: j ) of individuals are

in l. Alternatively, we can th ink of add i t ion as s u p r e m u m (un ion) in

the c o m p l e t e a tomic Boolean a lgebra of which the a toms are the or-

d e r e d pairs (I: j ) . T h e n the equa t i on def ines I as a s u p r e m u m of those

This is immediately meaningful only for finite or countable sets. Peirce was not at all precise about how he handled other sets. With Cantor's theory of ordinal numbers, however, we can well order the set by ordinal numbers into a (transfinite) sequence, and then Peirce's notation makes sense using (transfinite) matrices and is indeed a general notation. However, this is beyond the level of precision of the 1883 paper. One can alternatively not use well ordering and simply use index sets and indexed families as a basis for the discussion.

I 0 0 PEIRCE'S ALGEBRA OF RELATIVES

atoms (I: j ) such that the pair is in the relation. This last in terpreta t ion conforms very closely to Peirce's notation.

The modern definition of a relation as a set of o rdered pairs would replace the pair (I: j ) by the one-e lement set {(I: j )}. The Boolean algebra used would now be the power set 'P(A x B), the set of all sets of o rdered pairs with the first m e m b e r from A and the second m e m b e r from B. Then every relation as a set of ordered pairs is the union of its one-e lement subsets. Peirce did not clearly distinguish x from {x} (neither did Schr6der) , so one cannot tell which of the two Boolean algebras--with atoms (I: j ) or with atoms {(/ :J)}--is closer to Peirce, a l though they are, of course, isomorphic.

We remark that the calculus of relatives does not have a natural way of naming individuals other than this use of subscripts and corresponding capital letters. Whenever Peirce tries to talk about individuals by in t roducing absolute terms, the translation to relatives is very artificial.

5. 2. 2. Opera t ions on Re la t i ves

The s tandard modern approach to Boolean algebra is to treat "and," "or," and "not" as primitive notions. In his t rea tment of the Boolean operat ions in the calculus of relatives, Peirce does not do that. Instead, he treats numerical addition and multiplication as primitive notions, and then "and" and "or" are defined notions that he brings into his mathemat ical system by defining them in terms of ordinary addit ion and multiplication:

Relative terms can be aggregated and compounded. Using + for the sign of logical aggregation, and the comma for the sign of logical composition (Boole's multiplication, here to be called non-relative or internal multiplication) we have the definitions:

(l + b)~j - (l)q + (b)q

(/, b)ij = (l)q x (b)i j.

(p. 196)

The operat ions denoted by the plus sign and comma on the left-hand side of the equations are Boolean operators between relative terms, def ined in terms of arithmetical addition and multiplication, respectively, on the right. The coefficients (l)o and (b)o are numbers , ei ther 0 or 1. The addition, however, is the Boolean addit ion 0 + 0 = 0, 0 + 1 = 1 + 0 = 1 + 1 = 1 .

FROM PEIRCE TO SKOLEM XO1

Peirce says that, instead of (l)ij + (b)ij, addi t ion migh t be writ ten m o r e accurately as the two-level exponen t ia l :~

0 0 ( ,9 q + (l,) q.

The exponen t i a l 0 x is Peirce 's negat ion and works on all integers , so

using it twice is jus t the identity, where n > 1 is ident i f ied with 1. Tha t is, 0x= 1 for x =0 and 0x= 0 for x #: 0, so 0~ 0 for x =0 and 0~ 1

for x ~ 0. Thus, the Boolean "or" of a + b, taking the s t andard representa t ion of t ruth as 1 and falsity as 0, can be r e p r e s e n t e d as 0 ~

using the usual ar i thmet ica l defini t ions of addi t ion and exponen t i a t ion . W h e n Peirce says that 0 ~176 is m o r e accura te than (l)o + (b)o, he

robably means that, since 0 x normal izes every value of x to e i ther 0 or

, 0 ~ ( z) '' +'( ') " can r ep re sen t Boolean "or" wi thout assuming the nonar i thmetical rule 1 + 1 = 1.

Peirce 's def ini t ion of mult ipl icat ion is as a pointwise, scalar multiplication, the inne r product . He calls this p roduc t " internal mul t ipl icat ion," a t e rm that he first used technically in c o m m e n t i n g on Grassmann ' s

two mult ipl icat ions, in ternal and external , in 1877, in a br ie f p a p e r on Grassmann ' s vector calculus (Peirce 1877, p. 102). Grassmann ' s in ternal

and external mul t ip l icat ion are our familiar i nne r (scalar) and ou te r (exter ior) p r o d u c t in today's ex ter ior calculus. In his 1877 paper, Peirce viewed them as two distinct mult ipl icat ions of qua t e rn ions and wrote

Grassmann ' s a lgebra in his own system of l inear associative relative algebra (Peirce 1877, p. 103).

Peirce lists th ree principal formulas for nonrela t ive addi t ion and mul- t iplication and their converses:

I f l - - < s a n d b - < s , then l + b - - < s .

If s--< l and s - < b, t hen s - < l , b.

I f l + b - - < s , then l - < s a n d b - < s .

If s - < l, b, t hen s--< l and s--< b.

( l+ b ) , s - - < l , s + b , s .

( l+ s ) , (b + s) --< l , b + s.

(p. 196)

If we view --< as be ing a partial order, these formulas are the axioms

for a distributive lattice. In m o d e r n language, Peirce is saying that relations with these two opera t ions form a distributive lattice.

Peirce defines the relative p roduc t lb and its dual l t b, the "relative sum, as

s In the Harvard edition, this formula contains a typographical error. It appeared as

0 0 (t) q+ (b) q.

1 0 2 PEIRCE'S ALGEBRA OF RELATIVES

(lb)o = F,,, (l),,,(b),, j,

(l t b) 0 = II~{(/),~ + (b)~#}.

As in the case of nonre la t ive mul t ip l ica t ion and addi t ion , the logical ope ra t ions on the lef t-hand side of the equa t ions are de f ined in terms of ar i thmet ica l addi t ion and mul t ip l ica t ion on the right, where (l)~ x and (b)xi are e i the r 0 or 1.

Peirce recognizes explicitly that relative sum and p r o d u c t i n c o r p o r a t e a species o f universal and existential quant i f icat ion:

We now come to the combination of relatives. Of these, we denote two by special symbols; namely, we write

lb for the lover of a benefactor

and

l t b for lover of everything but benefactors.

The former is called a particular combination, because it implies the existence of something loved by its relate and a benefactor of its correlate. The second combination is said to be universal, because it implies the non-existence of anything except what is either loved by its relate or a benefactor of its correlate. (pp. 196-197)

Both are order -preserv ing opera t ions ,

if 1--< s, t hen l b - - < s b and 1 t b--< s t b;

and if b ---< s, t hen lb --< Is and l t b --< s.

(p. 197) The defini t ion of relative p roduc t is very familiar to a m o d e r n reader:

(lb)o = E,, (l),,,(b)~).

This is the def in i t ion of matr ix mult ipl icat ion. In his 1870 p a p e r Peirce viewed relative p r o d u c t as the natura l mul t ip l ica t ion in his system and d i s tanced h imsel f somewha t f rom the Boolean p roduc t . F rom a matrix- theore t ic po in t of view, this makes sense: the natura l ope ra t ions for the calculus of relatives, viewed as matr ix theory, are (1) relative p r o d u c t because it is matr ix mult ipl icat ion, (2) Boolean addi t ion because it is pointwise addi t ion of coo rd ina t e values, and (3) converse because it is mat r ix t ransposi t ion. It appears that we are seeing Boolean matr ix theory in Peirce 's 1883 paper. Peirce would have r ecogn ized (lb) O = I2,,(l)ix(b), 0 as matr ix mul t ip l ica t ion that will be e i ther 0 or 1.

Relative p r o d u c t and sum obey the associative law:

FROM P E I R C E T O SKOLEM lo3

I t (bt s ) = ( l t b) t s,

l(bs) = (lb)s.

Relative product does not distribute over relative sum, but Peirce gives two formulas in lieu of distributive laws:

l(b t s) -< lb t s,

(1 t b )s -< l t bs.

(p. 197) For every relative, Peirce has a negative, "which may be represented

by drawing a straight line over the sign for the relative itself' (p. 195). The negative includes every pair that the relative itself excludes. In addition, every relative has a converse, "produced by reversing the order of the members of the pair" (p. 195). Peirce's notat ion for converse is a curved line over the sign for the relative.

Peirce observes that De Morgan's laws hold with respect to both relative and nonrelative addition and multiplication,

l + b = [ , D , l , b = [ + D ,

I t b=[b, l b = [ t D

(p. 198), and he gives analogous laws for the converse:

g,

l t"~=/~ t l, l"s = b'/'.

(p. ~98) Peirce gives some interesting formulas connect ing relative and logical

product (and the dual cases)"

There are a number of curious development formulae. Such are

(l, b)s=IIt,{l(s, p) + b(s,/b)},

l(b, s) =IIt,{(l, p)b + (/,i0)s},

(l + b) t s = ~,t, l[/ t ( s+p ) ] , [ b t (s+fi]},

I t (b+ s) =I~t,l[(/+ p) t b], [(/+ i0) t s]l.

The summations and multiplications denoted by • and II are to be taken non-relatively, and all relative terms are to be successively substituted for p. (p. 198)

These formulas involve quantification over all relatives. Schr6der treats

~o4 PEIRCE'S ALGEBRA OF RELATIVES

t hem as t heo rems and uses them in developing second-o rde r quantification reduc t ion rules (Schr6der 1895, pp. 497ff). The first versions of these formulas a p p e a r e d in the last chap te r of Peirce (1880).

5.3. Syllogistic in the Relative Calculus

T h e next part of Peirce 's 1883 pape r (pp. 202-205) is devoted to demons t ra t ing that syllogisms can be deal t with entirely in relative calculus equat ions. Peirce provides four normal forms (in the a lgebra of rela-

tives) for universal proposi t ions, four normal forms for par t icular propositions, forms for six forms of affirmative premises occur r ing in the syllogism, and reduc t ion schemes for 21 forms of the Aristotel ian syl-

logism. The six forms of affirmative premises, he says, "are the propositions of the first o rde r re fe r red to in Note A." Note A itself is an a r g u m e n t for having quantif iers range over specific domains , that is, for i n t roduc ing the not ion of a s t ructure or a model . We here list the pr incipal points of Note A, with a br ief commentary .

First, citing Boole and De Morgan, Peirce out l ines various features of the universe of discourse:

Boole, De Morgan, and their followers, frequently speak of a "limited universe of discourse" in logic. An unlimited universe would comprise the whole realm of the logically possible. In such a universe, every universal proposition, not tautologous, is false; every particular proposition, not absurd, is true. Our discourse seldom relates to this universe; we are either thinking of the physically possible, or of the historically existent, or of the world of some romance, or of some other limited universe. (Peirce 1883b, p. 182)

The un l imi ted universe was that on which Frege based the semant ics

of his Begriffsschrift (1879). In addi t ion, Peirce points out that

Besides its universe of objects, our discourse also refers to a universe of characters. Thus, we might say that virtue and an orange have nothing in common. It is true that the English word for each is spelt with six letters, but this is not one of the marks of the universe of our discourse. (Peirce 1883b, p. 182)

The passage that follows is difficult to make out:

A universe of things is unlimited in which every combination of characters, short of the whole universe of characters, occurs in some object. (Peirce 1883b, p. 182)

FROM P E I R C E T O SKOLEM 1 0 5

This assertion is reminiscent of the power set axiom (the universe is unl imi ted if there exist always larger objects conta in ing all subsets, which might be called a combina t ion of characters if we cons ider syntactical formulas describing the subset). Peirce's exclusion of the whole universe of characters seems to point to some awareness of t rouble in allowing arbitrarily large sets of which one takes subsets. He says "short of the whole universe of characters," mean ing "not too large to get into trouble." It would be very remarkable if Peirce would have had a first

glimpse of the troubles of set theory back in 1882. On the o ther hand, if we in terpre t "character" as an atomic predicate

of individuals of the domain and "combinat ion" as finite conjunct ion , Peirce's s ta tement would say that every finite conjunc t ion of atomic predicates is satisfiable in the domain. "Unlimited" would have the con- nota t ion that whatever finite set of atomic predicates one gives, there is someth ing in the domain that possesses that set of propert ies . This is a constraint on the domain. Only nonempty atomic predicates would be allowed as characters, and every finite set of them would be satisfied by someth ing in the domain. (If infinite conjunct ions were allowed, we would get a primitive form of saturation.) In the above in terpre ta t ion , we concede that we have no idea what the phrase "short of the universe" means to Peirce technically.

Peirce continues:

In like manner, the universe of characters is unlimited in case every aggregate of thillgs short of the whole universe of things possesses in common one of the characters of the universe of characters. (Peirce 1883b, p. 182)

This seems to be a dual r equ i r emen t on the domain of predicates (characters). Here is a possible in terpreta t ion for finite domains. For any set of individuals from the domain , there is a predicate (character) that applies to the e lements of that finite set and to no o the r e lements of the domain. For infinite domains, our in terpre ta t ion would require that there be uncountab ly many characters. Peirce jus t does not say what he means.

In summary, in "Note A" Peirce observes that when we start applying logic, we are usually not applying it to everything. Instead, we begin with a domain of individuals in which we are interested, a specific universe. For each such domain , he argues that we use a specific set of predicates (characters). He goes on to argue that the predicates we use are built up from a class of basic predicates by a s tandard collection of

operat ions, Boolean and otherwise. This is a hint of the m o d e r n not ion of a s ignature of a model ; the

s ignature tells what domain of predicates is allowed, and the mode l tells

l o6 PEIRCE'S ALGEBRA OF RELATIVES

what d o m a i n of individuals is allowed. Peirce can thus be seen as posi t ing a weak p recu r so r of the no t ion of model .

5.4. Prenex Predicate Calculus

At the close of his 1883 p a p e r (pp. 205-206) , Peirce shows that Mitchell 's genera l iza t ion , which allows two-dimensional p ropos i t ions of a special sort ( that is, with two quantif iers , but writ ten in Mitchell 's fo rm) , can be wri t ten and der ived in the calculus of relatives. Peirce makes no such claim of representabi l i ty for m o r e genera l p ropos i t ions (since L6wen- he im later shows such claims not to be cor rec t in 1915, this is jus t as well), bu t he allows, in addi t ion to relative opera t ions within express ions , quant i f iers in front, of a very m o d e r n sort:

When the relative and non-relative operations occur together, the rules of the calculus become pretty complicated. In these cases, as well as in such as involve plural relations (subsisting between three or more objects), it is often advantageous to recur to the numerical coefficients mentioned. Any proposition whatever is equivalent to saying that some complexus of aggregates and products of such numerical coefficients is greater than zero. Thus,

I~;Ei/0> 0

means something is a lover of something; and

IIiHjlij> O

means that everything is a lover of something. (p. 206)

At this point , Peirce 's usage is such that variables occur explicitly in the subscripts; these are real m o d e r n quant i f iers appl ied to l o, that is, he is wri t ing (3i)(3j)l(i,j) and (Vi)(3j)l(i , j) . But he makes it c lear that l 0 is r e g a r d e d as a p ropos i t iona l func t ion r a the r than a formula , and that I~ and II are r ega rded as sums and products , respectively, of p ropos i t iona l func t ions that at any a r g u m e n t are s u m m e d (or mul t ip l ied) in the two-

e l e m e n t set. Tha t is, II~Ejl 0 is 1 (i.e., > 0) if and only if the two-ary func t ion l o, s u m m e d first over j to get a one-ary func t ion in i, t hen with a p r o d u c t o f these over all i, is 1.

W h a t we see here is the in t roduc t ion of quant i f iers as ope ra t ions on p ropos i t iona l funct ions of i, j f rom a domain , no t formal l anguage quant i f iers on formal expressions. T h e n , leaving ou t the = 1 (or equivalently, > 0), Peirce writes I]~I]i/o and I I ~ j / o as proposi t ions , which makes t h e m look and act like formulas , bu t in fact they are abbrevia t ions for an algebraic equality equal to 1:


We shall, however, naturally omit, in writing the inequalities, the > 0 which terminates them all; and the above two propositions will appear as

E;Ei l 0 and II;Ej lq.

(p. 207)

xo7

Peirce gives several examples of p ropos i t ions in this system: IIiF., i (1)o(b) q means that everyth ing is at once a lover a nd a b e n e f a c t o r o f someth ing ; I I iE i (l)q(b)j~ means that everyth ing is a lover of a benefactor o f itself; r, iF.,kIIi(l q + bik ) means that the re is s o m e t h i n g which stands to s o m e t h i n g in the relat ion of loving every th ing excep t benefactors of it. This last express ion, (lq + bik), is an ind i rec t way of intro-

duc ing a new, triple relative, tii k = lit + bik. His r e m a i n i n g examples all deal explicitly with triple relatives.

Thus , by 1883, Peirce u n d e r s t o o d the genera l no t ion of a p r e n e x formula . He u n d e r s t o o d the no t ion of a quant i f ie r r ang ing over the universe and the no t ion of the domain , and he u n d e r s t o o d how to rewrite a p r e n e x fo rmula as a p ropos i t iona l f o rmu la w h e n a cons t an t is i n t r o d u c e d as a n a m e for each and every e l e m e n t a quan t i f i e r ranges over in a specific in te rp re ta t ion . Existential quant i f ica t ion over i cor- r e sponds to call ing the e l emen t s over which it ranges by individual names i l, i 2, ..., e l imina t ing the quantif ier , subs t i tu t ing in succession each of i l, iz . . . . for i everywhere in the quant i f ier- f ree part , and taking the d is junct ion of these formulas , each of which is p r e n e x but has one less quant i f ie r at the front. If the leading quant i f i e r is H i , then , n a m i n g the e l emen t s of the d o m a i n over which j ranges as j l , j z , . . . , we omi t the universal quant i f i e r in each term, subst i tute each j~ for all the j ' s in the quant i f ier-f ree part, and replace by the con junc t ion . Since the re are a finite n u m b e r of quant i f iers in the prefix, we finally e n d up with a nes ted set of in f imums and s u p r e m u m s over the d o m a i n , app l ied in the e n d to proposit ional logic s ta tements , since all variables have b e e n rep laced by constants i k, jr, and so on. Thus , the validity of this proposi t ional logic s t a t emen t in this d o m a i n can be ques t i oned , bu t the s t a t emen t is infinitary. At this po in t Peirce uses the rules o f Boole ' s a lgebra, e x t e n d e d to infinitary dis junct ions and conjunc t ions :

When we have a number of premises expressed in this manner, the conclusion is readily deduced by the use of the following simple rules. In the first place, we have

E;Hj-< HIE;.

In the second place we have the formulae

xo8 PEIRCE'S ALGEBRA OF RELATIVES

{II;~(i)} {IIj~(j)} = II, l~b(i) �9 ~(i)}.

{II;~b(i)}lEj~b(j)} --<E;lth(i) �9 ~(i)}.

In the third place, since the numerical coefficients are all either zero or unity, the Boolian calculus is applicable to them. (p. 208)

If we work in one domain only and in te rpre t every quant i f ie r as e i ther a least u p p e r b o u n d or a greatest lower b o u n d (infinite dis junct ion of instances over the domain or infinite con junc t ion over the doma in ) , then every first-order s t a tement becomes an infinitary proposi t ional logic s t a t emen t (an infinitary Boolean a lgebra s ta tement ) . T h e n all logic deduct ions b e c o m e calculations using ord inary Boolean a lgebra rules plus the infinite distributive law. The re is also the infinite De Morgan ' s law; then every deduc t ion can be done by Boolean calculations, but infinitary ones. In fact, as Mitchell said, the calculations can be done by cancel ing conjunct ive hypotheses, add ing terms to disjunctive hypotheses, and then cancel ing terms after distributing.

Peirce concludes his pape r with an example of a deduc t ion in this system. The p rob lem he sets for h imself is to e l iminate the relative "servant" f rom the following two premises:

There is somebodywho accuses everybody to everybody, unless the latter is loved by some person that is servant of all not accused to him.

There are two persons, the first of whom excuses everybody to everybody, unless the unexcused be benefitted by, without the person to whom he is unexcused being a servant of, the second. (p. 208)

Peirce formalizes these s ta tements as

where ot denotes the triple relative "accuser t o - o f - , " E deno tes the triple relative "excuser to - of," and l, b, and s denote , as before, the binary relatives "lover o f - , .... benefac to r o f - , " and "servant o f - . "

First, Peirce claims that "the second yields the immed ia t e inference , l ' Ix~], ,I ' Iy~], ,(e.y x + ~,~b~)" (p. 208). This follows f rom the second premiss by three applicat ions of Peirce 's first rule, E i I I j -< IIjE;.

Peirce then says, "combin ing this with the first, we have ExE,,EyE~(e,,y~ + g~b,,x)(Ctx, ~ + sy~ly,)" (p. 208). This follows by applying Peirce 's second rule to the p roduc t of the two premises, identifying h with x, u with i, y with j, and k with v.

Lastly, appea l ing to the Boolean calculus, Peirce states:


Finally, applying the Boolian calculus, we deduce the desired conclusion,

x ~ u~y~ , , (6"uyx Ol ..... -It- e,,y,,/y,, + a .... b,, x).

The interpretation of this is that either there is somebody excused by a person to whom he accuses somebody, or somebody excuses somebody to his (the excuser's) lover, or somebody accuses his own benefactor. (p. 208)

lo9

T h a t is, d r o p p i n g the subscripts and ignor ing the I;'s, ExI],,I;yE,,(e,,y x + ~.,,b,,,,)(a .... + sy,,ly,,) reads

(e + ib)(o~ + sl).

Since in the Boolean calculus "and" and "or" distr ibute over one another , we can mult iply this expression out to obtain

ec~ + e sl + gbc~ + gbsl.

T h e te rm gbsl drops out, and we can replace esl by el and gba by bc~ to get

ec~ + el + bc~,

which is the result that Peirce obtains. In summary, the conc lud ing mater ia l of Peirce 's p a p e r is a p recur so r

of p r enex first-order predica te calculus, but wi thout func t ion symbols, constants, or equality. Peirce in terprets logical opera t ions in propositions as ar i thmet ical opera t ions in his semantics of proposi t ions . Thus, a r i thmet ic is the basis for in te rpre t ing proposi t ions. He also sees l inear a lgebra in the semantics of logic th rough Boolean matrices. This can be viewed as a reduct ionism, an algebrat izat ion and a r i thmet iza t ion of the semant ics of logic. This same kind of r educ t ion i sm was p resen t in Peirce 's fa ther ' s work. Peirce senior ident if ied complex n u m b e r s with matr ices using an e m b e d d i n g in 2 x 2 real matrices. Following in those footsteps, the younge r Peirce tried to build an in te rp re ta t ion of logic based on l inear a lgebra and ar i thmetic .

5.5. Summary of Peirce's Accomplishments in 1883

5.5 .1 . S y n t a x a n d Semant ics

In his 1883 pape r Peirce discovered the syntax of p r e n e x formulas of p red ica te logic. However, the m o d e r n genera l inductive def ini t ion of ( n o n p r e n e x ) formulas is absent. Instead, he gives an informal semant ics of p r enex formulas, based on an ar i thmet ical in t e rp re ta t ion of the log-

ical operat ions .


Peirce's purpose is to define a vocabulary for a language of the calculus of relatives. This was a big step. Previously, Peirce had mainly been interested in exploring identities and implications between terms built up of operat ion symbols, but in 1883 he makes a commi tmen t to spell out what the calculus of relatives is and how one puts things together in the system.

5.5. 2. Quantifiers

Quantif ier theory begins with Aristotle's syllogistic, and modern set theory and propositional theory begin with Boole. Boole's theory of quantifiers was practically nonexistent. Peirce's original theory used only relative product to capture (some) instances of quantification; we believe he thought that relative product was more expressively powerful than in fact it is. In his 1883 paper, Peirce recognizes that the universal and existential quantifiers have a semantics as sums and products over

the domain. In this paper Peirce interprets E and II arithmetically. He introduces

the quantifiers as operations on propositional functions of i, j f rom a domain, and not as formal language quantifiers on formal expressions. It is somewhat difficult to describe the difference between the two. Omit t ing the sign > 0, he writes II~Eil o as a proposition, which makes it behave like a formula, but in fact it is an abbreviation for an algebraic equality that is equal to 1. Of course, we do the same thing, typically when we write 4~, which is shor thand for the assertion "4~ is true"; the convention is there no less, just in different language.

This brings us back to the question of what Peirce's formulas mean:

l = ~ , ~ ( l ) , j ( I . j ) ,

(t + b) o = (t),~ + (b),j,

( t ,b) o = (l),j x (b),j.

They are best described as definitions of the Boolean matrix operat ions on the left-hand side in terms of the ijth entry on the right-hand side, where we are in the algebra of truth values 0, 1, and we use Peirce's interpretat ion of plus as inclusive "or" or as the least upper bound in the two-element distributive lattice of truth values. This can also be regarded as an operat ion on truth functions of i,j. This is quite a mode rn flavor, considering that Peirce has already laid out distributive lattices more or less axiomatically in the paper.

Similarly,

~i~jlii > 0

F R O M P E I R C E T O S K O L E M 1 1 1

says that the Boolean sum is nonzero, and hence is equal to 1 in the two-valued interpretat ion. At several points, including here, Peirce waf- fles on whether the operations are ari thmetic or Boolean.

The formulas on the very last pages of the paper, for example,

II, Ej(l),i(b),j, I I ,~ i (l),j(b)j,, F,,F,,IIj (lq + bjk ) ,

also look mode rn in form. We note that for these Peirce omits his j u d g m e n t sign, i.e., the > 0 sign.

In sum, what Peirce presents is part of the semantics of first-order logic for p renex formulas over the domain over which i and j range. The i and j act like individual variables, being indices ranging over a domain. The E and H mirror existential and universal quantifiers, being the cor responding least upper bound and greatest lower bound over indices over a domain. Since this is done uniformly for all domains, we almost have the Tarski semantics of quantifiers for first-order logic, but the inductive definition of formula and the inductive definit ion of truth value for a formula are not there.

Perhaps we should regard Peirce's 1883 paper as addressing not syntactical p renex predicate logic, but ra ther quantifiers as infinitary least upper bound and greatest lower bound operat ions on proposit ional functions.

5.6. Peirce's Appraisal of His Algebra of Binary Relatives

In his 1896 review of Schr6der 's Die Algebra der Logik and in an unpublished lecture from the same year, Peirce compared his algebra of binary relatives unfavorably to his "general algebra of relatives," i.e., the quantifier logic that he presents in his 1885 paper. In 1896 Peirce points out clearly that in the algebra of binary relatives one has complex and many identities, the representat ion of which for purposes of reasoning is neither easy to read nor easy to use. Peirce contrasts that with his general algebra of relatives, in which the quantifiers as sums and products are present, but where both the needed axioms and the reasoning are easier. He is unaware that, as Korselt showed later, the calculus of binary relatives is actually less expressive than the quantif ier calculus. Here is Peirce's own evaluation, in 1896, of his 1883 work:

Besides the general algebra of relatives in which a proposition is expressed by a "Boolian," or expression in the algebra of Boole with aggregation substituted for addition, and with indices to distinguish the individual cases, this "Boolian" being preceeded by a series of "quantifiers," or signs of serial products and times, I invented and published in a note in 1883, a special algebra for dual relations. It has the merit of dispensing with the indices and quantifiers, and thus


giving formulas relatively easy to write and to read; but its demerits are such that I never had a great liking for it. It consists essentially in introducing two "relative operations," in addition to non-relative multiplication and addition, and also two semi-logical special relatives, signifying "identical with -" and "other than -." The relative operations are called relative multiplication and relative addition. I indicate relative multiplication by writing the factors together, without an inter- vening sign. If l;j be taken to mean the individual i is a lover of j, and if bj~ means the individual j is a lover of k, then by (/b)ik I mean Ejlij" bjk, that is, i is a lover of some benefactor of k. I indicate relative addition by a dagger, to which I give a scorpion-tail curve in a cursive form. Then (l t b)ik means Ilil O v bjk, or /is a lover of everything but benefactors of k. For "identical with," I write l, only giving it the distinctive form of a heavy vertical line. For "other than," I write this with the obelus, and thus got a i. The objections to this algebra are as follows. First, it has four operations where two would suffice, which greatly complicates it. Second, it expresses plural relations and various other characters in most cumbrous form. Third, many of those relations which it expresses readily, it expresses in many different ways, so that there is a whole book-ful of equivalences of forms, mere for- malities. Fourth, it requires the constant introduction of 0, 00, l, i, in complicated ways, the meaning of which is far from evident, and load- ing the user of the algebra down with a great fardel of meaningless formulae. From all these objections the general algebra of relatives is free. (Peirce 1896, pp. 7-8)

So far, we have l e a r n e d that relatives, i n t r o d u c e d first by De M o r g a n (no t Boole) , were u n d e r s t o o d by Peirce, with relative p r o d u c t as the

basic a lgebra ic ope ra t ion . He u n d e r s t o o d that t he re was a l inear asso-

ciative a lgebra with o r d e r e d pairs as a basis, a n d tha t re la t ions h a d r e p r e s e n t a t i o n s as sums of these pairs, with coeff icients 0, 1. H e pu t

syllogisms into relative calculus and , fol lowing Mitchell , a d o p t e d quan-

tifiers with sum a n d p r o d u c t no ta t ions , bo th as an add i t i on to relative

calculus a n d as s o m e t h i n g to invest igate on its own. As we will see in

o u r analysis of Peirce 's s u b s e q u e n t paper , he m u c h i m p r o v e d Mitchel l ' s

n o t a t i o n by pu t t i ng the sums a n d p roduc t s all in f ron t in the style o f

calculus a n d infini te series, r a t he r than as subscripts. In o t h e r words,

Pei rce is now poised to deve lop what we would call a calculus of p r e n e x

formulas .

6. Peirce's Logic of Quantifiers" 1885

Introduction

Peirce 's p a p e r "On the a lgebra of logic: A con t r i bu t ion to the ph i lo sophy

of no ta t ion" was writ ten in the s u m m e r of 1884, while Peirce still he ld

an academic posi t ion at the J o h n s Hopkins University. It was pub l i shed

in the American Journal of Mathematics and was to be Pei rce ' s last technical p a p e r on logic to a p p e a r in a major scientific j ou rna l . The final two

sections of this paper , "First- intentional logic of relatives" and "Second- in ten t iona l logic," p resen t a p r e n e x form of first- and s e c o n d - o r d e r p red ica te logic, which is, in re t rospect , one of his mos t i m p o r t a n t sci-

entific achievements .

6.1. On the Derivation of Logic from Algebra

In the first sect ion of his 1885 paper, Peirce makes a cur ious and in-

teres t ing observa t ion abou t the na tu re of mathemat ics , which is an an- t e c e d e n t to some very inf luent ial papers of the m i d - 2 0 t h century, in particular, E u g e n e Wigner ' s "The u n r e a s o n a b l e effectiveness of mathematics." Peirce 's a r g u m e n t does no t deal with exactly the same ques t ion

as Wigner ' s , but parallels it:

It has long been a puzzle how it could be that, on the one hand, mathematics is purely deductive in its nature, and draws its conclusions apodictically, while on the other hand, it presents as rich and apparently unending a series of surprising discoveries as any observational science. Various have been the attempts to solve the paradox by breaking down one or other of these assertions, but without success. The truth, however, appears to be that all deductive reasoning, even simple syllogism, involves an element of observation; namely, deduction consists in constructing an icon or diagram the relations of whose parts shall present a complete analogy, with those of the parts of the object

II 3

114 PEIRCE'S LOGIC OF QUANTIFIERS

of reasoning, of experimenting upon this image in the imagination, and of observing the result so as to discover unnoticed and hidden relations among the parts. (Peirce 1885, pp. 212-213) ~

Peirce observes here that mathematical discovery is based on empirical generalization, which then must be backed up with deductive proof. One might naively assume, then, that the apparently much larger experimental repertoire of the physicist would be reflected in a much more interesting, more vibrant, more highly developed logical field. Peirce says, however, that is not what we observe; that in fact, even though mathematicians ' tools for gaining empirical knowledge about mathematics are limited to paper and pencil, employing geometric images, and nowadays to examining the results of computer experiments, mathematical theories appear to transcend their limited repertoire of experimental methods.

If one asks the question, "what tools do physicists have by which they gain knowledge about their subject?", the answer is that they can use sensing devices to make measurements and perform experiments, and can then use these results to frame laws and deduce their empirical consequences. These laws are in turn refined and justified by further observations and experiments. Thus, Newton's deduct ion of the motions of the planets was based on laws obtained by generalizing Galileo's observations of falling bodies and cannonball trajectories, and on Kep- ler's laws. Further experiments, such as those of Michaelson-Morley, led in turn to the refinements of Einstein, which were then verified by further experiments.

Peirce argues that the mathematician uses the same basic experimental method as the physicist. In formulating an explanation, the mathematician constructs examples and counterexamples, and gains new intuitions and insights. Possible theorems in mathematics are created by generalizing the observed relations between the parts of examples and counterexamples, which are themselves, as concrete objects of intuition, comparable to" the experiments of the physicist. If attempts to prove a proposed theorem fail, the mathematician looks to fur ther experiments, that is, constructs yet more elaborate examples and counterexamples, until the proposed theorem is refuted or proved on the basis of known theorems. This may take 300 years, as with Fermat 's conjecture. But the mathematician is ever hopeful. Of course, thanks to G6del 's incompleteness theorem, we now realize, as Peirce did not, that the mathematician 's work will never be done. There will always be mathematical truths our deductive methods cannot reach, however

Except where otherwise noted, all page numbers given in this chapter refer to Peirce (1885). in tile Collected Papers ~ Charles S. Peuce'(1933).


many expe r imen t s we make in the way of cons t ruc t ing examples and coun te rexamples . As for physics, will it ever be d o n e ei ther?

Peirce then states three objectives for his 1885 paper :

In this paper, I propose to develop an algebra adequate to the treatment of all problems of deductive logic, showing as I proceed what kinds of signs have necessarily to be employed at each stage of the development. I shall thus attain three objects. The first is the extension of the power of logical algebra over the whole of its proper realm. The second is the illustration of principles which underlie all algebraic notation. The third is the enumeration of the essentially different kinds of necessary inference; for when the notation which suffices for exhibiting one inference is found inadequate for explaining another, it is clear that the latter involves an inferential element not present to tile former. (p. 213)

The first objective is to ex tend logical a lgebra over the whole of its realm (logic). To Peirce, "logical algebra" m e a n t Boole 's a lgebra of logic. Since he had already ex t ended logic to include the rea lm of relatives, and Mitchell had e x t e n d e d it to conta in p r e n e x quantif iers , we can safely assume that Peirce here in tends to ex tend logical a lgebra to the realm of relative a lgebra plus quantifiers.

The second objective is to illustrate "the principles which under l ie all algebraic notat ion." Since in his 1870 pape r Peirce had already stated the laws that he felt should hold before an addi t ion or mul t ip l ica t ion symbol could be used for an opera t ion , this second objective can be viewed, in m o d e r n terms, as expla in ing the abstractness of axioms and what a mode l of abstract opera t ion symbols with axioms is (i.e., what an a lgebra is). This follows Peirce 's fa ther ' s abstract a p p r o a c h to l inear associative algebra.

The third objective is to make clear what rules of in fe rence are used for deduc ing consequences f rom premises, in algebraic terms if possible. It is here that we would expec t the r e a p p e a r a n c e of the not ion of a prepar t ia l o rde r f rom Peirce 's 1880 paper, in which the deduc t ion rules allowed him to have a transitive closure such that b deduc ib le f rom a is a prepar t ia l o rde r a_< b.

T h e r e is here an uns ta ted t heme that, if premises are writ ten algebraically and there is a desired conclusion, then algebraic t ransformations of the premises using a lgebra laws plus the in fe rence partial o rde r (i.e., a _< b if and only if a implies b) will lead to the des i red conclus ion if the latter is a consequence of the premises. This idea surely goes back to Leibniz, who m a d e an explicit proposal for a five-year team project to comple te ly formalize all logical deduct ive reason ing for use as a p roo f tool.


In this paper Peirce tackles I.eibniz's p rob lem, cons t ruc t ing an a lgebra of logic ex tend ing Boole, using addi t ional opera tors and algebraic laws and covering relative algebra and quantifiers. His poin t of depa r tu r e is algebra, and his approach will be to show that logical ideas can be e n c o d e d within algebra, a l though on those occasions where this does not work out in any simple or natural way, he will a u g m e n t the a lgebra in o r d e r to be able to a c c o m m o d a t e his unde r s t and ing of some logical construct . The result of this process will be, he hopes, a general iza t ion of a lgebra and logic that strongly resembles ordinary algebra.

6.2. N o n r e l a t i v e Log ic

6.2.1. Embedding Boolean Algebra in Ordinary Algebra

Algebra to Peirce was primari ly computa t iona l , and the idea of mak ing logic algebraic m e a n t to him conver t ing logical a rgumen t s into a computa t ional system. He thus begins with proposi t ions, and tries to e m b e d the Boolean algebra of proposi t ions into ord inary algebra, e x t e n d e d to inc lude two constants, r ep resen ted by v and f. He states:

According to ordinary logic, a proposition is either true or false, and no further distinction is recognized. This is the descriptive conception, as the geometers say; the metric conception would be that every proposition is more or less false, and that the question is one of amount. At present we adopt the former view. (p. 214)

This asserts that only two t ruth values will be allowed for t ruth functions, even though ar i thmetical opera t ion symbols are used in terms.

Peirce gives a very simple system:

Let propositions be represented by quantities. Let v and f be two constant values, and let the value of the quantity representing a proposition be v if the proposition is true and be f if the proposition is false. Thus, x being a proposition, the fact that x is either true or false is written

( x - f ) ( v - x ) = 0 .

(p. 214)

Peirce is cons ider ing the roots of equations; in this system, 0 is t rue and nonze ro is false; p roduc t is inclusive "or," and sum is "and." Thus ( x - f ) ( v - x) - 0 is the s t a tement that x is e i ther true or false, i.e., the law of the exc luded middle. Peirce is here descr ibing a funct ion f rom logical formulas to algebraic formulas, def ined by [x] ~ ( v - x ) , [ :~] ~ ( x - f), and [x A y] = [x] + [y] and [x v y] = [x][y].

FROM P E I R C E T O SKOLEM ax7

Peirce translates the logical s tatement "x implies y" into an algebraic formula by

(x - f)(v - y) = 0.

This means that either x is false or y is true. Peirce asserts that this algebra of logic is already powerful enough to

work syllogisms, and thus is substantially as good as Boole's. Peirce wants to prove the easiest syllogism (Barbara), viz., "if x is true, y is true" and "if y is true, z is true," then "if x is true, z is true." The manipulat ion he makes is to take

and

(x - f)(v - y) = 0 (1)

( y - f ) ( v - z) = 0, (2)

multiplying (1) by ( v - z),

( x - f ) ( v - y ) ( v - z) = 0, (3)

and (2) by (x - f),

( x - f ) ( y - f ) ( v - z) = 0, (4)

and then adding (3) and (4), which have the c o m m o n factors ( x - f ) and ( v - z), to obtain

( x - f ) ( v - z ) [ (v- y) + ( y - f)] = 0, (5)

finally canceling - y and y from (5) to arrive at

( x - f ) ( v - z ) ( v - f) = 0. (6)

Since v and f are never equal, he can divide both sides of (6) by their nonzero difference to obtain

( x - f ) ( v - z) = 0. (7)

This equation is simply his encoding of x implies z. In this example, Peirce uses exclusively equational reasoning, which

is one of the two ways to prove theorems in Boolean algebras. The other is based on partial order ing properties.

To explicate this, we give a translation of Peirce's a rgument into modern equational language. For this purpose, we need to reproduce the equational theory of Boolean algebras, in which Peirce seems to have been working. He needs only the special case for the algebra of propositional functions, but there is no change for abstract Boolean algebras. (We will use T and F for the modern 1 and 0 to avoid collision with his use of 0 for true and 1 for false, which is reversed from Boolean algebraic usage.)

I 18 PEIRCE'S LOGIC OF QUANTIFIERS

T h e ax ioms for a B o o l e a n a l g e b r a (B, v , A ,--,, T , F ) , based on these

o p e r a t i o n s r a t h e r t han o rder , r ead nowadays as follows:

I d e m p o t e n c e : a A a = a, a V a = a;

Commuta t iv i ty : a A b = bA a, a V b = bV a;

Associativity: (aV b) v c = a V ( b v c), (aA b) A c = a A (bA c);

R igh t Distributivity: (a v b) A c = (a A c) V (b A c), (a A b) V c = (a V c) A

(bV c);

Lef t Distributivity: a A ( b V c ) = ( a A b ) V ( a A c ) , a V ( b A c ) = ( a V b ) A

(av c); Tru th : a v T = T , a A T = a ;

Falsity: a v F - a, a A F = F;

C o m p l e m e n t : a A -',a = F, a V --,a = T.

We t h e n ob ta ins the De M o r g a n laws as t h e o r e m s :

--,(a V b) = -',a A --,b, --,(a A b) = --,a V --,b.

T h e c o m m u t a t i v e a n d associat ive laws al low us to i g n o r e the o r d e r o f

fac tors a n d d r o p p a r e n t h e s e s , the i d e m p o t e n c e laws al low us to d r o p

r e p e a t e d factors, a n d the De M o r g a n laws al low us to dr ive n e g a t i o n s

over the o t h e r ope ra t i ons .

T h e de r iva t i on rules for e q u a t i o n s are as follows:

Der ive a = a for all a; F r o m a = b a n d b - c, der ive a = c;

F r o m a = b, der ive b = a;

F r o m a = b , der ive c V a = c v b f o r a n y c;

F r o m a = b , der ive c A a = c A b f o r a n y c;

F r o m a = b, der ive "-,a = --,b.

T h e s e easily imply tha t

I f a = b a n d c = d , t h e n a A c = b A d , a V c = b v d , - - , a = - - , b .

T h e s e have as a special case post- o r p r e m u l t i p l i c a t i o n o f an e q u a t i o n

by any e l e m e n t .

Wi th this r e c o n s t r u c t i o n in m i n d , Pe i rce ' s p r o o f reads as follows:

T h e p r e m i s e s are

--,xV y =T , (1)

--,y v z = T,

f r o m which we mus t ob t a in the c o n c l u s i o n

--,xV z =T.

(2)


Take the s u p r e m u m of (1) on the left with z (i.e., p remul t ip l ica t ion) to obta in

zV (--,xV y) = zV T. (3)

By the t ruth axioms,

z v T = T. (4)

Applying transitivity of equality,

zV (--,xV y) =T. (5)

Take the s u p r e m u m of (2) on the left with --,x to obta in

-',x V (--,y V z) = --,x V T. (6)

By the t ruth axioms,

--,xV T = T. (7)

By the transitivity of equality, (6) and (7) give

-',x V (-',y V z) = T. (8)

Take the inf imum of (5) and (8) (that is, use the last of the equality rules listed above) to obtain

(z v (--,x v y)) A (--,x V (--,y V z)) = T A T. (9)

Apply the truth axioms, associative and commuta t ive laws, and properties of equality to obtain

((-,x v z) v y) A ((-,x v z) V-,y) = T. (10)

Apply the distributive law and a proper ty of equality to obtain

(--,x v z) v (y A --,y) = T. (11)

Apply the law of cont radic t ion ( complemen t ) and proper t ies of equali ty to obtain

(--,x v z) v F = T. (12)

Apply the falsity axioms and proper t ies of equality to obtain

--,xV z = T.

This is an exact mi r ro r of Peirce's a rgument . The alternative part ial-order style of p roo f for this p ropos i t ion is quite

different. For proposi t ional funct ions x and y, x < y is def ined to mean that whenever x is true, then y is true. This is the partial o r d e r of proposi t ional functions. Peirce's two premises are equivalent to x < y and y_< z. By transitivity of the prepart ial t ruth funct ion order ing , x_< z, which is equivalent to the conclusion desired. This is the form the


argument would take in Schr6der, who based the first volume of his Algebra der Logik on defining Boolean algebras by a partial order in which every two-element subset has a least upper bound and greatest lower bound, there is a 0 and a 1, the distributive law holds, and every e lement has a complement. This is, in other words, the partial-ordering path to Boolean algebras.

Here in Peirce the path is different. Boolean algebras are regarded as governed by identities that are in turn governing their operations of least upper bound (v) and greatest lower bound (^), which Peirce writes as plus and times, respectively. These two alternatives are still the main ways of defining Boolean algebras, and they are fully equivalent, as proved in Birkhoff's Lattice Theory (1948).

Peirce's goal in all of this is to expand algebra so as to contain logic. The introduction of the symbols v and f constitutes part of that expansion. The editors of the Harvard edition misunderstand this in their comment in a footnote to this section:

If this proposition [(x- f ) (v- x) --0] be added to the postulates of Boolean algebra and if the terms of that algebra be interpreted as propositions, a propositional calculus is secured. From an historical standpoint this is of tremendous importance. (p. 214)

This view is not justified. The law of the excluded middle is simply an identity true in all Boolean algebras, whether they arise from sets, propositions, or anything else. It is not special to "algebras of propositions." It seems as though the editors thought that this axiom constrains the Boolean algebra to be the two-element algebra of truth values 0 and 1, which they may have thought of as an algebra of propositions. However, the fact is that the Boolean identifies holding in any Boolean algebra are precisely the Boolean identities holding in the two-element Boolean algebra. No such identity as the one above distinguishes one Boolean algebra from another.

To see this, simply consider that an identity holding in a Boolean algebra holds in its two-element subalgebra. This is one direction in the proof. The other direction is a direct consequence of the Stone representation theorem. This theorem says that every Boolean algebra is a subalgebra of a direct product of two-element Boolean algebras. Any identity holding in a (the) two-element Boolean algebra holds in all products of two-element Boolean algebras. Any identity holding in a Boolean algebra also holds in any subalgebra. This proves that any identity holding in the two-element Boolean algebra holds in all Boolean algebras, as asserted. This same theorem can also be proved by algebraic manipulation. Therefore, the law of the excluded middle is just another identity for Boolean algebras.


Schr6de r ' s vo lume 1 def ines Boolean algebras as distr ibutive lattices with 0, 1 such that every e l e m e n t x has a c o m p l e m e n t "-,x. With this fo rmula t ion , the law of the exc luded middle exactly says that every e l e m e n t has a c o m p l e m e n t , and thus comple tes Sch r6de r ' s axioms for an abstract Boolean algebra. But this is not what Peirce is d o i n g here; he is simply showing how one writes the law of the e x c l u d e d midd le in a lgebra nota t ion . T h e r e is no special s ignificance to this.

Tarski 's theory of L i n d e n b a u m algebras shows that every Boo lean a lgebra is i somorph ic to a L i n d e n b a u m algebra of a p ropos i t iona l calculus; that is, is a h o m o m o r p h i c image of a free Boo lean algebra. Lin- d e n b a u m algebras are exactly the algebras of proposi t ions . Any distinc- t ion be tween genera l Boolean algebras and algebras of p ropos i t ions is thus a dis t inct ion wi thout a di f ference.

6. 2.2. Five Peirce Icons

In his t r ea tmen t of proposi t ional logic, Peirce restates the rules f rom his "algebra of the copula" in his 1880 paper, which he now calls "icons of algebra." The first of these icons is his principle of identity f rom 1880:

The first icon of algebra is contained in the formula of identity

x - - < x .

This formula does not of itself justit~, any transformation, any inference. It only justifies our continuing to hold what we have held. (p. 219)

Peirce 's pr inc ip le of illation, which is n o t h i n g o t h e r than the in t roduc- t ion and e l imina t ion rules for implicat ion, says "f rom x we can infer y" if and only if we can infer "x implies y." Applying this to the first icon, we get that f rom x we can infer x, which is roughly his own exp lana t ion .

The second icon restates the second rule of Peirce 's 1880 algebra:

The second icon is contained in the rule that tile several antecedents of a consequentia may be transposed; that is, that from

x -< (y-<z)

we can pass to y - < (x-< z). This is stated in the formula

Ix-< (y -< z)} --< {y --< (x-< z)}

(p. 219). Let us apply Peirce 's pr inc ip le of illation. F rom x ~ (y ~ z), we c o n c l u d e ( impl icat ion e l imina t ion) that with p remise x, we infer


(y ~ z). R e p e a t i n g the e l imina t ion , f rom premises x,y, we can infer z.

P e r m u t i n g , f rom premises y, x, we can infer z. Using impl ica t ion intro- duc t ion , f rom premise y, we can infer x ~ z. Repea t ing , we can infer

y ~ (x ~ z). This is roughly Peirce 's exp lana t ion .

Pei rce ' s p r o o f of m o d u s p o n e n s follows f rom (1) a n d (2):

By the formula of identity

(x--< y) - < (x--< y);

and transposing the antecedents

x - < {(x-< y) - < y}

or, omitting unnecessary brackets

x--< ( x - < y ) --< y.

This is the same as to say that if in any state of things x is true, and if the proposition "if x, then y" is true, then in that state of things y is true. This is the modus ponens of hypothetical inference, and is the most rudimentary form of reasoning. (p. 220)

T h e th i rd icon, he says, i n t roduces the image of a "chain o f conse-

quence" :

The third icon is involved in the principle of the transitiveness of the copula, which is stated in the formula

( x - < y ) --< ( y - < z ) - < x --< z .

According to this, if in any case y follows from x and z from y, then z follows from x. This is the principle of the syllogism in Barbara.

(p. 220)

T h e p r inc ip le of i l lation again allows this to be exp re s sed as the ru le

o f in fe rence : f rom x implies y a n d y implies z, infer x impl ies z.

T h e fou r th icon i n t roduces nega t ion :

We must now again enlarge the notation so as to introduce negation. We have already seen that if a is true, we can write x--< a, whatever x may be. Let b be such that we can write b--< x whatever x may be. Then b is false. We have here a four th icon, which gives a new sense to several formulae. (p. 221.)

No te tha t the four th icon is no t a p ropos i t i on in the l anguage . Rather ,

it says tha t if we have a p ropos i t i on b such tha t for all p ropos i t i ons x

we can infer b implies x, t hen the n e g a t i o n of b can be in fe r red . This

has a quan t i f i e r over all p ropos i t ions x. Its a lgebraic express ion is tha t


the zero of a Boolean algebra is the (unique) e lement less than or equal to all the elements of the Boolean algebra. Nowadays we would use the introduct ion and elimination rules for "not"; that is, from b infer false if and only if we can infer not b (without premises). This is in effect what Peirce does in practice.

The fifth icon is now called "Peirce's law":

A fifth icon is required for the principle of the excluded middle and other propositions connected with it. One of the simplest formulas of this kind is

{(x--< y) --< x} --< x

(p. 222). In terms of illation, or implication elimination, this icon says that if from x implies y we can infer x, then we can infer x absolutely.

Arthur Prior, in a paper published in the Journal of Symbolic Logic (1958), returned to a result of Mordchaj Wajsberg (1939), who showed that Peirce's five icons are a sufficient basis for the axiomization of the classical propositional calculus. ~ Wajsberg's system II. 93 has as axioms:

W1 CCpqCCqrCpr (i.e., [(p ~ q) ~ (q ~ r)] ~ (p ~ r)), W2 CpCqp(i.e., p ~ (q ~ p) ), W3 CCCpqpp (i.e., [(p ~ q) ~ p] ~ p), W4 COp (i.e., 0 ~ p),

and two rules of inference, substitution and detachment . Prior writes Peirce's five icons in Lukasiewicz's notat ion (in which

Cpq stands for "If p then q" and 0 is the false proposi t ion) as

P1 Cpp, P2 CCpCqrCqCpr, P3 CCpqCCqrCpr, P4 COp, P5 CCCpqpp.

Peirce does not call x - < ( y - < x) an icon, but proves it from his first and second icons (i.e., P1 and P2). Prior, however, includes both x - < {(x-< y) - < y} and x - < ( y - < x) in his list of axioms as, respectively:

P6 CpCCpqq, P7 CpC qp.

Wajsberg's axioms W1-W4 obviously match Peirce's icons P3, P7, P5, and P4. In earlier work based on Wajsberg, George Berry (19.52) had claimed that Peirce's icons P3, P4, P5, and P7 provide a complete axiomatization for propositional logic since Wajsberg's proposit ional cal-

See Hiz (1997) for all interest ing account of Peirce's inf luence on logic in Poland.

~24 P E I R C E ' S L O G I C OF Q U A N T I F I E R S

culus, which is complete, can be derived from these four Peirce icons using substitution and detachment . Prior observes that in order to derive P7 from P1 and P2, Peirce had to use an additional rule, q ~ (p ~ p), which Berry omitted. Prior concludes, however, that Peirce's derivation of P6 f iom P1 and P2 uses only substitution and detachment , and thus if the list of Peirce's icons is revised to P3, P4, P5, and P6, then Peirce's icons with substitution and de tachment as rules of inference do indeed form a complete axiom system for propositional calculus.

These are interesting results, but does the use of substitution plus de t achmen t to produce a Hilbert-style axiomatization reflect Peirce's true intent? Peirce proves x--< {(x-<y) --<y} from x--< x (P1) and {x--< (y-<z)} --< {y--< ( x - < z)} (P2). Peirce calls x --< { (x --< y) - -<y} m o - dus ponens, but as such this is just a single formula in his algebra and not a rule of inference. More generally, Peirce's style of p roof is easily captured by a natural deduct ion system, and this is the meaning of illation. As we have said in chapter 3, what Peirce does use is the principle (characteristic of natural deduct ion) that we can derive b from a if and only if we can derive "a implies b." This is seen most clearly in his presentat ion of his second rule in the 1880 calculus:

[T]he two inferences

X x

y and .'. y---<z

o~ Z

(2)

are of the same validity. Hence we have

Ix--< (y -< z)} = {y --< (x-< z)}. (3)

(p. 116)

The "if and only if" in (2) breaks up into the natural deduct ion rule, that if from premises x, y we can conclude z, then from x we can conclude z implies y; and if from x we can conclude z implies y, then from x, y, we can conclude z. The first is the rule of implication introduction; the second is the rule of implication elimination. This suggests that Prior's Hilbert-style formulation is not very faithful to Peirce, but ra ther that Peirce's icons form a natural deduct ion system (cf. Prawitz 1965) based wholly on introduction and elimination rules. In natural deduct ion systems there are no axioms, just introduction and elimination rules. Tha t is, one has only rules of inference of a specific kind. Thus, from Peirce's examples and text, the list of icons that Prior selects from Peirce should be regarded not as Peirce's axioms, but rather as a few obvious truths that can be established by introduct ion and elimination rules.

FROM P E I R C E T O SKOLEM x25

6. 2.3. Truth-functional Interpretations of Propositions

We n o w t u r n to e x a m i n e P e i r c e ' s a n t i c i p a t i o n o f t he t r u t h t ab l e m e t h o d

for t e s t i ng t a u t o l o g i e s . S u p p o s e we a re w o r k i n g o v e r t h e t w o - e l e m e n t

B o o l e a n a l g e b r a o f t r u t h values . S ince ( x - f ) ( v - y) = 0 m e a n s x - f =

0 o r v - y = 0, w h i c h m e a n s x = f o r y - v , Pe i rce , as h e s t a t e d at t h e

b e g i n n i n g o f his e x p o s i t i o n , c an set p r o p o s i t i o n s e q u a l to v o r f d i rect ly ,

a n d c o n s i d e r t r u t h f u n c t i o n s o f t h e s e p r o p o s i t i o n s . H e says t h a t h e

p r e f e r s this a p p r o a c h to his first sys tem:

But this notat ion shows a blemish in that it expresses proposi t ions in two distinct ways, in the form of quantities, and in the form of equations; and the quantit ies are of two kinds, namely those which must be e i ther equal to f or to v, and those which are equa ted to zero. To remedy this, let us discard the use of equations, and pe r fo rm no operat ions which can give rise to any values o the r than f and v. (p. 215)

P e i r c e h e r e i n t r o d u c e s t r u t h f u n c t i o n s o f p r o p o s i t i o n s :

Of operat ions upon a simple variable, we shall need but one . For there are but two things that can be said about a single proposi t ion, by itself; that it is true and that it is false,

x = v and x=f .

The first equat ion is expressed by x itself, the second by any funct ion, ~b, of x, fulfilling the condit ions

~ v - f 4)f = v.

(p. 215)

P e i r c e d o e s n o t d e v e l o p this i n s i g h t as far as h e c o u l d . H e d o e s n o t

wr i te o u t t r u t h t ab les for his logica l c o n n e c t i v e s , b u t i n s t e a d j u s t gives

a t r u t h - v a l u e analysis for a few f o r m u l a s . H e beg ins :

A proposi t ion of the form

x---<y

is true if x = f or y =v. It is only false if y = f and x =v. A proposi t ion written in the form

x - < y

is true if x = v and y = f, and is false if e i ther x - - f or y =v. (p. 224)

H e t h e n gives an e x a m p l e o f his m e t h o d :


Accordingly, to find whether a formula is necessarily true substitute f and v for the letters and see whether it can be supposed false by any such assignment of values. Take, for example, the formula

(x-<y) -< (y-<z) -< (x-< z).

To make this false we must take

(x-<y) =v

(y-<z) --< (x-<z) =f.

The last gives

(y-<z) =v,

Substituting these values in

(x-<y) =v,

we have

(v - < y) = v,

(x-<z) =f, x=v, z=f.

(y - < z) = v

(y- < f) =v,

which cannot be satisfied together. (p. 224)

Pei rce also suggests s implifying his calculus by r ep lac ing x - - < y by

+ y to ob ta in a p ropos i t iona l calculus of plus, t imes, a n d nega t ion :

This subjects addition and multiplication to all the rules of ordinary algebra, and also to the following:

y+ x;=y, y(x+ it) =y,

x + ~ = v , x ~ = f ,

xy + z = ( x + z)(y + z)

(p. 226). This allows Peirce to express Mitchel l ' s rules of i n f e r e n c e for

p ropos i t i ona l logic in a s imple form:

To any proposition we have a right to add any expression at pleasure; also to strike out any factor of any term. The expressions for different propositions separately known may be multiplied together. These are substantially Mr. Mitchell's rules of procedure. Thus the premisses of Barbara are

; + y and ~+z .

Multiplying these, we get ( ; + y)(3Y + z) = ;3~ + yz. Dropping 3Y and y we reach the conclusion ; + z. (p. 226)


6.3. F i r s t - O r d e r L o g i c

6.3.1. In f in i te S u m s a n d Products

T h e system Peirce presents in his sect ion on "First- intent ional logic o f relatives" is p r e n e x first-order pred ica te logic. Variables are wri t ten in a somewha t unusua l way, as subscripts r a the r than as a rgumen t s , and the re are no funct ions. However, since he later i n t roduces an equal i ty re la t ion and an equali ty axiom, he can replace an n-ary func t ion symbol by an (n + 1)-ary re la t ion symbol. Thus, he has a version f irst-order p red ica te logic that is expressively equiva lent to a f irst-order logic with funct ions , and conceptua l ly simpler. ( M o d e r n logic has terms, which increase the conven i ence with which one can express ma thema t i ca l ideas in the logic, but they do no t in fact qualitatively affect what one can in pr inc ip le express.) The genera l no t ion of n o n p r e n e x fo rmu la is missing in Peirce 's system, and so reduc t ions to p r e n e x fo rm are d o n e in English. Thus , even t h o u g h the expressiveness is the same as p red ica te logic, and Peirce knows how to write any th ing in p r e n e x form, he does no t yet have the genera l no t ion of a n o n p r e n e x fo rmu la a nd the rules for r educ t i on to p r e n e x form.

Peirce 's discussion of his "first- intentional logic" begins with a passage in which he summar izes the expressive power of Boo lean a lgebra as an a lgebra of m o n a d i c predica tes with a single variable:

The algebra of Boole affords a language by which anything may be expressed which can be said without speaking of more than one individual at a time. It is true that it can assert that certain characters belong to a whole class, but only such characters as belong to each individual separately. (p. 226)

Peirce p roposes to add to this system a t e c h n i q u e for m a k i n g quan- t ificational assertions, namely, infinite sums and products . T h e logic o f relatives cons iders s ta tements involving two and m o r e individuals at once , so indices are requi red . Taking first a d e g e n e r a t e fo rm of re la t ion, we can write x~y i to signify that x is t rue of the individual i, while y is t rue of the individual j. If z is a relative character , z o will signify that i is in that re la t ion to j. In this way, we can express re la t ions o f consid-

erable complexity. He goes on to say:

[I]n order to render the notation as iconical as possible we may use I~ for some, suggesting a sum, and II for all, suggesting a product. (p. 228)


Why does Peirce think this is suggestive? What is the m e a n i n g of "iconical" here? Earlier, he says:

I call a sign which stands for something merely because it resembles it, an icon. Icons are so completely substituted for their objects as hardly to be distinguished from them. Such are the diagrams of geometry. (p. 211)

Thus, "iconical" means that the nota t ion resembles in some significant fashion the thing that it represents . The diagrams of geometry , for

example , which are i n t ended to r ep resen t idealized points, lines, planes,

and regions in a hypothet ical Eucl idean plane, quickly b e c o m e the real objects of our discussion. Peirce 's type example for an icon is the role of a d iagram in the p roof of the Eucl idean proposi t ion. Within the p r o o f we lose the ability to recognize the d iagram as an e x e m p l a r of s o m e t h i n g m o r e general , and the d iagram itself becomes the object of real interest . The same thing happens with existential quant i f ica t ion and summat ion . Thus, using E for existential quant i f icat ion is iconic,

because in ar i thmet ic a sum of nonnega t ive terms is nonze ro if and only

if at least one term is nonzero , and this applies to a s u m m a t i o n of values

of a proposi t ional funct ion over a domain , so a sum of O's and l ' s over

the d o m a i n being nonzero exactly says that there is an e l e m e n t of the

d o m a i n satisfying the proposi t ional function. Peirce 's disciple Schr6der used the same notat ion, but with a strict

Boolean a lgebra in te rpre ta t ion , r a the r than a numer ica l one; Schr6der u n d e r s t o o d existential quant i f icat ion of a proposi t ional funct ion over a d o m a i n as an indexed infinite disjunction over e lements of the domain .

6. 3. 2. Mi tchel l

Peirce clearly states that, in his opinion, Mitchell i n t roduced an effective

nota t ion for quantif iers appl ied to Boolean expressions:

We now come to the distinction of some and all, a distinction which is precisely on a par with that between truth and falsehood; that is, it is descriptive.

All attempts to introduce this distinction into the Boolian [sic] algebra were more or less complete failures until Mr. Mitchell showed how it was to be effected. His method really consisted in making the whole expression of the proposition consist of tw6 parts, a pure Bool- ean expression referring to an individual and a Quantifying part saying what individual this is. (p. 227)

H e r e Peirce refers to expressions, not Boolean funct ions of the e l ement s


of the d o m a i n , a n d he says qu i t e c lear ly tha t b e c a u s e d o m a i n s can be

i n n u m e r a b l e , these are like sums a n d p r o d u c t s b u t not sums a n d p r o d -

UCts:

Here, in order to render the notation as iconical as possible we may use E for some, suggesting a sum, and II for all, suggesting a product. Thus E,x; means that x is true of some one of the individuals denoted by i or

~ix, = x; + xj + x k + etc.

In the same way, II;x; means that x is true of all these individuals, or

I - I i x i = x i x j x k , e t c .

If x is a simple relation, II;IIjxq means that every i is in this relation to every j, F, iIIi x 0 that some one i is in this relation to every j, IIjF~ix 0 that to every j some i or other is in this relation, Ei]2jxij that some i is in this relation to some j. It is to be remarked that I~,x; and II;x; are only similar to a sum and a product; they are not strictly of that nature, because the individuals of the universe may be innu- merable. (p. 228)

This is unl ike the ea r l i e r d i scuss ion in Pe i r ce ' s 1883 p a p e r (pp. 200if)

in t ha t f o r m a l e x p r e s s i o n s are p r e sen t .

In fact, Mi tche l l d id n o t arr ive at the general n o t i o n o f an a rb i t r a ry

s e q u e n c e o f quan t i f i e r s a p p l i e d to a B o o l e a n f o r m u l a ; he t r e a t e d two-

d i m e n s i o n a l f o r m u l a s in d e p t h , m e n t i o n e d t h r e e - d i m e n s i o n a l f o rmu la s

in passing, a n d does n o t a l l ude to fo rmu la s o f h i g h e r d i m e n s i o n t h a n

th ree . Pe i rce gives h im full c r ed i t n o n e t h e l e s s , a t t r i b u t i n g to Mi tche l l

t he n o t i o n o f a f o r m u l a in p r e n e x fo rm, a l t h o u g h we k n o w now tha t

this s h o u l d give an e q u i v a l e n t of every p r e d i c a t e f o r m u l a , as s h o w n by

the p r e n e x n o r m a l f o r m t h e o r e m .

6 .3 .3 . F o r m u l a s a n d R u l e s

Al l f o r m u l a s stay p r e n e x to the e n d of Pe i r ce ' s paper . Pe i r ce gives several

e x a m p l e s , all o f which are s e n t e n c e s ( lack ing f ree var iab les ) :

Let l 0 mean that i is a lover of j, and bq that i is a benefactor of j. Then

II~F~jl~jb 0

means that everything is at once a lover and a benefactor of something; and

13o PEIRCE'S LOGIC OF QUANTIFIERS

that everything is a lover of a benefactor of itself.

~,~kIIj(lij + bi~)

means that there are two persons, one of whom loves everything except benefactors of the o ther (whether he loves any of these or not is not stated). Let g~ means that i is a griffin, and c i that i is a chimera , then

means that if there be any chimeras there is some griffin that loves them all; while

means that there is a griffin and he loves every ch imera that exists (if any exist). On the o ther hand,

nir,g,(t,i + 6) means that griffins exist (one, at least), and that one or o the r of them loves each ch imera that may exist; and

njr,(g,t,~ + 6) means that each chimera (if there is any) is loved by some griffin or other. (p. 229)

T h i s is, o d d l y p o p u l a t e d , t he t h e o r y o f p r e n e x logic.

P e i r c e t h e n poses t he p r o b l e m : H o w d o e s o n e d e d u c e p r e n e x for-

m u l a s f r o m p r e n e x f o r m u l a s ?

We have now to consider the p rocedure in working with this calculus. It is far from being true that the only p rob lem of deduct ion is to draw a conclusion from given premisses. On the contrary, it is fully as impor tan t to have a me thod for ascertaining what premisses will yield a given conclusion . . . . I shall con ten t myself here with showing how, when a set of premisses are given, they can be uni ted and certain letters el iminated. Of the various methods which might be pursued, I shall here give the one which seems to me the most useful on the whole. (p. 230)

T h e first t h i n g h e d o e s is w h a t is d o n e in a u t o m a t i c t h e o r e m - p r o v i n g

today. H e " s t a n d a r d i z e s apa r t , " t h a t is, h e a r r a n g e s t h a t d i f f e r e n t p r e m -

ises have d i s jo in t i nd ices ( b o u n d va r i ab les ) :

First, the different premisses having been written with distinct indices (the same index not used in two proposit ions) are written together, and all the II's and ~:'s are to be b rough t to the left. This can evidently be done, for


(p. 231)

n , x , . n i x i = n , r l j x , x j

~,x,. njxj = ~,njx,xj

Z,x , . Zjxj = Z ,Zjx ,x j .

131

These formulas show that the con junc t ion of two p r e n e x formulas is

equ iva len t to a p r e n e x formula .

Next, Peirce gives rules for m a n i p u l a t i n g the quantif iers:

Second, without deranging the order of the indices of any one premiss, the rI's and l~'s belonging to different premisses may be moved relatively to one another, and as far as possible the I~'s should be carried to the left of the rI's. We have

n , E x,i = n i n , x,i

r, ,r, ,x,j= r, jr,,~,j

z , n j x , y j - - n , r , x , y , .

(p. 231)

For con junc t ions of p r e n e x formulas with disjoint b o u n d variables, per-

m u t i n g at the f ron t the quant i f iers f rom di f fe ren t p remises thus gives

equiva len t p r e n e x formulas. This is the re fo re a rule of d e d u c t i o n of p r e n e x formulas f rom p r e n e x formulas. Peirce also notes that

This formula [i.e., F,,Ilixiy j = IIjF.,ixiYj] does not hold when the i and j are not separated. We do have, however,

r, ,n j x,i - < n i r,, x, j.

(p. 231)

Thus, he knows that these quant i f iers c a n n o t be p e r m u t e d with im-

munity, bu t he realizes that at least one gets an impl ica t ion.

T h e word "quantif ier" occurs with a capital Q when Pei rce gives an-

o t h e r rule for d e d u c i n g one p r e n e x fo rmula f rom ano the r :

The next step, which will also not commonly be needed, consists in making the indices refer to the same collection of objects, so far as this is useful. If the quantifying part, or Quantifier, contains I2x, and we wish to replace the x by a new index i, not already in the Quantifier, and such that every x is an i, we can do so at once. (p. 232)

We can see f rom this rule that the indices do no t t ransla te perfect ly

x32 PEIRCE'S LOGIC OF QUANTIFIERS

into modern quantifiers, because they may be variables ranging over different domains (presumably specified in advance). Modern logic does not do this unless it is many-sorted" if we wish to restrict the domain of a variable to a subset of the universe, we add a predicate letter to the language and say (3x)(P(x) ^ ...). Here Peirce says that if index x runs over a class and index i runs over a larger class, then I2 x may be replaced by E i. Thus, the ranges of his indices can be distinct, unlike in modern predicate logic. This is in accord with long usage in algebra, however, and that is where Peirce takes his inspiration.

We note that in this section of his paper Peirce seems to use Boolean expressions that are recognizable as built up from atomic proposit ions exclusively, with no use of relative products or sums. There is therefore a shift of emphasis away from the relative operations and toward variables and quantifiers. Peirce never explicitly gives an inductive definition of formula, such as Frege's; nonetheless, it is all there.

In summary, Peirce's first-intentional logic is prenex logic. There is no formal definition of atomic formulas. There are no explicit conventions as to what variables to use. There is no proposed fixed list of rules for moving quantifiers about. The rules for simplifying Boolean expressions inside the quantifier-free body are likewise not explicit. When translating from English in his examples, Peirce always translates to a prenex formula. Since this is always possible, we have the expressive power of full first-order logic, but not all of its formulas. Peirce gives credit for the idea of prenex formula to Mitchell, a charitable and fairly accurate attribution for prenex two-quantifier statements.

Peirce's first-order logic is what we would today call pure predicate logic without equality. His presentat ion is remarkably modern; its only material additions are functions and terms.

6.4. S e c o n d - O r d e r Logic

Peirce's 1885 paper appeared during a period of active worldwide research on the foundations of mathematics. Cantor 's papers on the foundations of the real numbers (1872, 1874) had already appeared. De- dekind had published his theory of real numbers (1872), and the foundations of induction had been worked out independent ly by Frege (1879) and Dedekind (c. 1878; published in 1888). Yet to appear were Frege's foundational work of the 1890s, the paradoxes of set theory set forth by Russell (1903), and the first successful efforts to formalize set theory by Zermelo (1908), following Hilbert 's suggestions.

Peirce's 1885 paper was his first a t tempt to formalize higher mathematical notions in the higher order theory of relatives (his "second- intentional logic"). This work shows that Peirce was quite aware that his


logical l anguage was capable of cap tur ing significant d e v e l o p m e n t s of

c u r r e n t mathemat ics , provided that he used the h ighe r o r d e r version. In fact, a primitive set of axioms for set theory e m e r g e s t h r o u g h his icons.

In his "second- in tent ional logic" Peirce a t tempts to lay the founda t ions for a ma thema t i ca l u n d e r s t a n d i n g of the no t ion of a class or a set algebra. He lays out the proper t ies of classes vis-fi-vis individuals. Peirce states an early form of the axiom of extensionality: two e l emen t s are equal if and only if they be long to the same col lect ion of classes. He wants to talk abou t class fo rmat ion rules. O n e of his axioms is that to

every individual we associate the class that consists of only that individual.

He then goes on to talk about pair ing and o the r t radi t ional opera t ions

that one does with sets. Peirce begins his discussion of second- in ten t iona l ( second-order )

logic by in t roduc ing 1 o, the equality or identity relation:

Let us now consider the logic of terms taken in collective senses. Our notation, so far as we have developed it, does not show us even how to express that two indices, i and j, denote one and the same thing. We may adopt a special token of second intention, say 1, to express identity, and may write 1,). (p. 233)

"Token" is Peirce 's t e rminology for a predica te letter. Peirce recognizes that it is not e n o u g h jus t to add equali ty as a n o t h e r

o rd inary relat ion, but that equality conveys with it addi t ional conse-

quences , axioms of the form ViVj((i =j) ~ [x(i) ~ x(j)]), which is usually taken as an axiom of first-order predica te logic.

[T]his relation of identity has peculiar properties. The first is that if i and j are identical, whatever is true of i is true of j. This may be written

n,nj l i , ; + ~, + ~j}.

(p. 233)

Peirce also gives the o the r direct ion:

The other property is that if everything which is true o f / i s true of j, then i and j are identical. This is most naturally written as follows: Let the token, q, signify the relation of a quality, character, fact, or predicate to its subject. Then the property we desire to express is

II,IIi~k(lo + (lk, qkj)"

(p. 234)


This f o r m u l a says tha t if a p r o p e r t y k e i t h e r holds of i o r does n o t ho ld o f j, t hen i and j must be the same. :~ In this fo rmula , k is i n t e n d e d to r a n g e over una ry p red ica tes or classes and no t over individuals , and

qki m e a n s that k holds of i, o r in se t - theore t ic terms, i ~ k, so q is Pe i rce ' s a n a l o g u e to the m e m b e r s h i p re la t ion , ~.

Pe i rce t hen def ines equal i ty by giving Leibniz ' s p r inc ip le of the identity of indiscernib les . His use of tokens may signal the rea l iza t ion tha t a tomic fo rmulas are n e e d e d :

And identity is defined

1;j= IIk (qk;qkj + (]kiqkj)"

That is, to say that things are identical is to say that every predicate is true of both or false of both. (p. 234)

Leibniz ' s p r inc ip le of the ident i ty of ind iscern ib les is now the s t a n d a r d way of de f in ing equal i ty in h i g h e r o r d e r logic wi thou t equal i ty as a pr imi t ive symbol. T h e quan t i f i e r r anges over all classes (una ry re la t ions)

k. This is second-order , no t f i rs t -order logic. In f i rs t -order logic, equal i ty

is a b inary re la t ion (relat ive) , subject to the usual axioms. Equal i ty is not , in genera l , a de f ined c o n c e p t in f i rs t -order logic.

Af ter giving ax ioms for equality, Pei rce turns to q, his a n a l o g u e of the m e m b e r s h i p relat ion:

The properties of the token q must now be examined. These may all be summed up in this, that taking any individuals i 1, i2, i:~, etc., and any individuals jl,j2,j:~, etc., there is a collection, class, or predicate embracing all the i's and excluding all the j 's except such as are identical with some one of the i's. This might be written

(n.n,o)(n~n~) ~ (n. ~,, )n,q~;.(~;, + q,,.% + q,,.q,;,),

where the i's and the i"s are the same lot of objects. (p. 234)

Trans la t ed into se t - theore t ic te rminology, this f o r m u l a reads:

(vc~)(Vi~ ~ o~)(v/3)(vjt ~ E /3)

A(3k)((i~ ~ k) A [(Jt~ ~ k) ~ (3i~,~ ~ c~)(V/)(jo ~ l ~ i~ ~ /)]).

This special p r inc ip le is really a c o m b i n a t i o n o f things. To just ify it in

Z e r m e l o - F r a e n k e l set theory, we assume tha t i is an e l e m e n t with indices d rawn f rom a set, in which case to a rgue that we can col lec t the i's and ge t a set is an e x a m p l e of the ax iom of r e p l a c e m e n t . We can a r g u e the

Sin other words, ViYj3k( i=jv ['-,q(k, z) A q(k,))]), equivalently, qiYj[((qk)[q(k,j) =* q(k,i)]) = (i •j)].


s ame way for the j 's. T h e n we n e e d to ge t all the ind iv idua ls i t ha t a re

in the (var iable) co l l ec t ion c~ s e p a r a t e d f r o m all the ind iv idua ls j tha t a re n o t in tha t co l lec t ion . T h e n , if a is a set i n s t an t i a t i ng c~ a n d b is

l ikewise for /3 , t h e r e is a set k ( the s e p a r a t o r set) such tha t

aC_ k;

k c ~ b c _ a.

C o n s i d e r i n g these two cond i t ions , it s eems qu i t e obvious tha t k can be

c h o s e n to be a itself, so the only r ea son for this a x i o m seem s to be tha t

indiv iduals m i g h t be wi th in d i f f e r en t d o m a i n s .

As we have a l ready obse rved , Pe i rce uses d i f f e r e n t d o m a i n s for in-

dividuals , a n d the quan t i f i e r s (II,~IIi,~), (II~IIj~), a n d E k (/I~I~ i, ) serve to specify what d o m a i n his individuals c o m e f rom, s u p p o r t i n g o u r inter-

p r e t a t i on :

The II,~IIi~ ' shows that we are to take any collection whatever of the i's, and then any i of that collection. We are then to do the same with the j's. We can then find a quality k such that the i taken has it, and also such that the j taken wants it unless we can find an i that is identical with the j taken. The necessity of some kind of notation of this description in treating of classes collectively appears from this consideration: that in such discourse we are neither speaking of a single individual (as in the non-relative logic) nor of a small number of individuals considered each for itself, but of a whole class, perhaps of an infinity of individuals. (p. 234)

Pe i rce goes on to state fou r axioms, which he ident i f ies as icons, tha t

look very m u c h like par ts o f Z e r m e l o set theory. T h e first o f these ax ioms says tha t any ind iv idua l can be c o n s i d e r e d as a class:

II, EkII) qk,( gtk) + 10).

(p. 235) In o t h e r words , for all i t he r e exists a r e la t ion k tha t ho lds for i a n d

ho lds for no o t h e r individual . Pe i rce calls this f o r m u l a " the n i n t h icon."

As far back as in his 1870 paper , Pe i rce has w a n t e d to wri te ind iv idua ls

as relatives, a n d this a x i o m does that. In o t h e r words , if we have an

indiv idual , we also have the class c o m p r i s i n g exact ly tha t o n e individual ;

tha t is, for any ind iv idua l i, we can m a k e use o f the class {i}, a n d k is

tha t s i ng l e ton {i}.

T h e n e x t a x i o m ( the t e n t h icon) says that , given any class, t h e r e is

a n o t h e r tha t inc ludes all the f o r m e r l y e x c l u d e d indiv iduals a n d exc ludes

all the f o r m e r l y i n c l u d e d individuals:

x36 PEIRCE'S LOGIC OF QUANTIFIERS

This axiom claims the existence of a c o m p l e m e n t for every class. The next axiom (the eleventh icon) says that, given any two classes,

there is a third that includes all that e i ther includes, and excludes all that both exclude:

n,n., n, (q,,qk, + qm,qk, + gh, glm,glk, )"

(p. 235) This is union. These three axioms toge ther are enough to get the cardinal nu__mber

2__Namely, take i, and obtain, usin__g (9) and (10), {i}, {{i}}, a___nd {i} and {{i}}. Take the un ion of {{i}} and {{i}} by (11) to obtain {{i},{i} }, thereby ob ta in ing the cardinal 2.

We can obtain a similar pair ing from Peirce's axioms using Wiener ' s (1914) construct ion. That is, take a, b, and get {a} and {b} by (9), which a r e d i s t i n c t by (9) and (10). Then get {a,b} by (11), and {a,{a, b}} by (11) again.

The next axiom (the twelfth icon) describes a class that, given any two classes, includes the whole of the first class and any one individual of the second class that is not inc luded in the first, and no th ing else"

n,n~n.~n~lq. + 4~. + qk,(qkj + glo)}"

(p. 235) This axiom appears to be saying that, given any class and any indi-

vidual not in the class, we can add that individual to the class; in o the r words, a very weak successor axiom. At a bare m i n i m u m , this axiom describes a kind of union. It is extensionally the same as the eleventh icon, and it is difficult to see why Peirce includes it.

Peirce then proves that, supposing we are given that every proper ty is e i ther true of i or false of)', then i and j must be the same (according to the Leibniz identity)"

To show the manner in which these formulas are applied let us suppose we have given that everything is either true of i or false of)'. We write

Ilk (qk; + qki)-

The tenth icon gives

H, F~ (qt,(lk, + (b,qk,)(qo(t~i + (lOqk~).

Multiplication of these two formulae give

II, F-,k(qki (l,i + qo (lki),

or. dropping the terms in k


II~ (,~. + q0)"

Multiplying this with the original datum and identifying I with k, we have

137

II, (q,,qkj + glk,gAj).

No doubt, a much more direct method of procedure could be found. (p. 235)

Pe i r ce ' s susp ic ion tha t t h e r e is an eas ie r p r o o f is co r rec t . We of fe r a

very easy p r o o f ( n o t given by Pe i rce ) : C o n s i d e r the p r o p e r t y "no t equa l

to i." This p r o p e r t y does n o t h o l d of i, so it m u s t be false o f j . H e n c e

j is equa l to i, q.e.d.

Pe i rce t h e n de f ines the n o t i o n tha t the ca rd ina l i ty o f a class a is less

t han o r equa l to the ca rd ina l i ty o f a class b. To do this, he i n t r o d u c e s

a new t o k e n r, which s tands for " re la t ion" :

Just as q signifies the relation of predicate to subject, so we need another token, which may be written r, to signify the conjoint relation of a simple relation, its relate and its correlate. That is, %; is to mean that i is in the relation a to j. Of course, there will be a series of properties of r similar to those of q. But it is singular that the uses of the two tokens are quite different. Namely, the chief use of r is to enable us to express that the number of one class is at least as great as that of another. (p. 236)

T h e t o k e n r is t echn ica l ly s e c o n d - o r d e r - - i t takes two ind iv idua l var iab le

a r g u m e n t s a n d a r e l a t i on a r g u m e n t - - b u t to quan t i fy over r is th i rd-

o rder . N o t e tha t o~ is no t a r e la t ion , bu t a var iab le tha t r a n g e s over

re la t ions .

Pe i rce expresses tha t the ca rd ina l i ty o f a is less t h a n o r e q u a l to the

ca rd ina l i ty o f b as

+ b~),~,(6,~, , + d h + 1,,,)}.

(p. 236)

T h a t is, t h e r e exists a r e l a t ion c~ such tha t if i is in the class a, t h e r e

is a j in the class b for which i is in the r e l a t i on c~ to j , a n d if h is also

in a a n d in the r e l a t ion ol to j , t h e n i = h. In o t h e r words , t h e r e is a

p a i r i n g o f every m e m b e r of a with a d i s t inc t a n d s e p a r a t e m e m b e r o f

b.

Pe i rce obse rves tha t he can express this p r o p o s i t i o n m o r e s imply by

us ing an a tomic r e l a t ion symbol c for o n e - o n e c o r r e s p o n d e n c e :

The best way to express such a proposition is to make use of the letter


c as a token of a one-to-one correspondence. That is to say, c will be defined by the three formulae,

I I~II . I I , , I I ,o ( ~ + ~,,~,, + ~.~,~ + 1 .... ) ,

n o n . n , , n , , , ( e o + ~.o,o+ ,~ . . . . + ~.,,),

II,.E.E,,E,o(c,~ + r.~,,r.~,oi~,o + r.~,or , .... i.,,).

(p. 236)

The first fo rmula says that if ot is in c, then it is one-to-one; the second says that if c~ is in c, then it is single-valued; and the third fo rmula says that if c~ is both one-to-one and single-valued, then it be longs to c.

Using the token c, Peirce reexpresses his propos i t ion as

which is, m o d u l o notat ion, exactly our m o d e r n express ion for "a has cardinal i ty lesser than or equal to b":

~ f v i Y h [ ( i ~ a) ~ [f(i) e b] A ([f(h) =f(i)] A [(h e a) ~ (i = h)])];

in o t h e r words, fis a one-to-one funct ion f rom a to b. Peirce uses the not ion of one-one c o r r e s p o n d e n c e to def ine the no-

tion of a finite collections, in an in teres t ing example that conc ludes his 1885 paper.

In his 1881 paper "On the logic of number ," Peirce e x a m i n e d a flawed p r o o f of De Morgan (1847) that r equ i red add ing a hypothesis, finiteness, to cor rec t the flaw in the proof. The e r ro r occu r r ed in De Morgan ' s theory of "the syllogism of " t ransposed quantit ies." In his 1885 paper, Peirce re turns to De Morgan ' s example in o rde r to isolate a p roper ty of finite collections. Peirce states:

In an appendix to his memoir on the logic of relatives, De Morgan enriched the science of logic with a new kind of inference, the syllogism of transposed quantity. De Morgan was one of the best logicians that ever lived and unquestionably the father of the logic of relatives. Owing, however, to the imperfection of his theory of relatives, the new form, as he enunciated it, was a down-right paralogism, one of the premisses being omitted. But this being supplied, the form furnishes a good test of the efficacy of a logical notation. The following is one of De Morgan's examples:

Some Xis Y For every X there is something neither Y nor Z; Hence, something is neither X nor Z.

The first premiss is simply Ec, x,,y,,.


The second premiss may be written

From these two premisses, little can be inferred. To get the above conclusion it is necessary to add that the class of X's is a finite collection; were this not necessary the following reasoning would hold good (the limited universe consisting of numbers); for it precisely conforms to De Morgan's scheme:

Some odd number is prime; Every odd number has its square, which is neither prime nor even; Hence, some number is neither odd nor even.

x39

Now, to say that a lot of objects is finite, is the same as to say that if we pass through the class from one to another we shall necessarily come round to one of those individuals already passed; that is, if every one of the lot is in any one-to-one relation to one of the lot, then to every one of the lot some one is in this same relation. (pp. 237-238)

We will write Per ice ' s fo rmal iza t ion a little differently, in o r d e r to be t t e r u n d e r s t a n d his c o m m e n t a r y .

T h e first hypothes i s is E,,,x,,y,,. T h a t is, (a e X) A (a e Y).

T h e nex t hypothes is is says tha t t he re exists a func t i on f s u c h tha t (1)

d o m a i n f is X a n d (2) for all x in X, f(x) is no t in Y u Z.

T h e claim, which Peirce a rgues is no t valid because t he r e is a n o t h e r

hypothes i s r e q u i r e d , is that the re exists a y such tha t y is n o t in X u Z.

Pei rce ' s c o u n t e r e x a m p l e a m o u n t s to this: Fix a in X n Y. Now c o n s i d e r

f(a), where f(t) - t 2. T h e n f(a) is no t in Y by (2). T h e r e f o r e a g: f(a). C o n s i d e r f ( f ( a ) ) ; f ( f ( a ) ) ~ a because the same a r g u m e n t applies. But

f ( f(a)) ~ f(a) because f is one- to-one , a n d if f ( f (a)) - f ( a ) , this implies f(a) = a, which is a l ready known to be false. O f course , we can only fo rm

f ( f (a)) if f ( a ) i s in X. Now th ink of a as 0 a n d th ink of f as the successor re la t ion . T h e n

e i t he r we can fo rm the full s equence , a, f(a), f ( f(a)) . . . . within X, a n d t h e r e f o r e X is infinite, or we cannot . If we c a n n o t f o rm tha t se-

q u e n c e - i n o t h e r words, the re exists a k such thatfk(a) is n o t in X - - h e r e

is o u r e l e m e n t : fk(a) ~ Z. T h e r e f o r e , f*(a) is the e l e m e n t c l a i m e d to

exist.

Now, in l ight of this analysis, we will c o n s i d e r De M o r g a n ' s a n d Pei rce examples .

De M o r g a n gives the fol lowing example :

Suppose a person, on reviewing his purchases for the day, finds, by his counterchecks, that he has certainly drawn as many checks on his banker (and maybe more) as he has made purchases. But he knows

14o PEIRCE'S LOGIC OF QUANTIFIERS

that he paid some of his purchases in money. (De Morgan 1847, p. 168; quoted in Peirce 1885, p. 237)

The value of the checks that De Morgan's purchaser has written is greater than the merchandise that was bought out of that amount , or it is at least as large, and he knows there is something that he brought home that he paid for in cash. He infers then that he has written checks for something else except that day's purchases.

Let us consider how this a rgument works. The most interesting case is when the purchaser can construct a one-to-one correspondence between checks and purchases but knows that there is something he did not purchase with a check. He can then take the item that he did not purchase with a check and ask, "what check was associated with this?" and determine what that check actually purchased. That, in turn, will give rise to an associated check, and the purchaser will de termine what was actually purchased by that check. Eventually this process must ter- minate, and thus purchaser ends up with an object whose associated check cannot be paired up with an object that it purchased, and that is the check that was written for something other than a purchase.

Peirce, on the contrary, assumes that he has made infinitely many purchases and has written infinitely many checks, and that he paid cash for the first item. in general, Peirce assumes that he wrote the ith check for the (i + 1)st item. This gives him De Morgan's hypotheses but not De Morgan's conclusion. In order to arrive at De Morgan's conclusion, Peirce must assume, in addition, that the number of purchases is not infinite, that is, that De Morgan's set X does not have an infinite subset. Peirce is exactly correct in identifying this assumption.

Even though Peirce is exactly correct, he may be being slightly unfair to De Morgan. De Morgan may have written from the perspective that did not admit the possibility of reasoning rigorously about infinite collections. Thus, the hypothesis that a collection is finite may have been simply one of the rules of thought for De Morgan, not worth stating explicitly. Peirce is writing later than De Morgan, after Cantor 's initial theorems on convergence of tr igonometric series, and hence Peirce is willing to think about the possibility of sets as infinite. Peirce's examples require the construction of infinite sets, and he acts as though one can reason about them, a claim which was for a long time difficult to justify.

De Morgan's example is interesting regardless of its shortcomings because it appears to anticipate Dedekind's foundat ion for induction. De Morgan's example describes a function f which is the successor function. The successor function is simply a one-to-one function on the natural numbers that is not the identity. All one needs to generate the natural numbers is a one-to-one function f t h a t is not the identity function. We can then take a point x such that f (x) ~ x, call it 0, and f


gene ra t e s the natura l numbers . A c o u n t e r e x a m p l e to De M o r g a n ' s claim involves a set that has e m b e d d e d in it a copy of the integers . It is in te res t ing to specula te on w h e t h e r D e d e k i n d was s t imula ted to deve lop his cha in theory f rom u n d e r s t a n d i n g De Morga n ' s e x a m p l e and, also what was wrong with it.

Peirce clearly u n d e r s t o o d what was wrong with De M o r g a n ' s example . He p r o c e e d e d to def ine the no t ion of a finite col lect ion, us ing the now well-known p r o o f that every one- to-one c o r r e s p o n d e n c e is on to w h e n its d o m a i n and range are finite:

Now, to say that a lot of objects is finite, is the same as to say that if we pass through the class from one to another we shall necessarily come round to one of those individuals already passed; that is, if every one of the lot is in any one-to-one relation to one of the lot, then to every one of the lot some one is in this same relation. This is written thus:

IIalI.X;,,E, II,{ea + ~u + x,,r.a~ + x,(~, + r,a,)}.

(p. 238)

This is Peirce 's def in i t ion of finite col lect ion, which is very close to the no t ion of D e d e k i n d finite. Peirce 's analysis a p p e a r e d th ree years be fo re D e d e k i n d ' s publ ica t ion and is clearly i n d e p e n d e n t of Dedek ind .

In sum, Peirce presents , in his 1885 paper , the p red ica te logic of p r e n e x formulas , with all the natura l rules he cou ld th ink of. He recogn ized that if he wan ted equality, he would have to pu t it in with axioms, and he r ecogn ized that he n e e d e d re la t ion symbols. Thus , Peirce presents p r e n e x quant i f iers appl ied to Boolean c o m b i n a t i o n s of a tomic formulas rok.. .. T h e r e is no use of r e p e a t e d indices, that is, no r;i, and no use of nes ted quantif iers .

By 1885 Peirce had the full equiva len t of m o d e r n p red ica t e logic with identi ty [ identi ty allows us to say r ( x , x ) as r (x ,y ) A x = y], bu t p r e n e x fo rm only, and simplif icat ion or d e d u c t i o n rules for p r e n e x fo rm alone, separa t ing the ability to separa te and r e n a m e variables a nd the Boo lean calculat ions inside the formula . In his sect ion on s e c o n d - o r d e r logic, Peirce gives axioms for cer ta in set- theoret ic n o t i o n s m u n i o n , s ingle ton , and c o m p l e m e n t . Taken together , they give pa i r ing a nd successor, and a par t of Ze rme lo set theory. Peirce def ines one- to -one c o r r e s p o n d e n c e and uses it to express that one class is at least as grea t as a n o t h e r and to def ine the no t ion of finiteness. It seems likely that some of the works of C a n t o r or D e d e k i n d were known to Peirce at the t ime that he wrote this sect ion of his paper , and that he was trying to code some of the early ideas of set theory into the h ighe r o r d e r calculus o f relatives.

S c h r 6 d e r e x t e n d e d this further , but with few new ideas, in his treat-

PEIRCE'S LOGIC OF QUANTIFIERS

ment of Dedekind chains in volume 3 of his Algebra der Log~k (1895, pp. 346--384). L6wenheim, in his 1940 paper on Schr6der's relative calculus, tried to argue that Peirce and Schr6der were able to incorporate mathematics into the higher order theory of relatives. L6wenheim also argued that it is natural and convenient to do mathematics in the higher order theory of relatives; unfortunately, this is patently false, as the great work involved in decoding some of Peirce's formulas has shown. Very simple facts become opaque when expressed in the higher order theory of relatives. Eventually, Tarski and Givant, in their monograph A Formali- zation of Set Theory without Variables (1987), did produce a fully formal version of the theory of relatives that captures most of useful set theory, but their t reatment often seems artificial, in that it does not follow the usual ways of developing and expressing mathematics.

7. Schr6der's Calculus of Relatives


Ernst Schr6der was a German disciple of Peirce, best r e m e m b e r e d for his choiceless p roo f of the Schr6der-Bernstein t h e o r e m that if each of two sets can be m a p p e d one-one into the other, then there is a one-to- one onto map between them. But there is much also of interest in his three-volume Vorlesungen iiber die Algebra der Logik. It offers the first exposit ion of abstract lattice theory, the first exposi t ion of Dedekind ' s theory of chains after Dedekind, the most comprehens ive d e v e lo p me n t of the calculus of relations, and a t rea tment of the founda t ions of mathematics in relation calculus that L6wenheim in 1940 still t hough t was as reasonable as set theory. Schr6der ' s concept of solving a relational equat ion was a precursor of Skolem functions, and he inspired L6w- enhe im ' s formula t ion and p roof of the famous t heo rem that every sentence with an infinite model has a countable model , the first real the- o rem of m o d e r n logic.

Schr6der acknowledged his debt to Peirce in the in t roduc t ion to the tirst volume of his Algebra der Log~k:

vor allem durch die Arbeiten des Amerikaners Charles S. Peirce und seine Schule ... [above all, the work of the American Charles S. Peirce and his school]. (Schr6der 1890, p. iii)

and repeatedly says he owes everything to Peirce. Before Peirce caught on to predicate logic from his s tudent Mitchell, he was primarily interested in developing logic and mathemat ics using the identities and inequalities of his relative calculus, in which relative p roduc t and sum are the available special cases of existential and universal quant i f icat ion covered by the calculus. Mitchell in t roduced one- and two-quantifier statements; Peirce first adop ted these as an addi t ion to the relative calculus, and then d r o p p e d the relative calculus in their favor. Peirce

143

144 SCHRODER'S CALCULUS OF RELATIVES

usually treated quantifiers in a notation that indicates that the existential quantifier is defined semantically for one domain at a time as a least upper bound of propositional functions over all instantiations in that domain. Similarly, universal quantifiers are defined as greatest lower bounds.

Schr6der developed Peirce's relative calculus much further and much more systematically than did Peirce. Schr6der considered quantifiers (or, at least, sums and products equivalent to quantifiers for a fixed domain) in first- and higher order logic. He unders tood that there are notions such as countability that are beyond basic relative calculus (and also beyond first-order predicate logic). Alwin Korselt's example in L6w- enheim's 1915 paper demonstra t ing that "condensed" relative calculus (i.e., without quantifiers) is less expressive than first-order logic came out of these considerations. It is this emphasis on the relations between assertions and the domains in which they hold that is new in Schr6der.

Along with the systematization of Peirce's calculus of relatives, Schr6der 's other major contr ibution to the development of logic was the application of the calculus of relatives to specific problems of mathematical interest, in particular to Dedekind's work on justifying definitions by induction. Schr6der showed that Dedekind's chain theory could be carried out in the calculus of relatives, and converted the latter into a possible foundation for mathematics. Schr6der should also be regarded as the forgotten originator of what we now know of as Skolem functions, through his notion of "solving" relational equations.

Schr6der 's Vorlesungen iiber die Algebra der Logik was published in three volumes in 1890-1895. A fourth volume was printed posthumously in 1910. His work is prolix to an extreme, running to over 2,000 pages. Because no summary is available in English, we give a short outline of the contents of Schr6der 's three books and their relation to Peirce and L6wenheim. Translations of Schr6der 's Lecture IX, volume 3, on De- dekind's chain theory and translations of portions of Lectures I, II, III, IV, XI, and XII of volume 3 are given in Appendices.

Because the language of Schr6der 's lectures is very obscure to the modern eye, they are described here in modern terms.

7.1. Die Algebra der Logik: Volume 1

In volume 1 of his Algebra der Logik, Schr6der introduces a fully axiomatic t rea tment of partially ordered sets, lattices, and Boolean algebras, and gives class calculus as his main example. The theorems proved are intended to be applied to the lattices encountered in volume 2, which deals with propositional logics, and in volume 3, which is devoted to relatives.


Schr6der begins the book with a historical and philosophical introduct ion to logic and its meaning, with extensive acknowledgments of Peirce (pp. 1-125). 1

Lecture LBDiscusses the inclusion relation (subsumption) , inference, and implication, preparatory to the abstract partial o rder ing concept in t roduced in Lecture II.

Lecture II .mDefines nonstrict partial orderings abstractly. In his paper of 1880, Peirce had already given a brief abstract deve lopment of nonstrict partial orders and lattices. Schr6der first defines the reflexivity of a relation as Principle 1 (p. 168) and then defines the transitivity of a relation as Principle 2 (p. 170). He follows Peirce in taking a nonstrict partial order to be a reflexive, transitive relation. (He does not, however, introduce the term "nonstrict partial order"; this is later terminology.) He follows Peirce in defining equality and strict partial o rder in terms of the given nonstrict partial order. Thus, two elements a and b are equal (a = b) if each bears nonstrict partial order to the other, while a is less than b ( a < b) if a bears nonstrict partial order to b but b does not bear nonstrict partial order to a. This is often the way the same material is taught in beginning algebra, so the Peirce-Schr6der convention of taking nonstrict partial orders as basic has, for the most part, become standard notation in mathematics. Schr6der also defines what a least e lement 0 and a greatest e lement 1 are in a partial order (pp. 184, 188). However, the notion of an equivalence relation is absent.

Lecture III.mDefines, given a partial order, a + b as the least upper bound of a and b, which we write a v b, and which Schr6der calls identical sum (p. 196). He defines the product ab as the greatest lower bound of a and b, which we write a ^ b, and calls it identical p roduct (p. 196). He attributes these definitions to Peirce. Schr6der does not coin the word "lattice" for a partial order with least upper and greatest lower bounds for each pair of elements; this was first done by Birkhoff (1933). Earlier, Dedekind (1897) called the same notion a "Dualgruppe" [dual group]. However, Schr6der then deduces lattice laws unde r the assumpt ion that the operations are defined on the partially o rdered set, writing dual results in two columns when the roles of sum and product (least upper bound and greatest lower bound) are reversed. He thus unders tood the duality principle for general lattices very well. He leaves infinite least upper bounds and greatest lower bounds to a later volume; they do not occur here.

Lecture/E.~Interprets the lattice language for the special case of the algebra of classes of Boole.

Lecture V.mDerives many lattice laws from the lattice axioms. Schr6der

~In this section, except where otherwise noted, all page numbers refer to Schr6der (1890), voi. 1.

146 S C H R O D E R ' S CALCULUS OF RELATIVES

warns the reader that no t rea tment of negation for lattices has yet been given by titling this chapter, roughly, "Theorems not involving negation, pure theorems on multiplication and addition" (i.e., greatest lower bound and least upper bound) .

Lecture V/.mDiscusses the question, raised by Peirce (1880), of whether the distributive law is a consequence of the lattice axioms. This law, of course, holds for classes and propositions, but does it hold for all lattices?

Or, in modern notation, is (A A C) v (B A C) = (A V B) A C?. The title of this chapter starts out "Nonprovability of the second inclusion of the distributive law . . . . " T h e inclusion from left to right holds in any lattice; the question is whether the right side is contained in the left in any lattice. From the point of view of Dedekind in mainst ream m a t h e m a t i c s at that time, the answer is "obviously not." Dedekind (1897) had as his first examples of nondistributive lattices the lattice of subgroups of a

group, which is rarely distributive; he knew this much earlier. From the point of view of a modern s tudent of mathematics , the

simplest example of a nondistributive lattice is the lattice of subspaces

of a vector space of a dimension of at least two. The operat ions are then as follows: A v B is the space spanned by the union of A and B;

A A B is the intersection of A and B. If A, B, and C are subspaces spanned

respectively by a, b, and a + b, and a and b are linearly i n d e p e n d e n t

vectors, the distributive law fails because a + b ~ (A v B) A C, but a +

b ~ (A A C) V (BA C). Also known to the modern s tudent is the fact

that a lattice is nondistributive if and only if it has one of the two nondistributive five-element lattices as sublattices. We also know that the distributive law and its dual are equivalent, based on the lattice axioms. This information is presented in Birkhoff's Lattice Theory (1948); we omit his discussion. Schr6der comes back to the Dedekind examples later in his treatise. (See Schr6der 's appendix, Anhang IV, pp. 617-632, for the lattice of subgroups.)

Lecture V/I.EDefines a complemen t of a to be a b such that a v b =

1, a A b = 0. Schr6der shows that in a distributive lattice with 0, 1, if an

e lement has a complement , that complemen t is unique (p. 299). Fur ther on (p. 303) he adds the postulate that every e lement has a complement . This accounts for the title of the lecture, which is roughly "Negation for domains, with a postulate."

After 303 pages, Schr6der finally has the mode rn abstract definition of a Boolean algebra as a distributive lattice with 0, 1 such that every e lement has a unique complemen t (it is assumed that 0 :g: 1). This is almost certainly the first complete axiomatic definition of a Boolean algebra, al though Peirce came close to it in the papers we analyzed earlier.

Lecture VIII.mGives many more theorems about negation (comple-


ment) , with the notation d for the complement of a, as well as several examples in the calculus of classes.

Lecture IX.mGives a series of examples of how to solve sets of logical conditions using Boolean algebra simplifications.

Lecture X.~Studies Boolean polynomials. Schr6der gives Boole's famous law of expansion for Boolean polynomials f (p. 409):

f(x) =f(1) �9 x + f(0) �9 x I,

where x I stands for the negation (complement) of x. He also states (p. 415) the multivariable version, which is equivalent in propositional logic to giving the truth table for proposition f

Lecture X/.mWorks on the elimination problem for finite sets of Bool- ean equations, i.e., the problem of expressing their solvability in terms of their coefficients. First Schr6der conjoins them to get one equation, then he reduces the result to a normal form, either conjunctive or disjunctive. The possible solutions can then be read off from this normal form. In modern language, he is computing the representat ion of an e lement in the finite Boolean algebra generated by all variables and constant coefficients as a sum of atoms and using this representation to give conditions for the solvability of the Boolean equations.

Lecture X/I.--Discusses, in analogy with arithmetic, the possible interpretations of the inverse of addition and multiplication (p. 479).

Lectures XIII and X/V--These lectures give examples and make com- parisons with the work of Peirce and others such as W. StanleyJevons, Hugh Mac Coil, and John Venn.

7.2. D/e Algebra der Logil~ Volume 2

Volume 2 deals with propositional logic and its truth functions, treated as lattice theoretically as possible.

Lecture XV.--This lecture begins with an introduction to and discussion of concepts of propositional logic, but gives no formal system. One suspects the reason for this is that the load usually carried in modern works by the laws of inference and axioms is here carried by the fact that the truth functions form a Boolean algebra. This makes all the theorems on partial orders, lattices with 0, 1, and Boolean algebras from volume 1 available at once. This algebraic apparatus, applied to Boolean algebras of truth functions, is the apparent basis of volume 2.

The most interesting aspect of this lecture is that it is the first ment ion in Schr6der 's treatise of product, II, and sum, E (p. 25). 2 The first is the greatest lower bound and the second is the least upper bound in

In this section, except where otherwise noted, all page numbers refer to Schr6der (1890), vol. 2.

x48 SCHRODER'S CALCULUS OF RELATIVES

the Boolean a lgebra of t ruth funct ions. These symbols are absen t f rom vo lume 1. Today, we would def ine the no t ion of a least u p p e r b o u n d l o f an i n d e x e d family {a~} of e l emen t s of a partially o r d e r e d set (P, <) such that l is the least u p p e r b o u n d of {a~} (1) if for all i, a~_< l, and (2) if p e P and for all i, ai _< p, then l < p. T h e n , if a least u p p e r b o u n d exists, it is unique . But S c h r 6 d e r does not do this.

T h e Boolean algebra 'P(A) of all subsets of a set A is a c o m p l e t e a tomic Boo lean algebra; that is, all i n d e x e d families have greates t lower b o u n d s and least u p p e r bounds . T h e i somorph ic Boolean a lgebra {0, 1 }a of t ru th func t ions on A has the same proper t ies . S c h r 6 d e r makes effective use of this fact. If A = B x B, we get the Boolean a lgebra of t ru th func t ions o f two variables on the d o m a i n B, and this leads to the t r e a t m e n t o f relatives by matrices ( t ruth funct ions) in vo lume 3.

T h e r e is no ques t ion that Sch r6de r ' s II is a greates t lower b o u n d o p e r a t o r over a lattice of t ruth funct ions wheneve r it is used, and no t a universal quant i f ie r in a formal language. Similarly, his ~ is a least u p p e r b o u n d o p e r a t o r and no t an existential quant i f i e r in a formal language . But for a fixed doma in , the classical semantics of these same quant i f ie rs can be de f ined using t ru th funct ions on doma ins and these two ope ra t ions on truth funct ions. This is what S c h r 6 d e r seems to have done , thus bypassing formal languages and thei r syntactic quant i f iers in favor of algebraic ope ra t ions on t ru th funct ions on a fixed doma in .

For instance, on pages 26-27 S c h r 6 d e r in t roduces the ope ra t i ons II and E to r ep re sen t the semant ics of quantif iers; namely, "For every x in ou r d o m a i n " is expressed as II x, or "p roduc t over x," and "For at least o n e x in ou r doma in" is expressed as ~x, or "sum over x."

S c h r 6 d e r also in t roduces IIx.y and E x.y as, respectively, p roduc t s and sums over all x and y. He refers to the variables x and y as p r o d u c t or s u m m a t i o n variables, and uses t ru th funct ions of x and y r ang ing over the domain . He in terpre ts the quant i f iers in the l anguage as corre- s p o n d i n g to these ope ra t ions on t ru th funct ions on a fixed domain . This is ev ident by page 40.

In the discussion that follows, we have chosen to al ter Sch r6de r ' s choice of examples slightly, subst i tut ing a m o d e r n fo rmula of two variables, with y as parameter , for his fo rmula in one variable, which leads to a less clear illustration.

Fixing the p a r a m e t e r y, S c h r 6 d e r takes sums over all ins tant ia t ions of the variable x in the fo rmula 4~(x, y) by names of e l emen t s o f a doma in , and if f(x,y) is the c o r r e s p o n d i n g t ruth func t ion on the doma in , he represen t s it by ~j(x,y), m e a n i n g the least u p p e r b o u n d of all values o f f ( a , y ) as a ranges over the d o m a i n (where 1 represen ts t ru th and 0 falsity). Thus , he obtains a t ru th func t ion of one variable, y, over the same doma in , which co r r e sponds to (3x)rb(x,y). This makes pe r fec t sense, since the set of all two-variable t ruth funct ions on a fixed d o m a i n

FROM PE IR C E TO SKOLEM 149

is indeed a complete distributive lattice; it has least upper bounds and greatest lower bounds for arbitrary subsets.

To repeat, the semantics of quantifiers have been identified with operations on truth functions on the domain; infinite products and sums (greatest lower bounds and least upper bounds of infinite sets of truth functions) are used to define the truth function that results from the truth function for a formula when a variable of that formula is quantified. But no general concept of formula emerges. An arguable interpretat ion of the material in Lecture XV is that it is an exposition of the algebra of truth functions of one or more variables on a fixed domain that uses infinite least upper bounds and greatest lower bounds to interpret quantifiers for that domain.

It is also interesting to note that the program of volume 1 to define everything abstractly has not been continued. Namely, the general concept of least upper bound and greatest lower bound of arbitrary subsets of a partially ordered set do not emerge. These operat ions are used only for a particular kind of lattice, truth functions or power sets, where it is obvious that they can always be performed. In 1872 Dedekind published a clear exposition of greatest lower bounds and least upper bounds in his construction of the real numbers. Schr6der, however, did not follow up on Dedekind's work and define the not ion abstractly after volume 1, al though he had read the relevant papers of Dedekind.

Lecture XVI.mVerifies many rules of the propositional calculus directly for the truth functions.

Lecture XV/I.mConsiders the meaning of syllogistic in Boolean terms, using Boole's ideas and other notions. No use is made of I] and II for interpret ing quantifiers. This is rather a discussion of the content of older work that looked at particulars and generals in Boolean algebra terms.

Lecture XVIII.mThe lion's share of the attention in this lecture is devoted to the reduction and enumera t ion of Boolean functions. This follows previous work by others, including Peano. It is not clear why this is of any logical interest, a l though Howard Aiken found it worth- while to publish tables of Boolean functions fifty years later at Harvard in connect ion with switching circuits. Much of the remainder of volume 2 is devoted to summaries of the work on propositional logic by others such as Mitchell, Ladd Franklin, and Mac Coil, and on the condit ions of solvability of propositional logic problems.

7.3. D/e Algebra der Logil~ Volume 3

Lecture/.reintroduces relatives as binary relations on a fixed domain. Schr6der discusses ordered pairs, written (a :b) instead of (a, b), and

1 5 0 SCHRODER'S CALCULUS OF RELATIVES

relatives as sums (least upper bounds) of ordered pairs. Nowadays we would say that the relative is the union of singleton sets {(a: b)}, not of pairs (a: b). But it matters little, since we have isomorphic complete distributive Boolean algebras, generated by either as atoms. In a review in 1895, Frege did not understand that this was the Boolean algebra in which Schr6der was working. He criticized Schr6der for confusing objects and their unit sets, which is not correct. Rather, Frege read in the wrong (isomorphic) Boolean algebra. (An English translation of this lecture is given in Appendix 1.)

Lecture//.--Follows Peirce in defining the operations of the calculus of relatives. Relatives have six operations: the three Boolean operations of addition, multiplication, and negation, and three operations of their own, viz., relative product, relative sum, and converse. Schr6der introduces the two-valued matrix of a relative over a temporary universe set 1, another language for a truth function defined on pairs from the set. Hence, all theorems from volume 1 on Boolean algebras apply, since the matrices (truth functions) for the relatives form a Boolean algebra. There is a perfect correspondence between the Boolean relative operations and the Boolean operations on matrices, that is, between relative product and matrix product, and between converse and transpose.

In Schr6der's treatment of sums and products, which we know to be least upper bounds and greatest lower bounds from volume 1, he gives the two defining formulas

where u is an indefinite (variable) relative and f is a function in the algebra of binary relatives (p. 36). :~ Schr6der says that he will use the schema he has given "predominantly, if not exclusively" for individuals as well as relatives (p. 41). He further says that when sums and products range over individuals, he will write the indices to ~ and II as subscripts to the right of the symbols instead of beneath them. Thus, these are operations in the complete atomic Boolean algebra generated by individuals, and also in the complete Boolean algebra generated by ordered pairs of individuals. Thus, he really did know what Boolean algebra he was working in. (An English translation of part of Schr6der's Lecture II is given in Appendix 2.)

Lectures III and /E .mThese lectures are devoted to the mechanics of many algebraic identities of the calculus of relatives.

Throughout volume 3, Schr6der translates formulas from the pure calculus of relatives into their relative coefficient form in proving algebraic identities. He calls this "giving the coefficient evidence" (p. 65)

~ In this section, except where otherwise noted, all page numbers refer to Schr6der (1895), vol. 3.

FROM PEIRCE TO SKOLEM 15 1

and justifies it by fundamental stipulations (5)-(14) from Lecture II (pp. 22-32). Peirce first gave these formulas in his 1883 paper, in which the formulas were matrix-theoretic representations of the basic operations of the relative calculus. By 1885 Peirce was interpret ing these formulas as expressions of the Boolean and relative operat ions in first- order predicate logic. Schr6der seems to adhere to calculus of relatives as presented in Peirce's 1883 paper and interprets any identity in the relative calculus as at once representing on the right- and left-hand sides binary relatives and propositions.

Once Schr6der has translated a formula from the calculus of binary relatives into its relative coefficient form, he appeals to the laws of proposit ional logic ( A u s s a g e n k a l u l ) to justify algebraic manipulat ions on the Boolean part, and then translates the result back into the calculus of relatives. He says that he considers the coefficient evidence as "exclusively valid" in his theory, and that a theorem in the algebra of relatives cannot be accepted as certain unless it has been proved in this way (p. 65).

Lecture III, section 7, "Proofs of the Basic Laws," provides many examples of this method. Schr6der 's proof of the distributive law, a; (b + c) = (a; b + a; c), is one example. First, he translates the left-hand side of the formula into coefficient form and then, regarding {a;(b + c)}0 as a proposition, performs a series of transformations that he justifies by appeal to his fundamental stipulations laid down in Lecture II"

{a; (b + c)}~) = I]ha~h(b + c)j0 by stipulation (12) [(a; b)q = I;ha~hbhi];

= Ehaih(bh~ + Chj) by stipulation (10) [(a + b) o = a o + bo];

= Eh(a~hbhj + a~h%) by distributive law for propositions;

= F, ha~hbhj + Eha~hchj by distributivity of sum signs;

= (a; b)o + (a; c) o by stipulation (12);

= (a ; b + a ; c) !i by stipulation (10).

Applying stipulation (14), which states that two relatives are equal if and only if they agree on their corresponding coefficients-- that is, (a = b) = I Io(a O = bi i ) - - -gives a; (b + c) = (a; b + a; c) and completes the proof.

Peirce notes in his review of Schr6der's volume 3 that Schr6der uses Peirce's "general logic" in Schr6der's development of the calculus of binary relations, but Peirce does not say whether he believes Schr6der to understand that Peirce's general logic is (first-order) predicate logic:

My general algebra of logic (which is not that algebra of dual relations, likewise mine, which Professor Schr6der prefers, although in his last volume he often uses this general algebra) consists in simply attaching


indices to the letters of an expression in the Boolian [sic] algebra, making what I term a Boolian, and prefixing to this a series of"quantifiers," which are the letters II and Z, each with an index attached to it. Such a quantifier signifies that every individual of the universe is to be substituted for the index the rI or E carries, and that the nonrelative product or aggregate of the results is to be taken. (Peirce 1896-1897, pp. 282-283) 4

A m o d e r n ma themat i c i an would say that Schr6der is t ransla t ing the

re la t ional t h e o r e m to be proved into its ma themat ica l defini t ion, and then p r o c e e d i n g by ord inary ma themat ica l reasoning to derive the desired t heo rem. However, the ord inary ma themat ica l der ivat ion that re-

suits is a derivat ion by the algebraic rules of E and II f rom Peirce 's (first-

o rde r ) p red ica te calculus. (An English t ranslat ion of par t of Schr6der ' s

Lec ture III is given in Append ix 3.)

Lecture V.mFormulates the not ion of a genera l solut ion x - f ( u ) to a relative equa t ion F(x) = 0. Schr6der is here consciously imi ta t ing the

l anguage in a lgebra of a genera l solut ion to an algebraic equa t ion that will have the coefficients as parameters . What he wants is a (mult ivalued)

f i lnct ion f(u), expressed in his calculus, the range of which is the set of

values x such that F(x) = 0. Such an f is a relative and may be empty. He does discuss the fact that the value of f (u) is no t relevant; any choice

will do. He can use a binary relat ion instead of a funct ion to allow all

these values. What is i n t ended is that f should be a funct ion variable r ang ing over all funct ions fl such that for all u for which fl (u) is def ined, F(fl(u)) = 0. Schr6der is thus using a function variable, f, rang ing over funct ions with values relatives, but rang ing over only those funct ions f such that F(fl(u)) = 0 is identically satisfied (and over all of these funct ions) .

Sch r6de r also restricts the te rm "el iminat ion theory" to the case in which f(u) is descr ibed by a te rm built up f rom all the relat ion constants and variables o the r than x occur r ing in F. These are the relative symbols that are to be r ega rded as coefficients and pa rame te r s when the relative

symbol x is r ega rded as an unknown. He then substi tutes F(f(u)), where

f is a variable funct ion symbol with range as above. The p r o b l e m is then

r e d u c e d to solving F(f(u)) = 0, where x has been e l iminated . But u is

essentially all the relat ion variables in F o t h e r than x. Take one of them, y. Repea t the process, ge t t ing g(z), where z is a re lat ion variable o the r

than x and y, and g(z) is a genera l solut ion to F(f(g(z))) = 0 involving

only variables o the r than x and y. Con t inue until all variables are re- p laced by f, g, h . . . . . T h e n we get a relative equality with no rela t ion variables. It now is built up f rom f g, ... and the original relative con-

4 All page citations from this work are from the CoUected Papers of Charles Sanders Peirce (Hartshorne and Weiss 1933).


stants. With the range off , g, ... over f~, g~ . . . . now accep ted , the or iginal e q u a t i o n has a solut ion if and only if this final equa t i on with variables f g, ... has solut ions f~,g~ . . . . over the ranges of these variables as def ined .

W h e n r e g a r d e d as a func t ion of the p a r a m e t e r relatives in the express ion for F, this comes very close to i n t r o d u c i n g a Skolem funct ion ,

f(u), with value relative x, with the relatives in Fas pa ramete r s . It is not,

however, a Skolem func t ion over the d o m a i n of individuals. (An English

t rans la t ion of Schr6der ' s Lec ture V is given in A p p e n d i x 4.)

7.3.1 Peirce's Attack on the General Solutions of Schr6der

The no t ion of a genera l solut ion f was round ly a t tacked by Peirce in sections 10-12 of "Exact logic," his review of vo lume 3 of Schr6de r ' s

Algebra der Log~k (Peirce 1896-1897, pp. 320-326) .

Peirce says that Schr6der ' s concep t ion of a solution to a relative equa-

tion (or a first- or h igher o rde r logic s ta tement , for that mat ter ) is silly:

The general problem, according to [Schr6der], is "Given the proposition Fx=O, required the 'value' of x0," that is, an expression not containing x which can be equated to x. This "value" must be the "general root," that is, it must, under one general description, cover every possible object which fulfills a given condition. This, by the way, is the simplest explanation of what Schr6der means by a "solution problem"; it is the problem to find that form of relative which necessarily fulfills a given condition and in which every relative that fulfills that condition can be expressed. Schr6der shows that the solution of such a problem can be put into the form (F., x =fu), which means that a suitable logical function (f) of any relative, u, no matter what, will satisfy the condition Fx =0; and that nothing which is not equivalent to such a function will satisfy that condition. He further shows what is very significant, that the solution may be required to satisfy the "adventitious condition" fx--x. This fact about the adventitious condition is all that prevents me from rating the value of the whole discussion as far from high. (Peirce 1896-1897, p. 325)

Peirce is he re re fe r r ing to Schr6de r ' s discussion in Lec tu re V of vo lume

3 (pp. 161-165) . He says that what Sch r6de r calls a so lu t ion would be

ana logous to an algebrais t i n t roduc ing a formal func t ion of the coef-

ficients of a f i f th-degree algebraic equa t ion with l ead ing coeff ic ient 1

by simply saying that it chooses a solut ion to the equa t i on with the given

coefficients. In his op in ion , this empty exercise is jus t what S c h r 6 d e r is

doing:

Let us see how this "rigorous solution" would stand the climate of

154 S C H R O D E R ' S C A L C U L U S OF R E L A T I V E S

numerical algebra. What should we say of a man who professed to give a rigorous solution of algebraic equations of every degree (a problem included, of course, under Professor Schr6der's general problem)? Take the equation x ~ + Ax 4 + Bx :~ + Cx 2 + Dx + E=0. Mul- tiplying by x - a we get

x 6 + ( A - a)x 5 + ( B - aA)x 4 + ( C - aB)x :~

+ ( D - aC)x 2 q- ( E - a D ) x - aE=O.

The roots of this equation are precisely the same as those of the proposed quintic together with the additional root x = a. Hence, if we solve the sextic we thereby solve the quintic. Now our Schr6derian solver would say, "There is a certain funct ion , f u, every value of which, no matter what be the value of the variable, is a root of the sextic." And this function is formed by a direct operation. Namely, for all values of u which satisfy the equation

u6+ ( A - a)u "~ + ( B - aA)u 4 + ( C - aB)u :~

+ ( D - aC)u 2 + ( E - a D ) u - a E = O

f u = u, while for all other values, f u = a. Then, x = f u is the expression of every root of sextic and of nothing else. It is safe to say that Professor Schr6der would pronounce a pre tender to algebraical power who should talk in that fashion to be a proper subject for surveillance if not for confinement in an asylum. Yet he would only be applying Professor Schr6der's "rigorous solution," neither more nor less. (Peirce 1896--1897, p. 326)

We n o t e tha t in the e x a m p l e tha t Pe i rce gives to show tha t S c h r 6 d e r

is silly he is actually p r o d u c i n g a Sko lem func t ion , a n d so was Schr6der .

T h a t is, Pe i rce says tha t what S c h r 6 d e r is d o i n g is like tak ing the universa l

ex is ten t ia l s t a t emen t ,

For all c o m p l e x a0 . . . . . a 5, t h e r e exists a c o m p l e x x such tha t x ~ +

ao x4 + .. . + a 5 = O,

a n d i n t r o d u c i n g a func t i on symbol f ( a o . . . . . a s ) a n d the s t a t e m e n t ,

For all c o m p l e x a 0 . . . . . a~, (f(a,, . . . . . as)) 5 + a o ( f ( a , , . . . . . as)) 4 + ... +

a 5 = O,

a n d cal l ing this solving the e q u a t i o n .

Thus , Pe i rce is a t t ack ing S c h r 6 d e r for i n t r o d u c i n g the first f o rma l

S k o l e m func t ions , in the guise o f re la t ions . S c h r 6 d e r is saying tha t o n e

can i n t r o d u c e func t ion (really re la t ion) symbols in a p r e n e x f o r m u l a

such tha t the or ig ina l relative e q u a t i o n is satisfiability equ iva l en t to the

universa l quan t i f i ca t ion of the quant i f i e r - f ree f o r m u l a o b t a i n e d by put-

t ing in such func t i on symbols. S c h r 6 d e r was d o i n g this for the relat ive


calculus, not first-order logic, but he coded elements of the domain as relatives, so this is a small difference.

Peirce had no insight that it might be impor tant to be able to modify a s tatement by finding a simpler equisatisfiable s tatement as a step toward unders tanding whether and when the original s ta tement is satisfiable. But we can see that Schr6der 's was a great, and generally un- recognized, step forward toward the Skolem-L6wenheim theorem and modern model theory. We quote this commentary from Peirce's review not to devalue Peirce, but to point out that our interpretat ion of Schr6der as having int roduced Skolem functions for equisatisfiability is also Peirce's contemporary reading of Schr6der, a l though Peirce did not recognize what he was seeing. Later, L6wenheim took his notat ion from the same passages of Schr6der. This notation is used nowhere else by anyone who has been remembered by history. Assuming that he read what Peirce read, we can plausibly surmise that L6wenheim came to a very different conclusion about the usefulness of finding simpler equisatisfiable statements. This provides a concrete link between Schr6der 's solution and elimination method and L6wenheim's fundamenta l argument.

7.3.2. Lectures VI-X and Dedekind Chain Theory

Lecture VI is devoted to massive identities in the calculus of relatives, Lecture VII to questions of inverses to the fundamental operations, and Lecture VIII to the simple problems of f n d i n g general solutions.

Lectures IX and X develop Dedekind's theory of chains entirely in the language of relatives. Schr6der thinks of relations as generalized functions and generalizes what Dedekind did, analyzing the basis of proofs by induction on the integers. The applications of this theory are to justify inductive definitions with the domain the nonnegative integers.

Lecture IX on chains is the culmination of Schr6der 's three-volume work. This is the most subtle part of mathematics he carried out in the second-intentional relative calculus. Since none of Schr6der 's Algebra der Logik has been available to English language readers and this is its high point, a complete translation of Lecture IX is provided in Appendix 5. Here we sketch the historical background.

The earliest axiomatic development of the theory of integers we know of is that of Herman Grassmann (1861). Grassmann started out by assuming that we are given operations of addition and multiplication, satisfying the identities

15 6 S C H R O D E R ' S C A L C U L U S OF RELATIVES

x + O = x,

x + (y+ 1)= (x+ y) + 1, xO = O,

x( y + 1) - xy + x,

with suitable axioms for 0, 1, the induction axiom, etc. Assuming these, Grassmann proved by induction the various identities

for integers, such as the associative, commutative, and distributive laws. One can certainly proceed in this way, taking the operat ions of plus and times as given and taking the identities listed above as axioms. (Indeed, when number theory is developed in first-order logic, there is little choice in the matter; see Kleene's Introduction to Metamathematics [1952] for a discussion of this development.)

But does this mean that one must add axioms every timea new function, such as exponent iat ion, is introduced? For example,

x ~ = 1,

~,+ 1 x = x ( x " ) .

Are we to assume that exponent ia t ion is given, and that these are merely axioms about this given, from which fur ther propert ies of exponentiation are derived? Do we then need to repeat this for each new function? In that case, ari thmetic would be based on a constantly increasing set of axioms.

The problem is that such a set of equations are not explicit definitions. In an explicit definition, the term defined occurs on the left side of the definition but does not occur on the right side. That is, explicit definitions do not define something in terms of itself, since to do so is circular.

The identities for plus and times, as well as exponent ia t ion, are ar- chetypal circular definitions; that is, they are not definitions at all. They can be taken as axioms, or they can be taken as identities that need to be proved. Considering that there are many functions one wants to have in n u m b e r theory and that it is not very satisfactory to add new identities for them as axioms whenever a new function is required, it is clearly impor tan t to see whether these identities, which look like circular definitions, can be replaced by noncircular definitions.

The first person to publish the device needed to do this was Gottlob Frege, in his Begriffsschrift (1879, w 3). This work contains his theory of finite sequences, based on his definition of integer, e m b e d d e d as the third section of the seminal paper that in t roduced full quantif ier logic for the first time. But the "concept notation" Frege in t roduced for quantifier logic is difficult to read, and his work, except for some reviews, appears to have been completely neglected until Russell revived interest


in it in the late 1890s. In 1879 Frege did not emphasize justifying inductive definitions of functions, but he did int roduce the apparatus required for such a treatment. In his Grundgesetze, volumes 1 (1893) and 2 (1903), he takes inductive definitions up in more detail. Note that this is also the period in which Schr6der wrote his Algebra der Logik.

The work that, i ndependen t of Frege, finally in t roduced exactly the right notion and used it to give a more fundamenta l definit ion of functions on the integers was that of Dedekind (1888). To repeat, except for the neglected paper of Frege, before Dedekind 's m o n o g r a p h it was not unders tood that the principle of proof by induct ion does not im-

mediately justify definition of functions by induction, because the latter are simply circular definitions until a device for breaking the circle is supplied. That device was present in Frege (1879) and was rediscovered

and fully exploited in an explicit way by Dedekind in 1888. Dedekind finally broke the circle and gave explicit definitions of plus, times, etc.,

using his concept of chain. To put Dedekind 's work in the broader context of the evolution of

the foundat ions of mathematics, we note that Dedekind, following in

the footsteps of one of his mentors, Karl Weierstrass, emphasized building exact set-theoretic definitions of mathematical concepts. In a series of lectures in 1858, Weierstrass emphasized the construct ion of the real numbers from the rational numbers, and the rational numbers from the integers. Dedekind defined ideals set theoretically, a great advance over Kronecker, and defined the real numbers set theoretically as cuts, a simpler definition than Weierstrass's or Heine 's or Cantor 's . In his 1888 paper on the foundations of the integers, Dedekind focused on defining the integers themselves set theoretically. Thus, he starts with the definition of a finite set as a set in the smallest collection of sets containing the null set and closed under the operat ion of adding one

e lement to any set in the collection. Dedekind saw that all mathematical objects could be constructed by

set operat ions from simpler sets. In this he preceded Cantor, who succeeded in giving an intuitive set theory in which all mathemat ica l constructions are set constructions. This concept ion of set was axiomatized for the first time by Ernst Zermelo (1908), and for the first time in a

satisfactory way by Thora l f Skolem (1923) and A. A. Fraenkel (1922). Peano's development of foundations (1888-1889) was an a t tempt to

formulate mathematical systems on a set-theoretic basis. He cited Grass- mann, but did not recognize the difficulty with inductive definitions,

so far as one can see, because he used definitions of functions by induct ion freely, without breaking the circle in their definition. He simply wrote down the identities that would uniquely characterize the functions and then assumed there were functions satisfying these identities, ba-


sically as axioms. For example, in his Arithmetices principia, Peano defines addition

Definition a,b, e N. D .a+ (b+ 1) = (a+ b) + 1.

(18.)

Note. This definition has to be read as follows: if a and b are numbers, and if (a+ b) + 1 has a meaning (that is, if a+ b is a number) but a + (b + 1) has not yet been defined, then a + (b + 1) means the number that follows a + b. (Peano 1889, p. 95)

We believe that Schr6der was the first person, o ther than Dedekind, to justify definitions of functions on the integers by induction. Schr6der 's Lecture IX is the first publication on the subject after De- dekind. He makes Dedekind 's justification of inductive definitions by the method of chains the focal point of the lecture. Schr6der translates Dedekind 's set-theoretic t rea tment of chains line-by-line into the second- intentional calculus of relatives. With this, Schr6der shows that the second-intentional theory of relatives is sufficient to develop n u m b e r t~heory.

Frege's Grundgesetze (1893, 1903) also contains a t rea tment of definitions by induction, following the lines he had laid out earlier (Frege 1879), but this was unapprecia ted until the time of Russell, when definition by induction resurfaces in Whi tehead and Russell's Principia Mathematica, volume 90 (1913, p. 81 ff).

How and where is the problem of definition by induction handled in the modern set-theoretic foundat ion of mathematics in first-order logic? It is buried as a special case of the theorem on justifying definitions by transfinite induction on the ordinal numbers, as the special case in which the ordinal numbers are limited to the integers. The first place this t rea tment appears is in John von N e u m a n n (1923). Von N e u m a n n (1923, 1928)justified definitions by induction on the transfinite ordinal numbers , thus complet ing the foundations of Cantor 's work on set theory along Dedekind 's lines. Von Neumann ' s justification of proofs by transfinite induction is a simple extension of Dedekind 's work from the integers to the transfinite ordinal numbers, exactly generalizing De- dekind 's chains. This t rea tment also appeared in G6del 's monograph , Consistency of the Axiom of Choice and the Continuum Hypothesis with the Axioms of Set Theory (1940). It is the basis of G6del 's t rea tment of arithmetic, finite and transfinite.

Justifying definitions of functions by induction is indeed a subtle point. As late as the 1920s, as eminen t and deductively precise a n u m b e r theorist as Edmund Landau admits, in the introduct ion to volume 1 of


his Vorlesungen iiber Zahlentheorie (1927), that this gap had to be po in t ed out to h im by his s tuden t Grandjot , who discovered it anew wi thout r e fe rence to Dedekind . At first Landau threw Grandjo t ou t of his office because he could not see the gap!

The Pei rce-Schr6der t h e m e that h igher in ten t iona l relative calculus can be a full founda t ion for mathemat ics recurs twice in later mathematical history. First, L6wenhe im (1940) m a d e the claim that the relative calculus was jus t as suitable for a founda t ion of ma thema t i c s as set theory. Second, the t h e m e of Set Theory without Variables of Tarski and Givant

(1987) is that a form of binary relat ion calculus is a d e q u a t e as a foun-

da t ion for all of mathemat ics , and uses no variables.

In this respect Schr6der was highly sophis t icated c o m p a r e d to o thers

of the time. In contrast , Peirce did not seem to u n d e r s t a n d the necessity of justifying defini t ions by induct ion. He does not make any just i f icat ion

in any of his papers , simply using defini t ions by induc t ion as if they requi re no just if ication. But by 1903, wi thout even m e n t i o n i n g definitions by induct ion, Peirce begrudgingly acknowledges Dedek ind ' s chain

theory in the con tex t of discussing Schr6der:

The nearest approach to a logical analysis of mathematical reasoning that has ever been made was Schr6der's statement, with improve- ments, in a logical algebra of" my invention, of Dedekind's reasoning (itself in a sort of logical form) concerning the foundations of arithmetic. But though this relates only to an exceptionally simple kind of mathematics, my opinion---quite against my natural leanings toward my own creation--is that the soul of the reasoning has even here not been caught in the logical net. (Peirce 1903a, p. 344)

As late as 1905, in "Analysis of some demons t r a t ions c o n c e r n i n g defini te positive integers," Peirce acts as if addi t ion were obviously def ined by its recurs ion equat ions (Peirce 1905b, p. 282). He still seems not to have absorbed what Dedek ind did. Schr6der , on the o t h e r hand , seems to have u n d e r s t o o d Dedek ind ' s a r g u m e n t exactly:

Although some of these propositions may occasionally be used later, the main purpose of stating them, and for us here the only purpose of listing them ... is to prepare and make possible the proof of the proposition of complete induction ~59, which contains no circular argument.

...It goes to Mr. Dedekind's credit to be the first to have stripped the proof procedure, widely used and known by the name of "inference from n to n + 1," of its arithmetic additions, to have peeled out its logical core, and to have formulated the "proposition of complete induction" as a proposition of general logic, which can be represented

x 6 0 SCHRODER'S CALCULUS OF RELATIVES

and understood independent of any number concepts and even before the series of numbers is introduced. (Schr6der 1895, p. 355)

In sum, Frege was ahead of Peirce and Peano, and also Dedekind , in

giving a full t r e a tmen t of the integers and logic in 1879. Section 3 (pp.

55-82) of his Begriffsschriftsets forth the theory of finite sequences , which is equivalent to Dedekind ' s chain theory, a l though Frege lays out this

theory within his own logical system, as it was expressed in the previous two chapters of his book. But in 1879, nine years ear l ier than Dedek ind (1888), Frege thus has publ ishing priority for the appara tus of justifi- cat ion of defini t ions by induct ion.

Frege (1879), however, does not seem to state Dedek ind ' s basic the-

o rem, even if he has the appara tus for proving it; namely, that if f is a funct ion on A to A and w is the integers, then there is a funct ion g : w x A to A such that

g(O, a) = f(a),

g(n + 1, a) = f(g(n, a)),

for all n in w.

This is usually written J"(a) = g(n, a); i.e., the def ini t ion of i tera t ion of

f This is the pr inciple that gives all the ar i thmet ic funct ions that we usually use, and in par t icular the primitive recursive functions.

Frege ' s ant ic ipat ion of chains is acknowledged by Dedek ind in a le t ter to Keferstein (Dedekind 1890b):

Frege's Begriffsschrift and Grundlagen der Arithmetik came into my pos- session for the first time for a brief period last summer (1889), and I noted with pleasure that his way of defining the nonimmediate succession of an element upon another in a sequence agrees in essence with my notion of chain. (quoted in van Heijenoort 1967, p. 101)

7.3.3. Lectures X I -XI I and Higher Order Logic

Lecture XI begins with a study of the algebraic rules for sums and

produc ts over a doma in or over the pairs f rom that domain . Schr6der

goes on to treat many algebraic rules govern ing sums and products . He points out that the n u m b e r of terms in a sum or p roduc t can be uncountable :

The method would be to operate with infinite (or unlimited) multiple products I-I, even with one whose H-sign could possibly form a continuum (in case we would write it down in detail); for example, if we assign to each point of the line a II corresponding to some product


variable specifically chosen. For such products and sums we may also without hesitation transfer and apply the inference rules which are gua ran teed to us by the proposi t ion scheme, based on the dictum de om~'lio

This is probably the first time in mathematics that this is done. I will therefore guide the s tudent heuristically along the path on which the m e t h o d first occur red to me. (Schr6der 1895, p. 512) 5

161

H e o u t l i n e s his n e w i d e a as fol lows:

If we have a Ei of a I-I,, of a general term f( i , m), and we wish for some reason to push the E beh ind the II in an equivalent transformation, this is not immediately possible.

Because of I2II:(=HE, we could only do so by drawing weakened

conc lus ionsmif we would be satisfied with such a p rocedure . Other-

wise, no th ing hinders us from renaming the index of the II,,, in all

the o the r terms of the E;, that is, "to differentiate" all these indices as m, (m with the suffix i o t a )mwhereby we only have to r e m e m b e r that L changes in "parallel" with i.

This seems to suggest taking i itself instead of L as a suffix for m. Disregarding the fact that m i already has a fixed mean ing as relative coefficient of the e l emen t m in w 27, it still would not be correct . As we will soon s eemin case we s u c c e e d ~ w e may not choose for t a symbol which contains the name /----such as ~0(i).

This will have the advantage that we can now push each single II

ranging over an m, to the front, in front of our ~. We can now justity

the impor tan t formula

E,H,,,f(i, m) = E,H,, , f ( i , m,) = H,(II,,,)Eff(i, m,),

H , ~ . , f ( i , m) = H,~, , , , f ( i , m,) = m,~ , 39)

by which we have at tained our goal of having pushed all II 's in f ront of the E's. (pp. 513-514)

S c h r 6 d e r e x p l a i n s his m y s t e r i o u s l-I, o p e r a t o r as fol lows:

If t (in parallel with i) has to run th rough a series of values,

~,2,:~ . . . . . we could explain the mean ing of the mysterious ope ra to r in

f ront o f t h e last E; in 39) by writing it in the ordinary way, explici t lymby not men t ion ing the general term or factor, in the form of

H , ( H . , , ) = H.,IH.,2II.,3"'" or H.,1.,~.,~... = HH,,.,

and define it as p roduc t symbol for a (possibly unl imited) "mult iple product ." And then

~ In this section, except where otherwise noted, all page numbers refer to Schr6der (1895), vol. 3.

x62 SCHRC)DER'S CALCULUS OF RELATIVES

II,(E,,,) = E,,,Em2E,,3"'" or F',,,~,,2,,~... = ~;n,,,,

would be nothing but the summation symbol to indicate a "multiple sum." (p. 514)

S c h r 6 d e r is h e r e quan t i fy ing over all func t ions in the sense tha t m I is

all poss ib le values the fu n c t i o n can take o n 1, m 2 is all poss ib le values the f u n c t i o n can take on 2, etc., a n d he explici t ly says tha t L is b o u n d to i, so m, is the ju s t the c o r r e s p o n d i n g value d e p e n d i n g on i, w h e r e i

r anges over the in tegers a n d L r anges over the same d o m a i n tha t i r anges

over.

T h e var iable i can r an g e over any d o m a i n . In S c h r 6 d e r ' s e x a m p l e , i

a n d t can r a n g e over the c o n t i n u u m :

If the t i n parallel with i has to run through a cont inuum of values, such as all the points of a line, we can no longer write the meaning of II,(II,,,) explicitly. Arithmetic allows us, however, to name them all and differently by assigning to each of those points a real number from an interval. For example, we could let m, be the number corresponding to point t. (p. 515)

Thus , S c h r 6 d e r cou ld have the s t a t e m e n t "for all real n u m b e r s , some-

t h i n g is t rue ." Sums a n d p r o d u c t s h a d always b e e n t aken over d i sc re te

d o m a i n s ; he is h e r e tal~ing a p r o d u c t over a c o n t i n u u m of values. We m a k e special n o t e o f the fact tha t in f o r m u l a 39), S c h r 6 d e r is

sugges t i ng s o m e t h i n g very like a Sko lem func t ion , twenty-five years be-

fore Sko lem. In m o d e r n no t a t i on , the first e q u i v a l e n c e tha t S c h r 6 d e r

asserts in f o r m u l a 39) is

(3x)(u ) = (vf)(3x),p(x, f(x)).

This equ iva lence , in fact, holds: if t h e r e is an x such tha t for all y

,p(x, y) is t rue , t h e n for all f take tha t very s ame x; t h e n s ince ~0(x, y) is

t rue for all y, it will be t rue for y =f(x) . Conversely, we a s sume for all

f we have an x such tha t ,p(x,f(x)) holds. Why does t h e r e have to be a

s ingle x? S u p p o s e tha t this were n o t t rue. T h e n for all x t he r e is a y

such tha t r is false. T h e n we c o u l d de f ine a func t ion , i.e., for all

x t h e r e is a y for which ~0(x,y) is false. If we m a p this x to this y, tha t

will be f, a n d for this f t he re will no t be a s ingle x for which r T h e s e c o n d equ iva l ence in f o r m u l a (39) says tha t t h e r e exists a S k o l e m

func t i on ,

(Vx)(3y),p(x, y) = (~lf)(Vx),p(x, f(x)).

( S c h r 6 d e r does n o t have the Sko lem f u n c t i o n exactly. H e can say t h e r e exists m~, t h e r e exists m2, etc., bu t the t rue dua l fails: he c a n n o t swap

"for all" a n d "exists" because he does n o t say tha t t h e r e exists a func t ion ;


instead, he says that for all t there is a function value: i.e., he writes II, Emf(i,m) = II, E,,,f(i, m,) = II,(Em,)II,f(i, m,). He thus completely succeeds only in the first case, which is exactly the reverse of the Skolem function.) Schr6der did not introduce a new function symbol ex tending the language, as did Skolem in his later papers, but went to second~ order logic with a quantifier ranging over functions on the domain to the domain.

In mode rn textbooks there are two first-order predicate languages, one with the function symbol and one without it. The s ta tement (Vx)(3y)~,(x, y) is satisfiable in some model (of the language without the function symbol) if and only if the s ta tement (Vx)(~,(x,f(x)) is satisfiable in some model (of the language with the function symbol). To get from the first to the second and choose an interpretat ion o f f usually requires the axiom of choice. What Schr6der did instead was to go to second- order predicate logic over the same domain by allowing a variable F ranging over all functions on the domain to the domain. This is familiar from the classical and effective descriptive set theory of Nicolas Lusin and of Stephen Kleene, but it seems to be new with Schr6der and perhaps Peirce. That is, a function variable occurs in Schr6der ' s formulation, and a function constant in the modern version.

For Schr6der, the first-order s ta tement (u is satisfiable in some model if and only if the second-order s ta tement (3 a function F ) (u y)) holds. In both cases the equivalence is in second-order logic. In the first case, the statements asserted to be equivalent are both first order. In the second, one is first order and the o ther is second order. The equivalence is a s ta tement of what we now call semantics, in second- order logic. The points are subtle, as attested by the fact that in the late 1930s Tarski had to explain to Rudolf Carnap the distinction between syntax and semantics after he gave a formal semantics of truth for the first time, and Carnap had difficulty in seeing the source of the difference.

L6wenheim's paper uses precisely this form for put t ing together the countable model; that is, he forms the countable model and the required functions on the countable model in accord with the second equivalence, for arbitrary prenex statements of predicate logic.

Skolem begins his paper of 1920 in much the same way, but in his later papers gradually distills the extension of predicate logic by function symbols, again using it only semantically to obtain the Skolem-L6wen- heim theorem. It was left to He rb rand and G6del to work with syntax and get the completeness theorem for first-order logic.

In the very last part of Lecture XI, Schr6der outlines several methods for el iminat ing quantifiers. Schr6der 's technique of multiplying the gen-

t I eral term of a E h by (ljq + 0hi) is later used by L6wenheim to construct a binary tree of solutions of a first-order equat ion in the proof of his

16 4 S C H R O D E R ' S CALCULUS OF RELATIVES

famous theorem. Schr6der 's formula 111) is a theorem eliminating H, u

using a notat ion that L6wenheim employs, in a slightly modified form, in his 1915 paper:

II( u

II(E 111)

(p. 545) The sums I2~ and Ex range over any sequence of suffix values, such

as 1,2, 3, .... The II signs represent quantification over function spaces. In formula 116) (p. 551), Schr6der computes condensat ion (the elim-

ination of quantifiers) using the formula (i; a;j)hk, which stands in its own right as a binary relation, iaj(h,k), where juxtaposi t ion is composition. But then, allowing the domain of all binary relatives as a new domain, Schr6der gets that a binary relation r between binary relations i, j (with a fixed) by r 0 is (i; a ;j). Because this relation is de te rmined by a, Schr6der writes r O (a relation between relations on domain D de te rmined by relation a on D) as a o. Thus, if he allows the domain of all possible binary relations on the original domain, he gets this relation r (or a~j), and it is ttiis that Schr6der uses on page 551 to eliminate quantifiers. L6wenheim (1915) objects to this procedure of Schr6der 's explicitly:

Schr6der (1895, p. 551) declares that condensation can always be performed; but to carry it out he employs the formula aKx= (~; a'X) O, in which the elements of 11 are interpreted as relatives. (quoted in van Heijenoort 1967, p. 234)

L6wenheim dismisses Schr6der 's move up in type level as too trivial, and he views allowing a higher type elimination of quantifiers as inadmissible. However, this procedure is no different from introducing Skolem functions, which also eliminate quantifiers by adding something new, which L6wenheim readily adopts from Schr6der 's third volume.

Finally, in Lecture XII, Schr6der uses the quantifier rules of Lecture XI and relation algebra computat ions to construct within the relation calculus a theory of one-to-one maps and cardinal equivalence. L6w- enhe im used equations from Lecture XII as examples in his 1915 paper. In particular, L6wenheim proves his theorem 3 (which asserts that his theorem that every first-order statement with an infinite model has a countable model fails for higher-order logic) by showing that Schr6der 's definition of a one-to-one correspondence from Lecture XII provides


a c o u n t e r e x a m p l e to L 6 w e n h e i m ' s f i rs t -order t h e o r e m in the h igher -

o r d e r case. (Engl ish t r ans la t ions of par ts of S c h r 6 d e r ' s L e c t u r e s XI a n d

XII are given in A p p e n d i c e s 6 a n d 7.)

7.4. N o r b e r t W i e n e r ' s P h . D . T h e s i s (1913)

N o r b e r t W i e n e r ' s H a r v a r d d o c t o r a l d i s se r t a t ion (1913) gives the first

ax ioma t i c t r e a t m e n t of the ca lculus of re la t ions , p r e c e d i n g Tarski 's fa-

m o u s a x i o m a t i z a t i o n (1940) by m o r e t han twenty years. 6 In his thesis,

W i e n e r c o m m e n t s tha t S c h r 6 d e r ' s c o n t r i b u t i o n is m u c h m o r e s igni f icant

t h a n Russell a c k n o w l e d g e s in his Principles of Mathematics or in Whi te-

h e a d a n d Russel l ' s Principia Mathematica:

Russell says of Schroeder, "Peirce and Schroeder have realized the great importance of the subject the Mgebra of Relatives, but unfortunately their methods being based, not on Peano, but on the older Symbolic Logic derived (with modifications) from Boole, are so cum- berous and difficult that most of the applications which ought to be made are practically not feasible. In addition to the defects of the old symbolic logic, their method suffers (whether philosophically or not I do not at present discuss) from the fact that they regard a relation as essentially a class of couples, thus requiring elaborate formulae of summation for dealing with single relations .... "

Without any desire to belittle in any degree the magnificent work of Russell, I would like to raise the question whether the advances

which he had made in the Algebra of Relatives are of so sweeping a nature and mark such a radical departure from the direction of work pointed out by Schroeder as he then seemed to think . . . . it is an open question to me whether, in general, when Schroeder and Russell treat of the same subject, Schroeder is so much behind Russell after all. As to Schroeder regarding a relation as a class of couples, Russell explicitly affirms this very statement in his Principia Mathematica. It is true that Schroeder regards a relative as a sum of relatives which are of such a nature that they hold between the two terms of a unit couple, whereas Russell regards a relative as a class whose members are unit couples, but .... as I shall show, not only are these two expressions equivalent, but any operation on the one can be carried out on the other with exactly the same ease in an exactly parallel manner. (Wiener

1913, pp. 4-5)

W i e n e r po in t s o u t tha t the ca lculus o f p r o p o s i t i o n s as i n t r o d u c e d by

Russel l is a lgebra ica l ly e q u i v a l e n t to the ca lculus of classes as i n t r o d u c e d

by S ch r 6de r :

6 The introduction and last chapter of Wiener's thesis are reproduced in Appendix 8.


I intend to devote the first chapter of my thesis to a proof that the sets of postulates for classes and propositions given by Schroeder and by Russell respectively are equivalent, and that, in so far as the laws of the algebra of relatives coincide with those of the calculus of classes, their treatments of the algebra of relatives are also equivalent. (Wiener 1913, p. 11)

This is the Boolean opera t ions par t of the calculus of relatives. Wiener also remarks that the por t ion of Russell's system that corre-

sponds to the s tandard calculus of relatives (i.e., Boolean opera t ions and

relative p roduc t and converse), a l though done in a different axiomatic

order, is, for all practical purposes, a copy of Schr6der ' s calculus:

In so far as the subjects which they treat are identical, Schroeder and Russell are able, each on his own basis, to give equally accurate and rigorous accounts of them, which may always be translated step for step from the language of Schroeder into that of Russell. In very many cases a perfectly parallel translation may be made in the reverse direction, although certain of the ideas involved in the formulae of Russell must be paraphrased before they can be expressed in Schroe- der's terminology. The sole essential point of difference between their algebras of relatives lies in the fact that Schroeder conscientiously limits himself within the confines of what Russell calls a single type, and so is forced to do without many of the formula with which Russell finds himself able to deal. (Wiener 1913, p. 21)

With respect to expressiveness, Wiene r concedes a poin t to Russell: Russell 's system is m o r e expressive than Schr6der ' s , but this addi t ional power is outside the a lgebra of relatives. Within its i n t e n d e d d o m a i n of discourse, Wiener claims that Schr6der ' s system is equally expressive.

T h e r e is a n o t h e r aspect of Wiener ' s c o m m e n t that is worth not ing, which is that it is Russell's use of defini t ions to replace a compl ica ted

no t ion by a m o r e compac t symbolism that makes his work m o r e succinct

and readable . Schr6der, eager to make the poin t that his small n u m b e r

of connect ives suffice, systematically avoids definit ions. Thus, when Rus- sell discusses a topic he uses fewer a p p a r e n t symbols than Schr6der , but

only because he was willing to admi t defini t ional extens ions to his theory

and to take advantage of those defini t ions to give a m o r e natura l expression of the same mathemat ica l ideas.

The second issue, accord ing to Wiener, is proof- theore t ic power. Wie- ne r may have been one of the first to prove what is known as a con- servativeness result. In m o d e r n terminology, if a l anguage L l contains

a n o t h e r language L2, and T 1 is an L 1-theory con ta in ing T,e, an L2-theory, we say that T~ is a conservative extension of T, 2 if and only if wheneve r 4~ is a fo rmula in the smaller l anguage L2, then T,2 proves 4) only when


T~ does. Wiener 's claim is that, modulo a translation procedure , Russell's theory is a conservative extension of Schr6der's. This, from Wiener 's point of view, directly challenges Russell's claim that Schr6der is inadequate and Russell is original, because in the domain of discourse that Schr6der and Russell share, the two systems are equipotent . One needs the condit ion of a translation procedure because Schr6der 's and Rus- sell's languages are different, but Wiener shows how one can translate a Schr6der s tatement into a Russell statement, and as long as the Russell s tatement is of an appropriate form, one can effect the reverse translation as well.

Russell starts out with propositions and proposit ional logic (logic, not classes or sets) as basic, and his Principia Mathematica works according to that design. It begins with propositional axioms and develops propositional logic. It then applies this logic to proposit ional functions of one and more variables, applying the propositional calculus already developed to the more general case, and introduces quantifiers on propositional functions. Classes and the algebra of classes are developed as a consequence of the resulting theory of propositional calculus. Russell did this in a ramified type theory to avoid his paradox.

On the o ther hand, Schr6der, and Peirce before Mitchell, mostly thought of relations algebraically and developed the calculus of these relations quite extensively, generalizing Boole to relations. There is no hint that the calculus of propositions was regarded as a basis or that proposit ional functions were thought to be a prerequisite to the relation calculus. Mitchell, and Peirce following Mitchell, also developed the calculus of quantifiers on propositional functions. In some sense, they took classes as basic and derived results from that base, generalized to class functions (two-valued functions of pairs, triples, and so on), although this interpretat ion is difficult to prove.

Thus, Wiener is pointing out that at least without quantifiers it makes no difference whether one starts with propositional logic or class calculus; the end results are equivalent, and it is a question of choice which notions and axioms one starts with, which Russell regarded as arbitrary. Thus, Russell's claim to originality is not well-based. In making this argument , Wiener uses Schr6der as representing a more complete development of Peirce. The fine points of Schr6der 's view of the algebra of relations, broadly construed, as a full foundat ion of all of mathematics and one that incorporates logic, were beyond the scope of Wiener 's thesis, which is devoted to demonstra t ing the equivalence of the class and proposit ional basis, and thus the equivalence of Russell and Schr6der-Peirce for binary relations.

Wiener calls to attention another interesting er ror that has propa- gated to the present day. Russell implies that Schr6der confused mem-


bersh ip ( ~ ) and inclusion ( C ) . However, as W i e n e r indicates in the following passage, this is not an accura te read ing of Schr6der :

I shall discuss the z-relation and its absence in the treatment of the Algebra of Logic given by Schroeder. I shall show that the statement made by Padoa and implied by Russell to the effect that Schroeder confuses the z-relation and the C-relation is totally false . . . . I shall also show that Schroeder's symbolism involves the treatment of none of the notions which the z-relation is designed to embody, and that, therefore, he neither needs nor can express any hierarchy of "types" by his formulae, nor deal with relatives of different types. (Wiener 1913, p. 19)

In short , wherever we would say "x is an e l e m e n t of y," S c h r 6 d e r says

"x is a un i t class ( there are no smal ler d i f ferent f rom 0) and is c o n t a i n e d in y."

W i e n e r also asserts that Sch r6de r did not impose types bu t that he

had a typeless theory, even t h o u g h we would now dis t inguish the types of the propos i t ions in his a rguments . It seems that S c h r 6 d e r chooses a

d o m a i n , allows quant i f ica t ion over relat ions on that d o m a i n and over

re la t ions on relat ions over that doma in , and so on. Thus, he is using objects that have types, but he does no t always dis t inguish the f irs t-order

quant i f ie rs over individuals f rom the s econd -o rde r quant i f iers over relations, a l t hough we have seen that Sch r6de r does make this d is t inct ion in Lec ture XI and uses a no ta t iona l conven t ion , ~2 i versus I~, t h r o u g h o u t

the Algebra der Logik to dis t inguish the two. It is in connec~tion with this

po in t that his no t ion of "solution" is very close to Skolem funct ions. In the nex t chap t e r we will see how L 6 w e n h e i m uses Schr6de r ' s idea of

"solut ion" and re la ted ideas to analyze s ta tements in the calculus of relatives and d e t e r m i n e the cardinali t ies of the universes they satisfy.

8. L6wenheim's Contribution

Introduction

In 1915, Leopold L6wenheim published his paper "On possibilities in the calculus of relatives" in Mathematische Annalen, continuing work Schr6der left unfinished in his Algebra der Logih twenty years earlier. In the last two lectures of the final volume of his treatise, Schr6der distinguished between first-order and second-order formulas and separated his treatment of the elimination problem into first- and second-order cases. Schr6der knew that higher order statements can characterize uncountable structures (for instance, the second-order axioms for the real numbers, or the axiom that the domain of real numbers is not countable). Combining these examples from Schr6der, it was perfectly reasonable to ask whether there are also first-order statements that have only uncountable models. We conjecture that this question is the source of L6wenheim's theorem, and we base this conjecture on L6wenheim's detailed use of Schr6der's distinctive notation, which is quite unlike that of Frege or Russell.

It was also reasonable for L6wenheim, pursuing this question, to try to use the Schr6der elimination method, which introduces witness relations for universal-existential quantifiers, in order to simplify the problem to statements of a special form when these witness relations have been introduced. These are in fact universal sentences with the addition of witness relations. We conjecture that Skolem saw how to use function symbols rather than these relation symbols as witnesses, and that this was the origin of Skolem's proofs. We also speculate as to other possible origins of first-order logic in L6wenheim's paper.

Peirce introduced first-order (prenex) predicate logic in his 1885 paper and cleanly separated it from second-order predicate logic, also introduced in this paper, but he isolated no special properties of the first-order fragment and clouded his discovery by referring to his first-

169

17o LOWENHEIM'S CONTRIBUTION

order predicate logic as the "first-intentional logic of relatives" (Peirce 1885, p. 226).

Schr6der, who followed Peirce's work closely, cited Peirce's 1885 paper with its nascent full predicate logic (prenex) in the bibliography of his Algebra der Logik, but he seems to have used it in the first ten lectures of volume 3 (1895) only to perform computat ions on relatives, continuing to state propositions and build his system with binary relatives, with prenex quantifiers added on. Schr6der 's relative logic with prenex quantifiers was taken almost entirely from Peirce's 1883c paper. Not until the eleventh lecture of volume 3 does the distinction of first- and higher o rder logic made by Peirce in 1885 play an important role in Schr6der 's work. At the outset of Lecture XI, Schr6der divides the quantifier reduction problem into the first- and second-order cases and develops methods for reducing sums and products ranging over individuals, distinct from those needed to reduce E's and II's over binary relatives u.

T h e first-order reductions are easier. Although his int roduct ion of first-order is definitely borrowed from Peirce, Schr6der did not isolate first-order predicate logic. He includes relative operations in the quantifier-free matrix of most of his formulas, and in fact eliminates first- o rder quantifiers in favor of relative operations between an individual

_

i and an arbitrary relative a (e.g., I l i a ; i = a ;0 'ct0). His focus was on the quantifier-free fragment of the calculus of relatives, and he thought he could "condense" all first-order expressions with quantifiers over individual variables to quantifier-free formulas in the relative calculus.

It was Schr6der 's emphasis on the quantifier-free f ragment of the calculus of relatives that caught the attention of L6wenheim and his fellow Schr6der disciple, Alwin Korselt. Korselt found that if the calculus of relatives is restricted to Schr6der 's four modules 0, 1, 0 ~, and 1 t, then it is not possible to express that there exist four distinct individuals using only the relative operations without quantifiers. Since the first-order f ragment with quantifiers obviously can express the s tatement that the domain has four distinct elements (viz., EhukO~,ok = 1), Korselt had proved that the first-order fragment of the calculus of relatives is more expressive than the condensed fragment.

L6wenheim reported Korselt's result as the first theorem in his 1915 paper, and L6wenheim's main theorem follows as a natural generalization of this result. In Korselt's example, adding first-order quantification over individuals to the fragment of the calculus of relatives restricted to the four modules 0, 1, 0', and 1 ~ resulted in a more expressive language; by adding finitely many more relatives to Korselt's fragment, L6wenheim found that he could only get to a countable domain: there is no axiomatization for a purely uncountable domain that can be expressed in the first-order fragment of the calculus of relatives.

Stated in the language of the calculus of relatives, L6wenheim's cel-


ebrated theorem about first-order logic simply fell out of what was for him an obvious technical generalization of something that was more interesting in the calculus of relatives, that is, a technique that enables him to say what the cardinalities of universes can be from the analysis of the form of an equation. It was left to Skolem to extract the essence of L6wenheim's theorem and state it in the language of first-order logic.

Since the appearance of L6wenheim's theorem, first-order logic has become impor tant because it is a good context in which to study mathematical structures. It is adequate to express existing set theory, and therefore existing mathematics. Working in first-order set theory has made it possible to solve many relative consistency and independence questions. Second-order set theory, with arbitrary propert ies ra ther than merely first-order expressible properties, and quantification over all of them, seems more natural, but has gone nowhere. Almost nothing is known about it still, except in the systematically studied case of intui- tionistic logic, where Per Martin-Lof instigated p rofound investigations.

The proofs of L6wenheim's theorem led to the use of first-order logic to deepen and generalize algebra by extending algebra to relational systems. The theory of prime and saturated models is an early example, the Morley theory of rank and Shelah's forking theory are later examples, and Zilber's work is a yet more recent example. Pursuing this successful form of universal algebra, based on methods traceable back to L6wenheim, has been a principal motivation for the study of first- order model theory for the last fifty years.

8.1. Overview of L6wenheim's 1915 Paper

Although we r emember L6wenheim's 1915 paper today chiefly for L6w- enheim's theorem, it in fact proved six theorems, all on the topic of expressibility. L6wenheim uses Schr6der 's notat ion and domain language th roughout his paper; for ease of unders tanding, we here translate his results into modern form.

Theorem 1 states that there is a s tatement in first-order logic with equality for which there is no statement in the usual (quantifier-free) calculus of relatives with the same models. This s ta tement is that the domain has at most four elements.

Theorem 2 states that if there is an infinite model of a first-order statement, then there is a countable model. Since the calculus of relatives as in tended here is the first-order f ragment with quantifiers over only individuals, that is, modern-day first-order logic, according to theorem 1, theorem 2 is more general than simply saying that every statement of the condensed calculus of relatives having an infinite model also has a countable model.

x72 LC)WENHEIM'S CONTRIBUTION

Theorem 3 gives a second-order statement that implies that the domain is uncountable. By theorem 2, this cannot have the same models as any first-order statement. (In the calculus of relatives, this is second intentional in Peirce's or Schr6der 's sense. A familiar modern example of this p h e n o m e n o n is the conjunction of the axioms for the real numbers: an ordered field in which every set bounded above has a least upper bound. However, this is not the example used by L6wenheim.)

Theorem 4 states that first-order monadic logic (that is, first-order predicate logic with unary predicate letters only) has a s tronger property than theorem 2. Namely, any statement having an infinite model also has a finite model. Later textbooks reproduce this theorem within the decision theorem for monadic logic. Theorem 5 is part of this discussion.

Theorem 6 states that the question of satisfiability for a s tatement of first-order logic that involves ternary or higher arity predicates can be reduced to the question of satisfiability for a carefully chosen statement of first-order logic involving only binary relations.

Thus, L6wenheim's paper is very well-integrated. The paper is concerned with the relation between statements and their models. It is the first paper with this emphasis, and is the beginning of model theory. It shows that first-order logic, but not the condensed calculus of relatives, can express that a model has at least four elements. First-order logic cannot constrain the models of a formula to be uncountable, but second- order logic can. A monadic first-order logic cannot constrain a s tatement to have only infinite models. Finally, satisfiability of first-order statements in general can be reduced to those involving binary relations alone. Of these results, we concern ourselves only with theorem 2, which is now called the L6wenheim-Skolem theorem.

8.2. L 6 w e n h e i m ' s Theorem

L6wenheim's proof of his celebrated theorem has been accused of having impor tan t gaps that were later filled by Skolem. Robert Vaught (1974), in a paper on early model theory, makes no claim of real shortcomings, but states that he was unable to follow the original proof in detail, as it seems to weave in and out of first-order logic. He therefore follows the outline, but not the spirit, of the proof in his explanation of L6wenheim's result. Hao Wang (1970), in his introduct ion to Sko- lem's Selected Works, makes no claim of a gap in the proof, but merely says that L6wenheim makes an excursion into infinitary logic.

We contend that, properly interpreted, there is no gap in the proof except for the same implicit application of K6nig's lemma to which Skolem appealed in his 1922 address (Skolem 1923). The problem is that modern readers expect to see Skolem's proof using function sym-


bols for quantifiers when they read L6wenheim's a rgument , whereas L6wenheim instead used second-order logic. His formalism was that of Schr6der for the second-order calculus of relatives. Tha t notat ion, in turn, derives from Peirce's notation for the calculus of relatives of the "second intention." We explicate L6wenheim's proof, following it line by line. But first we translate Schr6der 's notat ion for second-order logic, which L6wenheim employs, into the modern notat ion for second-order logic.

L6wenheim's T h e o r e m 2. Suppose that cb is a formula in first-order logic. Suppose that M is a structure such that cb is true in M. Then there exists a

countable structure M o such that cb is true in Mo. This is the s tandard L6wenheim-Skolem theorem, stated in the lan-

guage with which we are most familiar today. L6wenheim's proof is divided into two parts. In the first part of the

proof, L6wenheim shows that for any first-order s ta tement 4), there is a s ta tement 4)o in an ex tended language such that 4~ is true in some model M if and only if 4~0 is true in some model M0 of the ex tended language.

The ex tended language used by L6wenheim is not, however, a larger first-order logic obtained by introducing new function symbols, as in Skolem (1920), or new relation symbols, as in G6del 's thesis (1929) or Hilbert and Ackermann (1928). Rather, his ex tended language is a second-order extension of the original language, with the same signature, but allowing second-order variables and quantification. The second-order variables allowed are function variables, ranging over functions on the domain. This follows the convention of Kleene's m o d e r n higher recursion theory, ra ther than modern second-order logic, which uses set variables ranging over all subsets of the domain. But it makes no difference, because ei ther system easily translates into the other.

It is this recognit ion of L6wenheim's ex tended language as the second-order logic of the domain based on function variables over the domain that makes it possible to follow L6wenheim's p roof in detail. The obstacle to unders tanding was chiefly his notat ion for the function variables and functions. Readers expected to see first-order logic with an ex tended signature for the models, not second-order logic with the same signature; such was the powerful effect on later generat ions of Skolem's version of the proof using such additional function symbols.

The 4~0 L6wenheim uses in the ex tended language is of the form

. . . v , , , x # ,

where X is quantifier-free. Here the initial quantifiers of ~0 are existential function quantifiers over the domain M0, and the n i are elements of the domain M0. The function quantifiers are defn i te ly second-order

174 LOWENHEIM'S CONTRIBUTION

objects. This s ta tement says that there is a choice of functions on the domain that satisfy a first-order universal formula over the domain.

Unlike Skolem, who added function symbols to the base first-order language to obtain a larger first-order language, this is an outr ight second-order statement. To repeat, L6wenheim differs from Skolem by in t roducing second-order existential quantification over functions, ra ther than increasing the signature of the language by adding function symbols. To those trained in conventional first-order predicate logic, this seems like an odd thing to do, Showing perhaps a lack of under- s tanding of first-order logic on L6wenheim's part. We argue that this is not so. To those also trained in recursion theory or descriptive set theory, this is not at all odd. In the Kleene hierarchy, El sets of integers are

jus t sets defined by second-order formulas of this form from arithmetic. They have a recursive predicate X of integer and function variables. Similarly, the analytic sets of Nicolas Lusin and Mikhail Souslin are def ined by formulas of the same form, but there the domain is the real numbers , and X denotes an open set in the function space topology.

Like Lusin and Souslin, L6wenheim uses second-order formulas with existential prenex function quantifiers followed by universal quantifiers over the domain. Thus, this move, al though not what is expected in the cur ren t way of doing business in first-order model theory, is very much in accord with conventional practice in higher recursion theory and descriptive set theory. In addition, it is not as though such analytic set methods are absent in modern first-order logic; they are often used in m o d e r n model theory, just not at its very beginnings.

The proof that 4~ is satisfiable in some domain if and only if 4~0 is satisfiable in some domain must use the definition of satisfaction; that is, it involves an induction on the definition of satisfaction for formulas. In Skolem, this is first-order satisfaction over M and M0. L6wenheim, however, uses only second-order satisfaction over M. That is their difference. To state the result, L6wenheim had to have a very clear unders tanding of what it means for an arbitrary formula to be satisfied in a model. In this regard, he was a true precursor of Tarski. He appears to have exceeded the precise unders tanding of both Peirce and Schr6der in the use of a general notion of logical formula, together with rules for interpret ing such a formula in a domain.

Skolem, by sticking to first-order logic, may have come up with a c leaner form of the proof, avoiding second-order satisfaction, but his unders tanding of satisfaction was no bet ter than, and almost certainly derived from, a careful reading of L6wenheim.

We now turn to L6wenheim's s ta tement of his theorem and his proof. We begin with the s tatement of the theorem:

Theorem 2. I f the domain is at least denumerably infinite, it is no longer


the case that a first-order fleeing equation is satisfied for arbitrary values of t,~e relative coefficients. (p. 235) ~

175

L6wenhe im defines a f leeing equa t ion as:

A fleeing equation is an equation that is not satisfied in every 11 but is satisfied in every finite 11 (or, more explicitly, an equation that is not identically satisfied but is satisfied whenever the summation or productation subscripts run through a finite 11). (p. 233)

This is requires some explanat ion . First, the t e rm I 1 is the d o m a i n of individuals. A f leeing equa t ion is then an equa t ion that is satisfied

in every finite mode l but not in every infinite model . For example , in

the d o m a i n of natural numbers , cons ider

3 xYy (y <_ x).

This will be t rue only for those sets of natural n u m b e r s in which there

is some m a x i m u m e lement . W h e n e v e r the set is infinite, t hen there is no m a x i m u m . This s t a t emen t is the re fore satisfied for all finite sets

because every finite set has a max imum. If we write it in equa t iona l form, 3xVy(y < x) = 1, we have an example of a f leeing equa t ion . We note that a f leeing equa t ion guaran tees that for any d o m a i n of finite size, there will exist some defini t ion of its re lat ion symbols that makes the equa t ion true, but not every possible def ini t ion will make it true.

In this light, what t h e o r e m 2 is assert ing is that for every first-order

express ion for which there exists some def ini t ion of its re la t ion symbols that makes it t rue for all finite domains but false for some infinite domain , then, given a countably infinite domain , it c a n n o t be t rue there for all possible values of its relat ion symbols.

Now we move on to the proof. L6wenhe im begins:

For the proof we think of the equation as brought into zero form. We prove first that every first-order equation can be brought into a certain normal form. (p. 235)

His no rma l form is

E I I F = 0 (3)

(p. 237). The only difficult case of his no rma l form reduc t ion is case 4, in

which l i e is to be t r ans fo rmed into a Ell. He writes the equat ion:

In this chapter, unless otherwise noted, all page numbers refer to L6wenheim (1915~ in From Frege to Gddel (van Heijenoort 1967).

176 LI~WENHEIM'S CONTRIBUTION

I-[i~kA ik = ~ • IIiA iki kx

(p. 236). Ultimately, given all arbi t rary first-order express ion, he wants to move

all the I~'s ou t to the left and have all the II's inside. To achieve this, he shows he re how to do it for the small case in which the re are jus t two quant if iers , one II and one ~ inside it. First, cons ide r the lef t -hand

side. This first-order express ion is saying that for all i the re is some k

that makes Aik t r u e . An example of this, in m o d e r n form, is the ex-

press ion Vx3yR(x,y). This express ion asserts that for every x the re is

some witnessing y. For example , we can take a d o m a i n with th ree ele-

ments , {1,2,3}. Suppose we had de f ined R to be t rue on the pairs

(1, 1), (2, 1), and (3,2). Now, as this example shows, when the x of the

o u t e r quant i f ie r is chang ing f rom 1 to 2 to 3, its witnessing y is c h a n g i n g f rom 1 to 1 to 2. Thus, d i f ferent x's may have d i f fe rent witnesses. In this example , when x = 1 and x = 2, the witnessing y is the same. But

w h e n x =3, y =2. So, in genera l , wheneve r the re is an express ion

Vx3y, as x varies over the universe, as we assign d i f fe rent values to x, its

witnessing y may be different . Nex t we cons ide r the express ion 3yVxR(x,y). This express ion asserts

that the re is a y that witnesses for all x. So, for the def in i t ion of R that

is t rue for (1, 1}, (2, 1), and (3, 2), this s econd express ion will no t be true,

because there is no un ique y that witnesses for all of them. Thus, the

first and second express ions are not logically equivalent , and we c a n n o t jus t bl indly swap the quant i f iers and br ing the " there exists" outside. We mus t do s o m e t h i n g m o r e carefully.

W h a t L 6 w e n h e i m does is the following. To p r o d u c e a s t a t emen t equiv-

a len t to Vx3y to start with, all he knows is that there is a y, but it m igh t

differ for x, so he in t roduces a new quantif ier , a doub le summa t ion . He

writes this as

~p-~X I-Ii A i*i'

and explains that the k x u n d e r the doub le s u m m a t i o n means that k• is to run t h r o u g h all e l ements of the d o m a i n of individuals, and that the X on the r ight of the doub le s u m m a t i o n means that each of the k• is to run t h r o u g h all of these e lements . He gives an equa t ion to show what this means:

FROM P E I R C E T O SKOLEM X77

�9 . = ]2 AlklA2k,2Ask:~ . . . I I i (A i l + Ai2 + Ai3 + ") k,.~,~,k~...

= r, ~ E . . . AlklA2k,~Ask3 . . . k,-~.2.:~ .... k~-l.2.:~ .... ~zl.,,.:~ ....

(p. 236). This is an infinite distributive law. The left-hand side of this equat ion , II;(A~ + Ai2 + A~3 + "") , says that

for every i the fo rmula in pa ren theses is true. If we c o m p a r e this to the

left-hand side of the no rma l form equa t ion that we are trying to explain,

I-Ii~kAik , which asserts that for every i there was some witnessing k that

makes the relat ion A true, we recognize that II~(A~l + Ai2 + "") is asser t ing the same thing. It says that for every i there exists some k; e i ther

A~l o r Ai2 o r Ai3, and so on, ex t end ing to all the possibilities. The re fo re , the two express ions are logically equivalent. The s u m m a t i o n runs plus, plus, plus; it goes on as many times as the size of the universe. It will hold for all universes.

Now, r e tu rn ing to the last equat ion , we see that to the r ight of the first equal sign there is a summat ion . The te rm inside the s u m m a t i o n ,

Alk lA2k .2Ask . . . , . : , says t h a t A l k I and A2k,~ and A3k~ , and so on. The first subscript runs f rom 1, 2, 3, and so on, and the second subscript runs kl, k2, k3, and so on. What this asserts is that for 1, k I is the witness; for 2, k 2 is the witness; for 3, ks is the witness, and so on, where

kl, k2, k 3 . . . . are somewhere in the universe. Thus, the lef t-hand side of this equat ion , if true, means II,(A~ + A~2 + Aa3 + ""), and the r ight -hand

side is a s u m m a t i o n that runs over all possible values of kl,k2, k s . . . . .

They are then equivalent , because the left-hand side says that for every i the re is some witness, while the r ight -hand side assigns a witness to each possible i, and tries out all possibilities of assigning witnesses. Since this s u m m a t i o n runs over all the possible k l, k2, ks, this m e a n s trying out every possible k~, every possible k 2, and every possible k 3. Since if the lef t-hand side asserts that for every i there is some witness, the right- hand side says that if for every i there is some witness, then we mus t have some ass ignment for all of them. Thus, on the r igh t -hand side,

each te rm of the s u m m a t i o n gives one set of witnesses for all of them,

and a n o t h e r te rm gives a n o t h e r set of witnesses. Since the re is a way of

s imul taneously giving witnesses to all, some te rm in this s u m m a t i o n mus t be true.

For example , suppose again that the d o m a i n has th ree e lements ,

{1,2,3}. The r ight -hand side of the equa t ion is a s u m m a t i o n of

A lklAzk.A3k.~ for this example , r u n n i n g over all possible values of k~, k2, k 3. If we expand out the summat ion , we have A l ~ A z l A s ! or

A l l A , 2 1 A s 2 or A ~ A 2 1 A s s m a total of 3 x 3 x 3 = 27 d i f fe ren t terms. The last one will be A~sA2,~Ass . This says that e i ther it is t rue that 1 has a

x78 L ( ~ W E N H E I M ' S C O N T R I B U T I O N

witness 1, and 2 has a witness 1, and 3 has a witness 1; or 1 has a witness 1, and 2 has 1, and 3 has 2; and so on.

We again compare this to the left-hand side of the original expression, Il f . ,kAik. What that says is that for every i there is some witnessing k. So if I-IiF.,kAik is valid, then at least one term in the summat ion is the one that gives a valid assignment of witnesses. Conversely, if one term in this summat ion is valid, it means that this term is assigning witnesses correctly, and thus the original expression IliEkA~k must be valid.

Returning again to the last equation, after the second equality sign there are an infinite number of summations:

E E E ... A l k l A 2 k A 3 k ~ k1-1,2,3 .... k2-1,2,3 .... k~-,1.2,3 .... '

These infinite summations basically say the same thing, noth ing new: that each ki runs over all the possible ways of assigning witnesses.

Returning to the case 4 normal form equation,

I-Ii~]kA ik = ~ x IliA ik,, kx

the IliAik, term on the right-hand side is basically each of the products. Each term is a II. The outer summat ion tells how many II's there are. In o ther words, the outer summat ion runs over all the possible assignments of witnesses, and the inner product for each assignment, basically says that "this is a witness for this, a n d this is a witness for this, a n d this is a witness for this." L6wenheim uses a IIi because in each term he assigns a witness to everything. Overall, this equation tells us that whenever there is a I-Ii~kAik , we can equivalently write it as a summat ion of a product.

In the end, the normal form is

~nF=O (3)

(p. 237). Thus, if the initial assertion was that something equals 0, by this

simplification, it will now be converted into an equivalent assertion that says "there exists for all F equals 0." Whatever sequence of quantifiers we begin with, at the end we can make it a sum of a product. Sum can be represented by an existential quantifier and product by a universal quantif ier in the case of an infinite universe. That is all that the normal form part of L6wenheim's proof has to do.

Assuming that we have converted the equation into the normal form, ~ I IF (L6wenheim's equation [3]), L6wenheim begins the second part of his p roof by saying that we can drop the outer E:


If we now want to decide whether or not (3) is identically satisfied in some domain, then in our discussion we can omit the E and examine the equation

IIF=0. (4)

(p. 238)

179

Let us see why that is correct . We cons ider the following s ta tement :

3xVyR(x, y) = O.

To say that this equa t ion is identically satisfied for some d o m a i n means

that for any def ini t ion of the relat ion symbol R over that doma in , this

s t a t emen t holds. In English, this is saying that the assert ion " there exists x for all y R(x,y)" is false. This means that there does not exist any x

such that for all y, R(x,y) holds. These two s ta tements are equivalent .

The re fo re , the fact that there exists an x such that for all y, R(x,y) is

false is the same as saying that, no ma t t e r what x we choose , we will

never be able to satisfy the assert ion "for all y, R(x,y)," or "for all y, R(x,y)" is equal to 0. This last s t a t emen t is thus equiva lent to 3xVyR(x, y) =0.

Now if we fu r the r assert that the full s ta tement , " there exists x for all y R(x, y) = 0," holds for any defini t ion of R, then, by the a r g u m e n t above,

this is equivalent to saying that the second s ta tement , "for all y R(x, y) = 0," when the first quant i f ie r is s t r ipped off, holds for any value of R and any value of x.

S t a t emen t 1 is thus

3xYyR(x, y) = 0, (1)

while s t a t emen t 2 is

YyR(x,y) =0. (2)

In s t a t emen t 2 x is a free variable. But no ma t t e r what value we subst i tute for this free variable x, the subasser t ion YyR(x,y) is false. Rephras ing , we can say that for every x the second equa t ion will hold because the

first equa t ion holds.

Tha t is jus t what L6wenhe im says. Actually, we are saying a little extra,

namely, that the first equa t ion holds for any value of R, which in our

new vocabulary would read "for every value of x and for every value of

R the second equa t ion holds." Since the first equa t ion holds for every

value of R, the second one will also hold for every value of x and every value of R.

Now let us do the proof. We will use the example given in equa t ion (1). We d rop the " there exists x" for obvious reasons. T h a t means that

x8o LOWENHEIM'S CONTRIBUTION

whatever is left in ou r example is the second equa t ion . Now in the s econd equa t ion x is a free variable, and in the genera l case the re may be several free variables. If we look at L 6 w e n h e i m ' s equa t ion (4),

I-IF=0, (4)

(p. 238) there he has d r o p p e d all the existential quantif iers . But if we recall his n o r m a l form a r g u m e n t , the re could have been an infini te s e q u e n c e of existential quantif iers , gkEk~ ... p r e c e d i n g II. L 6 w e n h e i m t r a n s f o r m e d every equa t ion to a form in which existential quant i f iers were to the left and II stayed inside. He also i n t r o d u c e d the d o u b l e s u m m a t i o n quantif iers , which actually r e p r e s e n t e d an infinite s e q u e n c e of quantif iers . But we d r o p all of these convent ions , for the same logical reason. It would seem that there migh t now be infinitely many free variables that will dangle in his new equa t ion . But that is no t the case: if we take the example that he has given (p. 238), which is the e qua t i on jus t below his equa t ion (4):

t -

IIh.~4(Z-h, + z-j,j + 10) ZaZk,~ = O,

the re is a variable l, and there is a variable k~. T h e variable l comes f rom d r o p p i n g the existential quant i f ie r a l ready in the equa t ion , be fo re the no rma l form t ransformat ion . The variable k~ comes f rom i n t r o d u c i n g the doub l e summat ion . W h e n we say a free variable, we thus only m e a n of the type l, no t the k~ types that we had to i n t roduce as a result o f swapping II and E. But this means that the free variables in the e qua t i on are those variables that were b o u n d by existential quant i f iers in the or iginal equa t ion , and not the variables that were i n t roduced . All the variables that were i n t r o d u c e d were not o f the s imple types l; some were k with some subscript.

Now what we need to show is that this equiva len t equa t ion (2) is no t identically satisfied in a d o m a i n that has at least coun tab l e e lements . T h a t is what the t h e o r e m says. This means that equa t ion (2) is no t satisfied for arbi trary values of relative coefficients.

To do this, we will come up with a def in i t ion of R (where R is the relative variable in ou r example ) over the variables in ou r doma in , and

some def in i t ion of xsuch that u n d e r those def ini t ions equa t ion (2) does not hold. If we could do that, it would be the same as proving the t h e o r e m .

C o m i n g up with a c o u n t e r e x a m p l e means c o m i n g up with an R and an x: f inding an R that would be a c o u n t e r e x a m p l e to e qua t i on (1) is the same as f inding some o t h e r R and some x that will be a counte r - e x a m p l e to equa t ion (2). Since we have shown that (1) and (2) are equivalent , we will work with (2), and c o m e up with some def in i t ion of R and some value of x such that (2) breaks down. U n d e r that def in i t ion


of R and x, VyR(x, y) will equal 1. W h e n we talk in terms of ou r example ,

the second equat ion , we unde r s t and the p roo f with respec t to ou r example , but imagine that the same th ing holds in the genera l case, which

is HE Let us start by def in ing this R over our domain . Now II runs over all

the e l emen t s of the universe, so first we pick as many symbols as there

are free variables in R(x,y). H e r e we have jus t one, which is x. Now we pick one symbol; call it 1. This symbol 1 represen ts some e l e m e n t of

the domain . It may rep resen t the actual n u m b e r 1 of the domain , or

n u m b e r 2 of the domain , or n u m b e r 3, or anything; to avoid confusion,

we will call this symbol s 1. In general , we pick as many symbols s m as the n u m b e r m of free variables.

Now the quant i f ie r II runs over all possible values of y. In particular,

it runs over s 1. In general , II will run over all s 1, s 2, ..., up to Sm, where

m is the n u m b e r of free variables. So for each like a s s ignment to y, there is the quantif ier-free part, which is R(x,y). We have one value for

R(x,y) for each value of x and y. Next, let the free variable x be r ep re sen t ed by the symbol s 1, and let

y range over s~. II consists of several factors: for example , R(x, yl), R(x, Y2), where y runs over all the e lements of the domain . We call t hem

factors, since it holds that VyR(x,y) is the same as R(X, yl) and R(x, y2), where Yl and Y2 basically cover all the e l emen t s of the domain . In our case, we only cons ider those factors that range over the symbol set that

we already have. Let us write down all the factors of II that involve

symbols only f rom our cu r r en t symbol set. Since our example has only

one symbol, in our case there is only one, R(s 1, sl). Now we give a slightly m o r e compl ica ted example . Cons ider

3x3zYyR(x, z, y).

W h e n we d rop existentials, we have two free variables, x and z. Thus, we will take two symbols to begin with, s~ and s2; x will be s I and z will be s 2. We let y range over all the possible objects in our symbol set. Tha t means that y will be Sl and s 2. So we have two factors. The first factor

will be of the form R(s~, s 2, sl), and the second factor R(Sl, s2, s2). We see that for the genera l case HF, we can let the variables u n d e r

II run over all the possible values of the c u r r e n t doma in , the finite

domain , and write down all the factors. For clarity, we write down our modi f ied example . Equat ion (1) is now

3x3zqyR(x, z, y) = 0, (1)

and equa t ion (2) is

YyR(x,z,y) =0. (2)

Equat ion (3) is the English form of (2):

x82 L O W E N H E I M ' S C O N T R I B U T I O N

for all values of x and z it holds tha t for all y R (x , z , y ) =0. (3)

R(sl, s2, S 1) • R(Sl , s 2, s 2) = 0.

Equat ion (4) is

(4)

We are trying to show that equa t ion (2) breaks down for some x and z

and some defini t ion of R. Wha t does it mean for equa t ion (2) to hold identically? It means that

for any defini t ion of R and for any defini t ion of x, z, the equa t ion holds.

Wha t does it m e a n for equa t ion (4) to hold identically? (Note that

equa t ion [4] is only valid over our small domain , {s 1, s2}.) It means that

for any def ini t ion of R we pick, and for s I and s 2 having arbi t rary equa-

tions ho ld ing a m o n g them (in o the r words, s 1 may be equal to s 2 or s I may be distinct f rom s2), equa t ion (4) holds. We have two possibilities: s I and s 2 are ass ignments to certain variables, but those ass ignments can actually match. We can assign the same e l e m e n t to both x and z, or we

can assign them different e lements . So when we say that equa t ion (4)

holds identically, we m e a n that for any arbi trary equa t ion ho ld ing a m o n g s~ and s 2 and for any def ini t ion of R, the p r o d u c t

R(Sl, s2, S1) • R(s l, s 2, S2) is 0. O u r claim is that if equa t ion (4) holds identically, then equa t ion (2)

holds identically over the domain , namely, Sl and s 2.

We note that the full equa t ion (2) has one factor for each possible value of y. Since our domain is infinite, equa t ion (2) has an infinite n u m b e r of factors, one for each value of y. But if we limit our d o m a i n to these two e lements s~ and s 2, then the factors will look like R(x, z, s 1) and R(x,z , s2). We migh t also suppose that x and z have also been assigned s~ and sz, respectively.

Thus, in our equa t ion (2) we take two e lement s in the domain , assign x to be one of t hem and z to be the other, and let y range over these

two e lements , get t ing two possible factors in our domain , which is the

infinite domain . These two factors exist a m o n g the infinite factors we

would get. But all the infinite factors will have x equal to s~ and z equal

to s 2. The only coord ina te that is d i f ferent ia ted is y, and y now ranges over the whole domain , while x and z are s~ and s 2, respectively. Then ,

a m o n g those factors, we pick out jus t those in which y is only rang ing

over s~ and s 2. Now f rom what we have asserted, namely, that (4) is identically true,

this means that the p roduc t of these two factors in (4) is zero. These two factors also appea r a m o n g the infinite list of factors in (2). Since

(2) is a p roduc t of factors, if any factor is 0, the whole p r o d u c t is also

0. Thus, if we pick x and z to be any two e lements in our infinite domain , since (4) held for arbitrary equat ions between s I and s 2, however we assign x and z f rom our infinite domain , there will always exist two terms


in the product that will vanish. In that infinite product , there will be two factors that will vanish because if they could be satisfied by some definit ion of R, then we could have done that in (4) itself.

Now let us go back. What this is actually saying is that no mat te r how we assign x and z, whenever we consider these two terms in the product , where y also ranges over those values of x and z, we will get exactly these two factors. Thei r product will always vanish because (4) implies exactly that. If it so happens that x and z are both assigned to the same element , in that case there will be just one factor, because if s~ equals s 2, then both factors become the same. However, we have claimed that (4) holds for any value of s~ and s 2, whether they are equal or not. This means that no mat ter what happens, whether x and z are assigned differently or to the same element , one factor at least, or two factors in the infinite product , will always vanish, and so the whole p roduc t will vanish. In o ther words, for all y, R(x, z, y) equals 0. Therefore , having (4) identically hold for any arbitrary value of R implies that (2) holds identically for any arbitrary value of R, x, and z. If x and z had the same assignment, say Sl, claiming that (4) holds identically means that one factor will vanish, which is R(Sl, Sl, Sl).

Returning to (2), whenever we assign x and z to the same value of the domain, consider that term in this infinite product where y also takes the same value that we have assigned to x and z; then that factor will vanish. In this case, at least one of the factors in the infinite product vanishes. In o ther case, two of the factors vanishes, when x and z are assigned separately. Therefore , to conclude, whenever (4) holds identically, then (2) will hold identically.

This is not auspicious, because we are ultimately trying to show a counte rexample to (2), whereas we are getting the result that (2) holds identically. But note the main point: we have only claimed that if (4) holds identically, then (2) will hold identically. However, if (2) has come from some fleeing equation, it cannot hold identically. The claim is that if (4) vanishes identically, then (2) vanishes identically. But (2) cannot vanish identically because (2) vanishing identically actually says that the a rgumen t that if (4) holds implies (2) holds does not depend on whether our domain was infinite or how infinite, whether countable or uncountable . In fact, if (4) held identically, that would imply that equation (2) always held, no mat ter what the domain was.

Recall that we started with a fleeing equation, pe r fo rmed a normal form conversion, to obtain another equation. We know that a fleeing equat ion holds for all finite domains, but it does not hold for some infinite domain, by definition. So it is not true that (2) holds identically for all possible domains, because (2) is equivalent to the fleeing equat ion we t ransformed into (2). Rather, there must be some domain over which (2) fails. Therefore (4) must fail: (4) cannot hold identically. If (4)

t84 L O W E N H E I M ' S C O N T R I B U T I O N

could hold identically, then by our claim "(4) implies (2)," (2) would hold identically. But (2) cannot , so (4) cannot , by contraposi t ion.

Now what does it mean that (4) does not hold identically? This means that there exists some defini t ion of R and some equa t ion be tween s~ and s 2 such that the p roduc t of these two factors is not equal to 0; in fact, it is 1.

Let us in t roduce some notat ion. The p r o d u c t that we form over our symbol set of size 2, s~ and $2, has two factors in it: R(sl,Sz, Sl) and R ( s ~ , s 2, s2). We will call this p roduc t P~ ("1" to deno te the first step of the cons t ruc t ion) . Pl will have several forms: Pl', PI", and PI", and so on, d e p e n d i n g on what equat ions hold be tween s~ and s 2. Thus, P~' is the p r o d u c t u n d e r the equa t ion s~ = s 2, and P~" u n d e r the inequa t ion

S 1 ~ S 2.

In general , if there were several symbols, s~, s 2, s 3, up to s m, there would exist several possible equat ions and inequat ions be tween the s~'s, but only finitely many. For each possible equa t ion or inequat ion , there will be a version of P1, viz., Pl', Pl", P~", and so on, but only finitely many. This is crucial.

Now the fact that (4) does not hold identically, which we have jus t c la imed, because (2) does not hold identically, means that u n d e r one of these versions of P~, the p roduc t equals 1. Otherwise, if u n d e r all versions of P~ and u n d e r all defini t ions of R the p roduc t equals 0, then (4) would hold identically. Since that is not true, (4) not be ing true means that there exists some P~, say P~' or P~", such that this p roduc t is equal to 1.

Suppose that our equat ion was s~ = s 2. This means that there exists a def ini t ion of R over a doma in of a single e lement , because s~ equals s 2. If the equat ion that we are cons ider ing is s~ :~ s 2, then that there exists a def ini t ion of R would m e a n that there exists a def ini t ion over a d o m a i n of size 2. Whichever is the case, we have some def ini t ion of R and some equat ion that makes (4) fail.

Wha t we have c la imed so far is that there is a doma in of size 1 or 2 over which the original equa t ion does not hold identically. This is because (4) not hold ing identically is the same as the original def ini t ion not ho ld ing identically over our smaller domain .

We now increase this finite world. How does it grow? Let us go back to L6wenhe im ' s example after his equa t ion (4):

(p. 238). We note the term zk, in his example . In general , there will be several

k,'s in F; in fact, there may an infinite n u m b e r of k~'s. But these k~ subscripts d e p e n d on some variable i that is u n d e r the II quantifier. It


canno t be k z, since the I comes from the existential quant i f ie r p resen t in the original form of the equat ion.

In L6wenhe im ' s example , there is only one free variable, l, and he picks a single symbol, called "1." We call it s~; he calls it simply "1." L 6 w e n h e i m then writes the factor of the equa t ion in which II ranges only over 1:

el = 2~11(2~i1 "~- 2~!1 "1- lt11)2~21 = 2~11z21-

Now since all of h, i, j run only over 1, what we get is s etc. T h e r e is also a factor s which comes in because k i, in general , represents some e l e m e n t of the universe. It may be the same as 1, but it may also be different , because ki only d e p e n d s on i. To be safe, L 6 w e n h e i m has called this symbol "2." He then constructs the p r o d u c t PI and says, by the same a rgumen t , that this p roduc t P~ does not vanish for some definition of z's.

For each of those finitely many possibilities, there are several versions: PI', PI", etc. They are dif ferent versions of the same p r o d u c t PI u n d e r the d i f ferent equat ions, whe the r 1 equals 2 or 1 differs f rom 2. The n u m b e r s 1 and 2 are his versions of our s 1 and sz. We had only two possibilities for s 1 and s2: equality or inequality. However, in general , if there are several free variables, there will be several symbols and thus several possibilities of equality and inequali ty (a l though at most finitely many) . L6wenhe im has a large n u m b e r of pr imes because there are many possible free variables, but not an infinite number . T h e only free variables that will appea r in the equa t ion are those that come f rom existential quantifiers, which were leftmost in the original f leeing equation, and that will remain on the left even after his no rma l form trans- format ion . The new double summat ion quantif iers, which are introduced in the normal form t ransformat ion , will have variables of the type k with subscripts, but, as we said previously, these are not free variables. Thus, our only free variables are those that already existed as quant i f ied by existential quantif iers in the original equat ion. We c a n n o t have an infinite n u m b e r of free variables because, to begin with, we had a f leeing equa t ion that had only finitely many quantifiers. T h e in t e rmed ia t e infinite quant if iers arise because of the normal form t ransformat ions .

We have so far cons idered only those cases in which the variables u n d e r II range over 1. But we now have a n o t h e r symbol, "2," by which we can include o the r factors in which the variables u n d e r II range over both 1 and 2. With respect to L6wenhe im ' s example , this means that there will be more factors arising, in which the variables h, i, and j will run over both 1 and 2. We will then get several terms such as PI =

2~11(2~11 + i l l "~- ltll)72,Zl = 2~11Z21 , since it is now no longer the case that all

a 86 LOWENHEIM'S CONTRIBUTION

i, j, h vary over jus t one value. Since they vary over both 1 and 2, seven more factors will enter.

We now ask the same quest ion with respect to this new product , this larger set of terms in which there will be both k 1 and k 2. The k~ term we called "2," but there will now also be a term k 2, which we will call "3." The next set of factors will therefore have l 's, 2's, and 3's in the subscripts to z.

With respect to this larger p roduc t we can ask the same quest ion: is it t rue that it holds, i.e., does it vanish for all defini t ions of R? If it does, then by the same a r g u m e n t we can say the original equa t ion must also be identically vanishing, which is not possible. This means that there exists some defini t ion of the predica te z and some equa t ion ho ld ing be tween the e lements , d e n o t e d by the symbols 1, 2, and 3, such that this larger p roduc t equals 1.

We therefore assume that P~ does not vanish identically. From P1 we let the variables i, j, h range over both 1 and 2, and we get a new set of factors, which L6wenhe im calls P2 (P. 239). But P2 includes P1, because we have already cons idered all the cases in which h, i, j have r anged over jus t 1.

P2 is the p roduc t of factors we get when we allow II to range over this new set of symbols; in L6wenhe im ' s example , when we allow all h, i, j to range over both 1 and 2. In this case, we get k 2, which we call 3. This p r o d u c t L6wenhe im writes as:

P2 = P~(ll + 12 + 1'~2)(12 + 11 + 1'~2)(12 + 12 + 122)(21 + 21 + 1'~)

x(21 + 22 + 1'~2)(22 + 21 + 12~)(22 + 22 + 122)12" 32

= Pl(21 + 22 + 1'~2)12" 32

= (22 + 1'12) " 1! �9 12" 21 �9 32

(p. 239).

However, we now know that P2 has certain factors that include the factors of PI, because when h, i, j range over 1, whatever factors we get are a subset of the factors we get when h, i, j range over both 1 and 2 freely.

We can now ask the same quest ion for Pz. Does P2 vanish identically? In o the r words, for every equa t ion over 1, 2, and 3, and for every definit ion of the predicate z, does P2 vanish? If it does, then we must say that the original fleeing equa t ion vanishes, which is not possible. If P2 does not vanish identically, we will let h, i, j range over 1, 2, and 3, and now there will be someth ing called k~, which we will call 4. We will then get some more factors, and call the new p roduc t ~ , which L 6 w e n h e i m writes as


P:~ = (22 + 1'12)(23 + 1'13)(22 + 23 + 12.~)(31 + 1'12 )

x(31 + 3 3 + 1' 13)(33 + 123)

�9 11 �9 12" 13" 21 " 3 2 " 4 3

~87

(p. 239).

Again, we will have several possible equat ions be tween the e lements 1, 2, 3, and 4. For each of those finitely many possibilities, there are several versions of ~: ~' , ~", ~" , and so on, but only finitely many.

If ~ does not vanish identically, there exists some set of equa t ions ho ld ing between 1, 2, 3, and 4, and some def ini t ion of z such that does not vanish. And so on. We will never stop, because if at any stage it happens that for some P,,, P,, vanishes identically, this would immediately imply that the original f leeing equa t ion vanished identically. This means that we can go on and on and cons t ruc t P1, P2 . . . . , Pk . . . . . up tO P=: for any finite n we can cons t ruc t P,,. Moreover, at every stage k, we are able to make Pk not vanish for some def ini t ion of z's and for some defini t ion of the symbols that occur in Pk-

We now p roceed to the last stage of L6wenhe im ' s proof. We have an infinite sequence Pl, I~ . . . . . L6wenhe im says that if for some K all P," vanish, then the equa t ion is identically satisfied. This means the original f leeing equa t ion is identically satisfied, which is not possible, as shown by the p reced ing discussion. In P~", the v's are primes, and so it c anno t h a p p e n that all pr imes vanish. As L6wenhe im says,

If they do not vanish, then the equation is no longer satisfied in the denumerable domain of the first degree we have just constructed. (p. 240)

Tha t is, we cons t ruc ted a doma in using symbols 1,2, 3 . . . . . and it is at least d e n u m e r a b l e . Since we have m a n a g e d to cons t ruc t the sequence Pl, P2 . . . . wi thout s topping anywhere, this means that we have an infinite set of symbols 1,2, 3 . . . . . Now what is t rue over this domain? L 6 w e n h e i m is c la iming that the equa t ion we star ted with is not true over this infinite d o m a i n identically. Now he shows why. The overall strategy of the p r o o f was to come up with some defini t ion of the z's and some ass ignments to the existentially quant i f ied variables that would make the no rma l form equa t ion I I F = 1. If we could show this, it would imply the theo rem, and L 6 w e n h e i m has now m a n a g e d to cons t ruc t it. Let us see how.

First of all, L6wenhe im says that we have a d e n u m e r a b l e domain . To really get a cont radic t ion , that is, to prove the t heo rem, we need to assign values to the original free variables and assign values all the k~'s such that the p roduc t 1-IFequals 1; as L6wenhe im says,


For then among the Pl', Pl", Pl" ..... there is at least one Q1 that occurs in infinitely many of the nonvanishing P,~ as a factor. (p. 240)

By Q~ he means one of the finitely many P~""s.

Neglec t ing the pr imes for the m o m e n t , we recall that the p r o d u c t P~ itself existed as a part of P2, which was a par t of P3. This means that

Pl exists in all fu r ther products we construct: P2, P3, P4, and so on. Now at any stage Pk, there was some set of symbols that we had to hand, and

some equat ions a m o n g them. Cons ider the stage Pk. T h e r e were Pk',

Pk", and so on, d e p e n d i n g on what equat ions we cons ide red a m o n g the exist ing variables. These equat ions were equat ions for all the symbols

in Pk- In particular, they inc luded the symbols that existed in P~. This means that these equat ions also def ine some equat ions that existed

purely between the symbols in P~. In our example , for P~' we gave Sl = s 2, and then when we went to P2 we said that there would be some p r ime of P~ that would include the ass ignment s 1 = s 2.

We have c la imed that Pk does not vanish identically. This means that

for some set of equat ions of the symbols in Pk and some def ini t ion of z, Pk does not vanish. Now Pk includes, a m o n g all its factors, factors that

consist purely of symbols f rom P1. Whatever equat ions exist for Pk, we

can select the subset that jus t concerns itself with symbols in P1. For example , suppose we are cons ider ing all the possible equa t ions be tween

s~, s 2, and s3. These are s I - " S 2 and s 2 ~ s3; or s~ = s 2 and s 2 = s3; or s~ ~: s 2 and s 2 = s3; or s~ e: s 2 and s 2 ~ s:~. However, if we cons ider any of these four possibilities, it will also say some th ing about the s~ and s 2 equat ion . There fore , we jus t separa te out the por t ion that talks only abou t s~ and s 2.

L6wenhe im directs us to cons ider Pk and that set of equa t ions for which it does not vanish. The s t a t emen t that Pk does not vanish for some set of equat ions implies that there is some set of equa t ions over symbols

in P! for which P~ does not vanish, since Pl is no th ing but a subset of

all the factors in Pk. In o the r words, if a large p roduc t is not equal to

0, then every subset of it is not equal to 0. For example , suppose

x~ x xz x x~ x x 4 is not equal to 0; this means , in particular, that

x 1 x x 2 is also not equal to 0. This is a proper ty of product . L6wenhe im

is saying the same thing here. Since Pk involves all the factors of/]1, if for some ass ignment Pk did not vanish, u n d e r the same ass ignment , that

subset of the product , which only involves factors f rom Pl, also does not

vanish. If the original p roduc t did not vanish u n d e r some equat ions ,

then this subproduc t also will not vanish u n d e r that same set of equa-

tions, a l though only cons ider ing the relevant part. The re fo re , if Pk does not vanish for some defini t ions of z and some equat ions, then P~ does

not vanish for some defini t ion of z and some equat ions.

Now Pk does not vanish, Pk+~ does not vanish, and so on, so there is


an infinite s equence of these Pk, Pk+~ . . . . . such that each of t h e m does no t vanish for some def in i t ion of z and some e qua t i on be tween its symbols. Each one mus t con ta in as a s u b p r o d u c t some e q u a t i o n over /]i and some def in i t ion of z over P~, as jus t d e m o n s t r a t e d . However , the re are only finitely many possible equa t ions over P~ and finitely many possibilities o f z over P~. This means that it mus t be e i the r Pl' o r Pl" or P~", and so on. O n e of these mus t be occu r r ing as a s u b p r o d u c t of each Pk, because we have listed all the possibilities for m a k i n g P~.

T h e r e is one such Pl' or PI", e tc . , that occurs for Pk ". T h e r e will also be one such occu r r ing for "k+~P"k+~, because this is t rue for all k's', that is how the cons t ruc t ion con t inued . If at any po in t Pk did vanish, t hen the con- s t ruct ion collapses, so this process is ongoing . This m e a n s that for each of these levels, some P~' or P~" appears . However, this h a p p e n s infinitely many times, and we have only finitely many pr imes, which m e a n s that there must be some P~' or P~", etc., that is occu r r ing infinitely many times. This is precisely what L 6 w e n h e i m says:

There is at least one Q~ that occurs in infinitely many of the nonvanishing P~{~) as a factor (since, after all, each of the infinitely many nonvanishing P~(") contains one of the finitely many Pi (") as a factor). (p. 240)

T h e r e are infinitely many nonvan i sh ing P~{")'s, because K goes f rom 1,2, 3 . . . . , bu t there are only finitely many P~"l's, and o n e of t h e m mus t exist as a factor in each of those nonvan i sh ing factors.

We apply the same a r g u m e n t for P2. Cons ide r P2', Pz", Pz', etc. T h e r e is at least one Q2 that conta ins Q~ as a factor and occurs in infinitely many of the nonvan i sh ing Pk" as a factor. L 6 w e n h e i m then says (omi t t ing the phrase "that conta ins Q1 as a factor and") :

Furthermore, among P2', P2", P2" .... . there is at least one Q2 that occurs in infinitely many of the nonvanishing Pk" as a factor. (p. 240)

This is identical to the a r g u m e n t given above. T h e r e exists at least one Q2 that occurs infinitely many times as a factor. But no t only does there exist o n e such Q2; the re will exist one such Q2 that c o n t a i n e d that Q1 that o c c u r r e d infinitely many times for the P~ case. Now why is this true?

In the first stage, we isolated one Q1; Ql is o n e of the Pl""s such that it o c c u r r e d in infinitely many of the nonvan i sh ing Pk"'S. We took this Ql, and, a m o n g all the Pk"'s, we jus t took those Pk's for which Ql was a factor. We know that the re are infinitely many of them. We now clear away all the r e m a i n i n g Pk's.

In this new, smal ler world of Pk's, which is still infinitely large, we are

9 ~ LOWENHEIM:S CONTRIBUTION

g u a r a n t e e d that Q~ agrees everywhere, wherever this Pk vanishes because we have thrown away the cases in which Q~ did no t occur as a factor. To rephrase L6wenhe im ' s s ta tement ,

Furthermore, among Pj, P2", P2" .... . there is at least one Q2 that occurs in infinitely many of the nonvanishing Pk ~ [of the new world] as a factor (p. 240).

In this new world there are infinitely many Pk"'S that are vanishing. Since Pk" is a large p r o d u c t that consists of all the P2 in it, and since k is g rea te r than 2, then a m o n g the finitely many P2', P2", P2" the re will exist some Q2 that appears infinitely many times as a factor.

Even in this smaller world, because it is infinitely large, the re is some Q2 that appears infinitely many times, because there are only finitely many P2"2's. O n e of them will thus be such a Q2 that would a p p e a r infinitely many times in our new world. But ou r new world consists of those things in which Q1 is a factor, so this Qz would con ta in Ql.

Qz is in a larger world that conta ins the world of Q~, so Q2 will con ta in Q1. Cons ide r all the factors in P2; they inc lude all the factors in P~. Tha t is how we cons t ruc ted it. The re fo re , once we assign some def in i t ions o f

to P2, it also automatical ly assigns some def in i t ion of z for the factors that came f rom P1. Thus L 6 w e n h e i m says that when we choose a Q.2 in the new world, this means that it conta ins Q~ as a factor, and so on.

Similarly, in this new world we can again throw away those things that do no t con ta in Q2 as a factor, and so get a still newer world. This newer world is again infinite. By the same a rgumen t , we can choose a Q.~ there , and so on. O u r initial Ql was some equa t ion for symbols in P1 and some as s ignmen t to z over the world of P~. Wha t was Q2 ? Some equa t ion over the world of P2 and some ass ignment to z's over the world of P2. Wha t was Q37 Q3 was some nonvan i sh ing ass ignment to z's in P3 and some set o f equat ions . The fact that it was nonvan i sh ing means that Q~ is equal to 1. Similarly, Q2 includes Q1 and some o t h e r factors. Thus , each of these Q, is equal to 1. As L 6 w e n h e i m puts it,

Every Q, is 1; therefore we also have

1 = Q I Q , e Q 3 . . . ad infinitum.

(p. 240)

All o f the Q's that we pick this way are equa l - to 1, so thei r p r o d u c t is equal to 1.

Yet not ice what we have done . O u r QI assigned some values to some free variables and some of the ki's. Since Q2 i nc luded Ql, Q2 also m a d e some ass ignments to some m o r e ki's and some m o r e free variables, bu t


these assignments did not clash with the previous assignment because Q2 included Q,. In the same way, Q3 included Q2, so Q3 performs a fur ther assignment at the level at which our world is growing. Thus, Q3 now gives some assignment to z's that does not clash with the previous assignment and some assignment to those symbols such that it does not vanish. If we simply follow the assignment given by QI, Q2, and Q3, and go on similarly and keep assigning all the k~'s accordingly, we will get a large factor that is equal to 1. This means that we have p roduced an assignment for the free variables and for all the k~'s such that the product equals 1.

The product of these factors, which is nothing but IIF, thus equals 1. This means that we have managed to produce an assignment of relation symbols for which the fleeing equation we started with does not hold. This, in turn, means that the fleeing equat ion does not hold for arbitrary assignments, since we have just now shown an assignment for which it does not hold.

The a rgumen t for this proof is complete.

8.3. Conclusions

Our analysis shows in detail how L6wenheim first reduced any predicate logic prenex formula over a fixed domain to a simple disjunctive normal form, but with terms so notationally complicated that it is difficult to read and write them.

L6wenheim then argued, in the second part of his proof, that if we suppose that we have a single formula satisfied in an uncountable domain, then all the quantifiers range over it, so only one naming is needed for all the elements of that domain. If we distribute completely (using the axiom of choice to justify the distributive laws), we have a disjunction of countable conjunctions of atomic statements equivalent, by the process above, to the arbitrary predicate logic s ta tement over the uncountable domain in which it is true. The disjunction is, as we have explained, over a huge space. Since the s ta tement is satisfiable, one of the countable conjunctions is satisfiable, because the s ta tement and the disjunction are equivalent by the distributive laws.

Any one countable conjunct ion /s countable. Defining a countable model using exactly the constants men t ioned in this s tatement, declar- ing exactly the terms of the conjunct ion true, gives a countable model. The problem is that since we just changed to the domain of constants on this conjunction, the equivalence with the original s ta tement does not follow by distributivity. Rather, it must be proved directly, which L6wenheim does by verifying the original s ta tement on the conjunction- defined model, working through all the quantifiers from inside to out.

x92 LOWENHEIM.'S CONTRIBUTION

L6wenheim did not state this clearly. He simply showed that it is implied that in his tree of possible valuations, at no finite point can all the branches be blocked, since some branch of that length is part of a conjunction (of the sort ment ioned above) that is true. Today we would finish the proof by appealing to K6nig's infinity lemma, namely, that a finitely branching tree with infinitely many branches has an infinite branch. But K6nig's lemma was not published until 1926-1927. G6del, in his proof of the completeness theorem in 1930, says at the same point of the proof that it follows "by a familiar argument." What was the familiar argument? The answer is the so-called Bolzano-Weierstrass theorem, namely, that every closed, bounded infinite set of points on the real line contains a member that is a limit point. Every German mathematician learned the Weierstrassian presentation of calculus, in which the Bolzano-Weierstrass theorem is proved. We conjecture that this is G6del 's "familiar argument," and that it is the one L6wenheim had in mind as well. It is not just a similar proof. K6nig's lemma can be derived from the Bolzano-Weierstrass theorem in the form stated above, by a simple coding. In any case, this a rgument completes the proof of L6w- enheim's theorem.

In L6wenheim there are no new function symbols, just existential quantifiers ranging over functions. There is often a cont inuum of equally good choices of such functions when the domain is infinite. This fact probably led L6wenheim away from Skolem's later argument . Skolem had a finite number of function constants in his statement, each denoting a function on the domain. He could take a finite nonempty subset of the domain, close it under these finitely many functions, and get a countable (elementary) submodel of the original model that satisfies the statement.

This approach did not occur to L6wenheim. Instead, he used the existence of functions witnessing the existential function quantifiers repeatedly to assure that the finitely branching tree he was building had arbitrarily long branches and therefore an infinite branch. The infinite branch was labeled with a complete definition of a countable model satisfying the desired statement. This procedure does not lead directly to a countable submodel of the original model, but it does lead to a countable model. What L6wenheim had discovered was that using the hypothesis that there exists an infinite model of the s tatement with the existential prefix guarantees a "semantic consistency" property for the tree, which assures that it has arbitrarily long branches, an infinite branch, and a countable model described by that infinite branch.

One may ask why a second-order statement with a model cannot be proved to have a countable model by this method. L6wenheim in fact raises this question in the discussion following his proof of theorem 3. The reason is that no analysis of all possible ways of introducing Skolem


funct ions ( the index sequences) is offered for s econd-o rde r logic quantifiers. Tha t is, the tree of Skolem funct ions that are possible is in the first-order case a countably b r anch ing tree with every level finite.

If one is looking for witnesses for second-orde r quant if iers , one is

looking for an expression, (for every set x ) ( the re exists a set y), that has a c o n t i n u u m of choices of y. We would n e e d to have a very large,

c o n t i n u u m - b r a n c h i n g tree. This is some th ing L 6 w e n h e i m clearly never investigated. However, if the quest ion is whe the r the a r g u m e n t can be adap ted to prove someth ing , it can, as can all the later, s impler argu-

ments . To do so, one simply uses the not ion of the H a n f n u m b e r of a language. With every s t a t emen t that has a model , the re is a m o d e l of

least cardinality, usually some infinite cardinal. O n e takes the m a x i m u m of these cardinals over all s ta tements of the language. This is the H a n f number . The ana logous quest ion is thus: what is the H a n f n u m b e r of

second-orde r logic? If we want to es t imate it, we n e e d a transfini te a r i thmet ic apparatus . L6wenhe im probably did not know anyth ing abou t

transfinite numbers , since a lmost no one did at that t ime.

Wang 's (1970) discussion of L6wenhe im ' s p r o o f and the distributive laws in his in t roduc t ion to Skolem's Selected Works is similar to our in- te rpre ta t ion . Wang observes:

The use of "Skolem functions" seems to go back to logicians of the Schr6der school to which L6wenheim and Korselt belong. They speak of a general logical law (a distributive law) which, in modern notation, states:

Vx3yA(x, y) =- 3f VxA(x,f(x)).

(Wang 1970, p. 27) Wang remarks that L6wenhe im ' s a r g u m e n t is "less sophis t icated" than

Skolem's , a view that we do not share. However, Wang 's sketch of L6w-

e n h e i m ' s p r o o f agrees with our m u c h m o r e deta i led account :

Suppose each schema has solutions for each level k. Let E k be the finite set of solutions of level k and E be the infinite union of these sets El, E2, etc. Within E l there must be a solution Ql of level 1 which occurs as a part of infinitely many members of E, since each of the infinitely many solutions in E contains one of the finitely many member of E 1 as a member . . . . Hence we can take the union of Ql, Q2, etc. and give an interpretation of the given schema in N. (Wang 1970, pp. 27-28)

W h e r e are the m o d e r n Skolem functions? They are the witnesses to the existential funct ion quantif iers, de f ined by the final values of f leeing subscripts in the final model . What L6wenhe im does is to i n t roduce a


generic search for witnesses, which succeeds only at the end with the final model, and witnesses the second-order existential function quantifiers.

L6wenheim's proof is based on a search for better and better approximations to witness the second-order existential function symbols by trying larger and larger initial segments of the integers as domains, with trial definitions of the witnesses on that domain given as fleeing subscripts. The limit of these approximations is obtained by a K6nig- style tree argument.

Korselt and L6wenheim were the first to establish that, when compared side by side, relational algebra without quantifiers as originally conceived by Peirce was weaker than the first-order f ragment with quantifiers as extracted from (perhaps) Peirce and from the last volume of Schr6der 's Algebra der Logik. The source of the example in L6wenheim of a "fleeing equation" is difficult to trace. Does it emerge from a reading of Schr6der, or elsewhere? And why is nei ther Russell nor Frege mentioned? There is no trace in L6wenheim of a non-Schr6der origin; maybe there was none, even if Korselt and L6wenheim had read o ther writers.

It is natural to ask and difficult to determine whether ei ther L6w- enheim or his predecessor Schr6der had a discernible relation to the German algebraists, whose thought about abstract systems and their properties was coming into focus th roughout mathematics. In 1910 the German mathematician Ernst Steinitz published a work that became famous, in which he gave the first abstract definition of a field and proved the existence of an algebraic closure, i.e., if an equation is solvable, we can construct an extension field that contains the e lement that solves it and then obtain a maximal extension; the maximal extension is proved to be the algebraic closure.

We do not know if Steinitz's proof of the existence of algebraic closure is historically connected to the L6wenheim construction. However, Steinitz carried out his work on algebraic extension fields while he was Privatdozent at the Technische Hochschule Berl in-Charlottenburg from 1894 to 1910, and L6wenheim was a student at the Technische Hochs- chule Berlin-Charlottenburg from 1896 to 1900.

Mathematically, there is certainly some relation. Each constructs models out of symbols. The construction for adjoining a root of an irreducible polynomial to a field containing its coefficients is attributable to Leopold Kronecker, who did this as part of his reduction of the notions of mathematics to the theory of integers. He took a polynomial domain in x over the field (formal expressions) and reduced the domain modulo the ideal generated by the irreducible polynomial. In the quotient, which is an extension field of the original, the equivalence class of the indeterminate x is a root of that polynomial. (The language of ideals is Richard Dedekind's; the construction itself is Kronecker 's


[1882]). Steinitz applied this successively to adjoin all roots of all polynomials using the well-ordering principle, which was being used then by very few mathematicians. Every equation over the resulting field has a root in the same field. Thus Steinitz put in witnesses (roots) for all polynomial equations that occurred in the construction, and they are very formal expressions (polynomials with coefficients in the field constructed at the previous stage). In analogy, L6wenheim introduces witnesses to every first-order statement in an arbitrary first-order theory to build his countable model of that theory. Steinitz was witnessing existential statements asserting the existence of roots, while L6wenheim was witnessing arbitrary existential quantifiers at the front of arbitrary first- order statements in the theory. Perhaps Steinitz's earlier use of witnesses influenced L6wenheim, but this is only conjecture. More likely, the time was ripe for this kind of abstract construction.

The not ion of finding a solution to a relational problem is strong in Schr6der and is criticized by Peirce, and it seems to be the closest ancestor to the method of proof adopted by L6wenheim. The latter explicit me thod is definitely due to Schr6der and Peirce, using an informal definition of satisfaction of a prenex statement over a domain based on writing the quantifiers as unions and intersections and using a giant distributive law, which seems to appear nowhere else at that time.

We have placed the L6wenheim argument in a context: namely, the part of the a rgument that L6wenheim's commenta tors thought was infinitary really is, but it merely applies extended distributive laws; he also had no good notat ion for functions or function spaces, and had to use subscripts. Skolem realized that these functions could be in t roduced in the first-order language, especially since he was familiar with Schr6der 's notion of solving a relational equation by int roducing a function and then represent ing it by a relation. He recognized that if he did this, he would get an equisatisfiable statement involving function symbols, and when the function symbol is replaced by a relation, one gets the Skolem equisatisfiability form. This is probably the origin of Skolem functions.

8.4. Impact of L6wenheim's Paper

It is not clear that anyone read L6wenheim's theorem and L6wenheim's original proof before Skolem, except perhaps L6wenheim's colleague Alwin Korselt, whose own result showing that there are first-order formulas that cannot be expressed in the calculus of relatives (without quantifiers) was published as theorem 1 in L6wenheim's paper.

Hilbert was then the editor of Mathematische Annalen, in which L6w- enheim's 1915 paper was published, but it is difficult to de te rmine when and how Hilbert first became aware of L6wenheim's theorem. Since

196 LI~WENHEIM'S CONTRIBUTION

results from Lrwenheim (1915) are summarized in Hilbert and Ack- e rmann (1928), notably the decision method for monadic predicate logic, it is certain that either Hilbert or Ackermann, or Paul Bernays, who assisted with the book, had learned about and understood Lrw- enheim's paper by 1928.

Hilbert and Ackermann (1928) write that Lrwenheim's theorem showed that every expression that is universally valid for a countable domain of individuals has that same property for any other (i.e., uncountable) domain (Hilbert and Ackermann 1928, p. 80). They refer to Skolem's 1920 paper for a simpler proof of Lrwenheim's theorem. However, they make an interesting statement that suggests that whoever authored their discussion of Lrwenheim's theorem may not have read Lrwenheim's original paper; namely, they say that the theorem is stated by Lrwenheim in its dual form: i.e., every formula of the predicate calculus is either contradictory or already satisfiable in a countably infinite domain. This is not L6wenheim's own statement of the theorem but rather Skolem's, taken almost word-for-word from the first paragraph of Skolem (1920).

It is possible that Hilbert and his associates may not have read the Lrwenheim paper at all, but learned about it through Skolem's 1920 paper and through reports of Skolem's 1922 congress address, which apparently was widely talked about. They do not mention Skolem's 1923 paper, containing his second proof of Lrwenheim's theorem, which resembled very closely the proof given by Lrwenheim himself and also Grdel ' s later proof of the completeness theorem. Did Lrwenheim's paper have a referee, and if so, who? If his paper was not refereed, was it communicated to the editors of Mathematische Annalen by a prominent mathematician, and if so, which of the editors accepted it for publication? We have been unable to ascertain the publication details of Lrw- enheim's paper, since there are no extant records of it at Mathematische

A n nalen.

Skolem studied in Grt t ingen in 1915, and within the very year that Lrwenheim's paper was published, had already realized the implication of L6wenheim's theorem for set theory, namely, the relativity of set theory (the Skolem paradox), and had communicated his ideas to Felix Bernstein. 2 In the next chapter we will compare Skolem's two proofs of Lrwenheim's theorem and determine what new facets Skolem contributed to L6wenheim's original argument.

See Concluding Remark to Skolem 1923, in van Heijenoort (1967), pp. 300-301.

9. Skolem's Recasting

Thora l f Skolem was well versed in Schr6der 's work. Several of Skolem's earliest papers of the years 1913 to 1919 are devoted to problems in Schr6der 's algebra of logic. ~ He then, being possibly the only person active at the time other than Norber t Wiener who knew Schr6der ' s work more than casually, turned to a study of L6wenheim's theorem. We may assume that, because he knew Schr6der 's notation and methods, Skolem had no difficulty penetra t ing L6wenheim's text. In fact, it appears that, a l though L6wenheim's paper was published in 1915 in the premier mathemat ical journa l in the world, it was not until Skolem's paper of 1920 that L6wenheim's theorem received any attention.

Here we discuss two of Skolem's versions of his proof of L6wenheim's theorem, from 1920 and 1922 (the latter published in 1923) and take a brief look at a lacuna filled by another paper of Skolem's on the L6wenheim theorem from 1929.

Skolem's 1920 paper, "Logico-combinatorial investigations in the satisfiability or provability of mathematical propositions," begins with a theorem known today as the Skolem normal form (Skolem 1920, 2 p. 255). He proves (given here using more compact notat ion):

Given any s ta tement of first-order logic 4~, there exists a quantifier- free X with at most x I . . . . . x .... Yl, ...,Y,, free such that 4~ is satisfiable in some domain if and only if

( V X l ) " '" (Vx,,,)(~y~).." (3y,,)x

See especially "The structure of groups in the identity calculus" and "Untersuchungen fiber die Axiome des Klassenkalkfils und fiber Produktations- und Summationsprobleme, welche gewisse Klassen von Aussagen betreffen" in Skolem's Selected Work~ (1970).

Here and subsequently, all page citations to Skolem (1920) refer to the reprint in van Heijenoort, ed., From Frege to G6del (1967).

197

19 8 SKOLEM'S RECASTING

is satisfiable. Skolem's s t a t emen t of the L 6 w e n h e i m t h e o r e m is:

Every [first-order] proposition in normal frnvn either is a contradiction or is already satisfiable in a finite or denumerably infinite domain. (Skolem 1920, p. 256)

By "a con t rad ic t ion" Skolem certainly means "has no models" since the re are no syntactic proofs based on formal consis tency in e i the r Skolem's 1920 or 1923 papers on L 6 w e n h e i m ' s t heo rem; they are semant ic only.

Skolem's 1920 p r o o f of L 6 w e n h e i m ' s t h e o r e m uses the ax iom of choice. It is qui te simple. It is also no t L 6 w e n h e i m ' s proof , a l t hough the ideas are p resen t in L 6 w e n h e i m in a less clear form and in a l onge r proof . Skolem's approach , again in simplif ied language , is as follows:

Proof Since the first-order p ropos i t ion 4~ is satisfiable, so is its no rma l form, f rom the t h e o r e m above. The lat ter has a mode l M. By the axiom of choice, there are n funct ions f~ . . . . . f , of m a r g u m e n t s de f ined on M such that for all X~ . . . . . X m e M:

x ( x , , . . . , x .... f , ( x , . . . . . x . , ) . . . . . L ( x , . . . . . x , . ) ) .

If we take any fixed a e M, the closure of {a} u n d e r the funct ions f~, ... , f , is the des i red model . This is the smallest subset of M con t a in ing

a and closed u n d e r the funct ions f~ . . . . . f , . T h e fact that this subset is c o u n t a b l e seems evident to us today, but Skolem (1920, p. 258) appeals to D e d e k i n d ' s theory of chains to draw this conclus ion. This ends the proof .

Skolem's later 1922 p r o o f was devised to avoid using the ax iom of choice. We believe that L 6 w e n h e i m ' s tree p r o o f also did no t use the ax iom of choice, but the exposi t ion there is muddy.

In o t h e r words, as we see f rom Skolem, L 6 w e n h e i m ' s t h e o r e m can be p roved in Zermelo-Fraenke l set theory wi thout the ax iom of choice. It is no t surpr is ing that Skolem would e x p e n d cons ide rab le effort to prove this. He was one of the main p r o p o n e n t s o f the view that set theory is a theory in first-order logic and did no t s u p p o r t formal ized f irst-order set theory as an ul t imate f o u n d a t i o n of mathemat ics . In his

1923 paper , "Some remarks on ax iomat ized set theory," Skolem app l ied L 6 w e n h e i m ' s t h e o r e m to d e d u c e that if the axioms of set theory have a mode l , then they have a coun tab le model . This was to show that set theory can prove in first-order logic that there are u n c o u n t a b l e sets, such as the real numbers ; but if set theory has a m o d e l that is a set, it has a coun tab le mode l in which all sets of the mode l , and h e n c e the set of real n u m b e r s of the model , are countable . This is called Skolem's paradox . Skolem's 1923 p a p e r shows that this pa r adox can be o b t a i n e d wi thou t using the axiom of choice. (Of course, this is no t a paradox;


the def in i t ion of countabi l i ty has changed . O n e is the ex te rna l world 's

[meta theory ' s ] no t ion of countabil i ty; the o t h e r is the m o d e l ' s no t ion of countabili ty. This is what is r e fe r red to as the relativity of the no t ion

of cardinal in set theory, or the relativity of se t - theoret ic not ions . ) We will he re e x a m i n e the closure ope ra t i on used in Sko lem's 1920

p r o o f above in detail, n a m i n g by an in teger every e l e m e n t tha t en te rs

the m o d e l be ing cons t ruc ted . This leads directly to Skolem's 1922 proof . Suppose we are given a finite s equence of k e l emen t s of M. Wha t is

the n u m b e r of ways to form a subsequence X~ . . . . . Xm of those k e l emen t s

as values for the variables Xl . . . . . Xm in X? The answer is km. If we choose

(as in Skolem's 1920 proof) for each such s equence X 1 . . . . . Xm (by the

choice func t ion in his 1920 proof) one s e q u e n c e of n e l emen t s

Yl . . . . . Y,, f rom D such that X1 . . . . . Xm, Yl . . . . . Y,, satisfies X, how many Y's are i n t r o d u c e d a l together? If we have k names for the e l emen t s of

the initial sequence , we n e e d n(k m) new names for the Y's. Now we go

back to closing {1} u n d e r the choice funct ions that give Y's f rom X's. We have jus t seen that if we break the closure o p e r a t i o n into steps of

the above sort: �9 At stage 1 we i n t roduce 1 as a n a m e for a single e l e m e n t a of M,

set t ing c 1 -- 1. ~ At stage k + 1 we i n t roduce n(Ck) m n e w names. These are i n t r o d u c e d

n at a t ime as the first u n u s e d integers to be used as n a m e s of witnesses

Yl . . . . . 1I,, f rom M c o r r e s p o n d i n g to the X l . . . . . Xm cur ren t ly cons idered . Skolem's choice is thus to use successive in tegers as names. Every t ime

a new Yl . . . . . Y,, is i n t roduced , the first n previously u n u s e d in tegers are

i n t r o d u c e d as the names of the Y's. If we start with o n e e l e m e n t {a}, and let c I = 1 and ck+ l = c k + n(ck) m, we will have e n o u g h names a m o n g the in tegers 1,2 . . . . . c k to n a m e all e l emen t s o b t a i n e d in M for witnesses up to the kth level of the closure p r o c e d u r e as ou t l i ned above. This is how names of witnesses grow if we are bu i ld ing the m o d e l given in

Skolem's 1920 p r o o f by closing a o n e - e l e m e n t subset of M u n d e r the

witnesses p rov ided by the choice funct ions. This induct ive process names

every e l e m e n t of the closure. Relative to the choice funct ions , it specifies

a coun t ab l e s u b m o d e l of M. However, what if we are given no such choice funct ion? T h e set of

names i n t r o d u c e d and how they are o rgan ized is i n d e p e n d e n t of what

choice func t ion is used; in fact, it does no t d e p e n d on a choice func t ion

at all. It is simply a col lect ion of names with an o rgan iza t ion and in ten t

to n a m e e l emen t s of M as they migh t be used as witnesses. We thus

have a col lect ion of names suitable for witnesses Y, wi thou t having cho-

sen any witnesses. The object of Skolem's 1922 p r o o f is to make up a

coun tab l e m o d e l out of the names themselves ( in tegers) , with no use

of the choice funct ions. We will use the fact that the witness Y exists for a given X because

2 0 0 SKOLEM'S RECASTING

the normal form statement is true in M. But we do not use any uniform way of choosing Y as a function of X. We do not get a denota t ion in M for the names. We use the fact that statements are satisfiable in M, but do not name specific elements of M. This is how the axiom of choice is avoided.

This is also the first step toward syntactic model-building procedures , which had already been used by L6wenheim in his tree construction. We conjecture that Skolem went back to L6wenheim's original tree a rgumen t (isolated by K6nig in 1929 and known as the K6nig infinity l emma argument) when trying to find a proof that did not use the axiom of choice. We note that at the crucial point Skolem is no more explicit in his 1923 paper regarding the use of an a rgumen t in the style of K6nig's l emma than was L6wenheim. Skolem gives a full a rgumen t only in his 1929 paper "I]ber einige Grundlagenfragen der Mathematik" (Skolem 1929, pp. 227-274).:~

Skolem's s ta tement of the L6wenheim theorem in his 1923 paper is as follows:

I f a first-order proposition is satisfied in any domain at all, it is already satisfied in a denumerably infinite domain. (Skolem 1923, p. 293) 4

Skolem then remarks that L6wenheim, in his proof, must make a de tour into the nondenumerab le . What he means by this s ta tement is that there are a nondenumerab le n u m b e r of possible indicial sequences (or Skolem functions), and L6wenheim is always trying to show tha t , as

he goes on, there are some candidate indicial sequences (Skolem functions) left to work with. Skolem does not do this. He does not hun t th rough the completed set of all possible Skolem functions for one that works.

We will now work through the steps of Skolem's 1922 proof, giving commenta ry in more modern terminology. We use the normal form again,

( V X l ) "" ( V X m ) ( 3 y l ) "'" (: ty .) X(Xl . . . . . Xm, y~ . . . . , y,,),

X being quantifier-flee, true in model D. (Skolem writes X as U~, . . . . . . ~ . , , ..... y . )

Proof We have specified above the names of level k + 1 as an initial segment of the integers extending the previous names of level k. We consider ground instances of X, that is, we substitute integers for the

.s In Skolem's Selected Works(1970) . 4 Here and subsequently, all page citations to Skolem (1923) refer to the reprint in van

Hei jenoort , ed., From l"rege to G6del (1967).


va r i ab l e s in X, g e t t i n g p r o p o s i t i o n a l logic s t a t e m e n t s bu i l t u p f r o m

a t o m i c s t a t e m e n t s o f f i r s t - o rde r logic. S k o l e m e x p r e s s e s this as:

For the first step we choose x I = x~ ="" = x, , = 1. T h e n it must be possible to choose y~ . . . . . y,, among the numbers 1,2 . . . . . n + 1 in such a way that U~.~ ..... ~.>.~ ..... >.,, is satisfied. Thus we obtain one or more solutions of the first step, that is, assignments de t e rmin ing the classes and relations in such a way that Ul. 1 ..... ~.>.~ ..... y. is satisfied. (Skolem

1923, p. 294)

We n e x t a s soc ia t e wi th e a c h level k a s ing le f in i te d i s j u n c t i o n ~k o f

g r o u n d i n s t a n c e s o f X. T h i s is t he d i s j u n c t i o n o f t h e f in i te ly m a n y g r o u n d

i n s t a n c e s o f X o f t h e f o r m x ( X 1, . . . , X m, Y1 . . . . . Y,) s u c h t h a t

X1 . . . . . Xm is a s e q u e n c e f r o m {1 . . . . . Ck}" a n d Y1 . . . . . 11, is t h e c o r r e -

s p o n d i n g s e q u e n c e o f success ive i n t e g e r s i n t r o d u c e d at s t age k + 1 as

n a m e s to wi tness x ( X ~ . . . . . X .... Y1 . . . . . Y, ,) . S k o l e m e x p r e s s e s this as:

The second step consists in choosing, for x~ . . . . . x,,,, every pe rmuta t ion with repeti t ions of the n + 1 numbers 1,2 . . . . . n + 1 taken m at a time, with the except ion of the permuta t ion 1,1 . . . . . 1, already cons idered in the first step. For at least one of the solutions ob ta ined in the first step, it must then be possible, for each of these (n + 1) . . . . 1 permutations, to choose y~ . . . . . y,, among the numbers 1,2 . . . . . n + 1 + n ( ( n + 1 ) ' - 1) in such a way that, for each pe rmuta t ion x I . . . . . x,,, taken within the segment 1,2 . . . . . n + 1 of the n u m b e r sequence , the proposi t ion U,,~ ......... y~ ..... y, holds for a co r r e spond ing choice of

Yl . . . . . y,, taken within the segment 1,2 . . . . . n + 1 + n ( ( n + 1)"' - 1) .

Thus from certain solutions gained in the first step we now obtain certain cont inuat ions, which constitute solutions of the second step. It must be possible to cont inue the process in this way indefinitely if the given first-order proposi t ion is consistent. (Skolem 1923, p. 294)

E a c h s u c h d i s j u n c t i o n ~k is m a d e t r u e by a t r u t h v a l u a t i o n o f its a t o m i c

s t a t e m e n t s . T h i s can be s h o w n , u s i n g t he t r u t h o f t h e n o r m a l f o r m in

M, by a s i m p l e i n d u c t i o n . T h a t is, t he i n t e g e r n a m e s o c c u r r i n g in ~k

c a n be a s s i g n e d as n a m e s to e l e m e n t s in M in s u c h a way t h a t e a c h

i n s t a n c e o f ~ is t rue . Bu t s u c h a t r u t h v a l u a t i o n is n o t s i n g l e d o u t by

an), c h o i c e f u n c t i o n . Ra the r , f o r e a c h k, o n e is s h o w n to exist . S k o l e m

e x p r e s s e s this as:

In o rde r now to obtain a uniquely de t e rmined solution for the ent i re

n u m b e r sequence, we must be able to choose a single solut ion f rom a m o n g all those obta ined in a given step. To achieve this, we can always take the first from all of the solutions obta ined in an arbitrary step, once they have been o rde red in a sequence in the following way.

(Skolem 1923, p. 294)

202 SKOLEM'S RECASTING

We now have, in p r o p o s i t i o n a l logic, an inf in i te s e q u e n c e o f f inite d i s junc t ions ~,, each satisfiable by at least o n e t ru th va lua t ion T k on its

a t omic sen tences . In add i t ion , each d i s junc t ion ~k+l impl ies ~bk, in tha t

every va lua t ion m a k i n g the first t rue makes the s e c o n d true. S k o l e m

t h e n says:

The relative coefficients occurring in the given first-order proposition can be linearly ordered so that the relative coefficients formed within the segment 1,2 . . . . . n of the number sequence precede all new relative coefficients that are formed within the segment 1,2 . . . . . n + 1. For any two different solutions L and L' of an arbitrary step, write L < L ' i f a n d only if R0 is equal t o 0 i n L a n d L'. From L < L ' a n d L' < U it then follows that L < L"; we can also readily see that for two solutions L n and L',, of the nth step that are, respectively, continuations of the solutions L, and L', of the i, th step L,,< L',, implies L,_< L',. (Skolem 1923, p. 294)

At this po in t , we mu s t invoke the c o m p a c t n e s s o f p r o p o s i t i o n a l logic in s o m e fo rm in o r d e r to c o n c l u d e : s ince every finite subse t o f this

c o u n t a b l e set o f p ro p o s i t i o n a l s t a t e m e n t s is satisfiable, t h e n all a re satisfiable at o n c e by a single t ru th va lua t ion T. This T d e f i n e s the r e q u i r e d

m o d e l , a n d the d o m a i n is the set o f all in tegers o c c u r r i n g in at least

o n e ~bk. T h e a tomic re la t ions d e c l a r e d t rue in tha t d o m a i n are those

va lued t rue by T. O f course , the c o m p a c t n e s s t h e o r e m had no t b e e n f o r m u l a t e d in

1922. (Sko lem de l ive red this p a p e r as an address in Ju ly o f 1922.) W h a t d id S k o l e m offer as proof?. This is the o n e p o i n t on which the 1923

p a p e r is i n c o m p l e t e :

Let L~ .... L 2 . . . . . . . . L,,,.,, be solutions of the nth step. If we now form the sequence L1.1, L~.~ . . . . of the first solutions, we can verify without difficulty that they converge in the logical sense. For let L~.,, be a

n ta t

continuation of L,,,L, , (n> v). Then, if n '> n, a~ _< a , . But, since the number a~ can only have values 1 to % it must remain constant for all sufficiently large n. Thus we can obtain as "limit" the fact that the first-order proposition is satisfied in the domain of the entire number sequence, q.e.d. (Skolem 1923, p. 294)

An i n c o m p l e t e appea l seems to be m a d e at this p o i n t to inf in i ta ry

p r o p o s i t i o n a l logic, because we are l o o k i n g at the c o n j u n c t i o n of an

inf in i te n u m b e r of finite d is junct ions . T h e intui t ive bu t n o t p rec i se

a r g u m e n t wou ld be to write this us ing a d is t r ibut ive law as a d i s junc t ion o f inf in i te con junc t ions . Each of the c o n t i n u u m - m a n y inf in i te con junc -

t ions is a c o n j u n c t i o n o f o n e cho ice of an a tomic s t a t e m e n t f r o m each

~bk. Since the d i s junc t ion is n o n z e r o , a s ingle inf ini te c o n j u n c t i o n is


nonzero, and this defines the required truth valuation. Notat ion indicating something of this sort stems from L6wenheim's paper. Note that we are outside countable propositional logic entirely, because the infinite conjunctions are generally a cont inuum in number. This is the best we can do to conjecture as to what Skolem and L6wenheim may have been thinking, in the highly algebraic terms they had acquired from Schr6der. However, there is no easy way to fill in Skolem's argument here because the distributive law used (which is correct for sets) does not hold in these infinitary propositional logics. The compactness theorem does not hold for them, either. It is only the fact that the disjunctions are all finite that saves the argument .

The gap is filled in Skolem's 1929 paper by an a rgument in the style of K6nig's infinity lemma, which goes as follows. List all the atomic propositions involved, that is, all the instantiations by integers of atomic formulas occurring in some tkk. Call this list P0,P~, ..., usually infinite in number. Define a truth valuation T making P0 true if P0 is true under infinitely many of the T,,; otherwise make P0 false. This defines T(po). By induction, define T(pn+l) to be true if infinitely many of the Tk for which Tk(p0) = T(p,,) . . . . . Tk(p, ,) = T(p,,) make P,,+! true; otherwise T(p,,+l ) is false. Without difficulty, this is the right T.

Put in terms of K6nig's lemma, the truth valuations that value all the atomic statements of a ffk and make that s tatement true form, under extension, a finitely branching tree with infinitely many nodes. By K6nig's argument , there is an infinite branch, which defines the desired model. This is the proof given above.

G6del 's thesis (1929) proves the completeness theorem for predicate logic. Divorced from inessential details of notation, he takes the phrase "inconsistent" used in Skolem's (1920) formulation of the L6wenheim theorem in a semantic sense of having no models and checks that, if a formal definition of consistency relative to proof rules is substituted, exactly the a rgument just given still produces a model. That is, if a s tatement does not lead to a formal contradiction, then its Skolem normal form does not either, and the step-by-step construct ion of a model given above in Skolem (1923) still succeeds; every use of satisfiability in M is replaced by an appeal to the consistency of the normal form statement. Thus, the construction succeeds and proves that every formally consistent statement has a countable model. The transformation of the Skolem (1923) proof supplemented by the Skolem (1929) a rgument can be carried out by simply observing what formal deduct ion rules are needed to replace the use of elements of M.

What was an obscure tree a rgument in L6wenheim (1915) becomes a very simple closure a rgument in Skolem (1920), but in constructing it, Skolem uses the axiom of choice. It is t ransformed by Skolem (1923) into a choiceless argument , by transforming to a world of constants. He

204 SKOLEM'S RECASTING

justifies that the needed disjunctions are true by semantic in terpre ta t ion in the model M. In 1929 G6del t ransformed this p roof to observe that the hypothesis of formal consistency is sufficient to replace the hypothesis of t ruth in a model M. In this way the G6del completeness t heo rem was born.

H e r b r a n d (1930) took the p roo f and replaced all existential quantifiers by funct ion symbols and thus replaced the r e q u i r e me n t of satisfying the original s ta tement with satisfying an equivalently satisfiable universal statement. This he does using all the terms built up from a cons tant symbol using the funct ion symbols as his domain , or a Her- b rand universe. He proceeds after that to find, for a consistent universal s ta tement , a similar family of disjunctions as above. In o ther words, the world of constants (integers) used by Skolem is replaced here by the H e r b r a n d universe of terms. The existence of the single truth valuation and the model in which the universal sentence is satisfied if consistent are formula ted in terms of instantiations in the H e r b r a n d universe, not in the integers. He rb rand was a constructivist who did not accept the K6nig l emma as constructive. He did not accept the Bolzano-Weierstrass t h e o r e m either. Thus, his formulat ion and interests went in a different direction. We might say that, due to his constructivist prejudice, he refused to conclude the completeness theorem, and the credit went to G6del.

In the last paragraph of his 1929 paper, Skolem reflects on the relativity of set theory:

The important results of this work are the following: in the first place, the general set-theoretic relativity, which I have already demonstrated earlier. All concepts and theorems of set theory have meaning relative to the axioms and can through the translation from one domain to another be completely altered; especially notable is the relativity of the cardinality concepts. In the second place, the conjecture connected with it, that it cannot be possible to completely characterize mathematical concepts. In the third place, the knowledge that an otherwise contradiction-free theory also remains contradiction-free, when the law of the excluded middle axiom or axiom of replacement is appended, just as the introduction of undecidable sets and the formation of the intersection and the union, especially upper and lower limits of point sets, is possible without contradiction. On the other hand, all these formal extensions of mathematics cannot be used to settle problems which can be formalized without them. (Sko- lem 1929, p. 273)

The astute reader may note that there is an unstated assumption beh ind this paragraph. Skolem and Fraenkel had freed set theory of the second-order quantif ier "for all propert ies P of sets," as in Zermelo,


in favor of first-order expressible properties (classes), and it was becoming apparent that contemporary mathematics could be done in this purely first-order system. Skolem then reverses g round and assumes that every mathematical notion should be stated in first-order logic. Then, with the Skolem-L6wenheim theorem, he finds that set theory cannot specify the notion of cardinal, in that sets in a one-to-one correspondence in one model (by a correspondence in the model as a set) may not be in one-to-one correspondence in another model. Skolem then, having assumed that all formal theories really should be expressible in first-order logic without saying so explicitly, asserts a ra ther absolute noncharacterizability of mathematical notions such as cardinality.

Von Neumann and Bernays saw this gap, and when they in t roduced classes as well as sets, they essentially said that a collection of properties (of which the classes of a model are the extension) can be more or less independent ly specified, possibly larger than those expressible only by formulas with parameters as in Zermelo-Fraenkel set theory. However, this again is all first-order, and whether Skolem was "correct" or not depends on whether, in the future, the second-order version of set theory allowing all properties, however defined, plays a role. It has not yet, so Skolem is still, provisionally, historically correct.

Appendix 1" Schr6der's Lecture I

Introduction

T h e r e are some i m p o r t a n t me thodo log ica l points of Sch r6de r ' s work

that can be exp la ined by a careful r ead ing of this lec ture as a self-

c o n t a i n e d whole, wi thout impos ing the pre judices or l anguage of later

genera t ions . Peirce p r e f e r r ed impl icat ion i n t e r p r e t e d as i n f e rence as

the basis of logic, m u c h as in natura l d e d u c t i o n systems today. S c h r 6 d e r

explicitly acknowledges this, and claims to be deve lop ing the a lgebra

c o r r e s p o n d i n g to logic, no t logic itself. His a lgebra is based on e x t e n d i n g

the fo rma t of the identi ty calculus to a calculus of relat ions. T h e ident i ty calculus is formal Boolean a lgebra based on the Boo lean ope ra t ions and identi t ies alone. Thus , De Morgan ' s law and the distr ibutive law

are s t ipulat ions (axioms) for him. T h e rules of i n f e r ence are the inference rules for equality, and the reilexive, symmetr ic , transitive, and subst i tu t ion of equals for equals rules. In his a lgebraic setup, every proposi t ion is s tated as an identity, and proofs are essentially strings of identities, each ob t a ined f rom previous ones by these rules. This is in con t ras t

to Peirce 's p r e f e r e n c e for logic systems based on m o d u s p o n e n s and

i n t r o d u c t i o n and e l imina t ion rules r a the r than a lgebra systems based solely on identit ies. These two fo rmula t ions are, however, fully equiva-

lent. W h e t h e r one does cylindric or polyadic a lgebras based on iden-

tities, or p red ica te logic based on Hilbert-style axioms and m o d u s po-

nens, is a ma t t e r of taste and historical p r eceden t . S c h r 6 d e r had an

algebrais t ' s taste, Peirce a logician's. ~

S c h r 6 d e r has of ten been accused of no t be ing able to dis t inguish a f rom the set whose only e l e m e n t is a. But as S c h r 6 d e r says explicitly in

Some of Peirce's fundamental papers are in fact full of algebraic identities. He acquired an algebraic skill from his father, the preeminent American algebraist of his time. But however fascinating the algebraic identities were, Peirce nonetheless did not like using them as tile [bundation for logic.

207

208 S C H R O D E R ' S LECTURE I

the first lecture, he is primarily interested in doing algebra, not building up set theory. He takes for granted what Dedekind took for granted. In algebra one can start with objects a;, declared to be atoms, and form the Boolean algebra they generate, where finite supremums of atoms a~ are written a~ + "" + a,,. They generate a complete, completely distributive Boolean algebra, so one can define infinite sums of atoms as least upper bounds of infinite collections of atoms.

It is not stretching things to read Schr6der in this way. An earlier volume of his work was indeed a full-bIown exposition of abstract lattices, distributive or not, with an accurate t rea tment of least upper bounds and greatest lower bounds. This is the way he thought. If we start with a prespecified collection of atoms and build in modern terms the complete completely distributive Boolean algebra it generates, we get what Schr6der specified as the unary relatives of a domain with superscript 1. He does use the infinite distributive laws in later lectures, assuming them to be correct, but never takes them as necessary stipulations. This is probably a gap in his reasoning. But here it is sufficient to establish that in the algebraic way of building genera ted structures, the question of unit sets versus their elements simply does not arise.

Given a domain of atoms, Schr6der introduces the set of ordered pairs of atoms, without worrying about what an ordered pair is. Indi- vidual binary relatives are simply individual ordered pairs. Taking the o rdered pairs of the domain as atoms, he then builds the complete, completely distributive Boolean algebra they generate, now having least upper bounds of subsets of the atoms as the definition of binary relatives. Obviously, the abstract notion of a complete completely distributive Boolean algebra is not explicit, but using this notion, as in t roduced later by Tarski (who admits his intellectual debts to Schr6der) , does make it clear that the algebraic point of view works on pairs as atoms of generated Boolean algebras, and again, the question of the confusion of e lements and their unit sets simply does not come up. There is absolutely no analysis of the notion of ordered pair. One simply takes two atoms and puts a semicolon in between the symbols that denote them. This is a very algebraic way to proceed.

In Schr6der 's original text, Schr6der typographically distinguishes his comments from his main discussion by setting them in smaller type; we have done the same here and in the appendices that follow.

Page 1

Page 2


First Lecture

2 0 9

Introduction

w 1. Outline. The Operational Sphere of the Algebra of Binary Relatives

c~) It is a grandiose discipline, rich in expressions and powerful methods of inference, almost too rich in propositions (al though they are of matchless propor t ion) , into which I will try to introduce the reader.

Although the first beginnings of this discipline--with Augustus De Morgan- -da te back no further than to the middle of this century, the literature in this field is already of considerable bulk; the unders tanding of this discipline is made further difficult because of its dispersion in various, not easily accessible documents, and the differences between what I can only call "hieroglyphic" systems, which its founders used, but which changed often, even within the work of its main promoter , Charles S. Peirce. In addition to these two main creators, the discipline owes a great deal of support to the work of Mr. R. Dedekind. And it is now my task to " round up" to the present, so to speak, the totality of previous contributions.

It seems imperative--lest the overall view and the beauty and rigor of the whole be lost-- to separate sharply the various perspectives from which this discipline can be considered because of the almost unlimited number of directions into which it is capable of being developed, the mult i tude of applications in highly diverse fields--to which the notions "finiteness" [Endlichkeit], "number" [Anzahl], "function," and "substitution" belong as much as, for example, the term "human relation- sh ips"mand because of its dual nature as an algebra, on the one hand, and as a developed form of logic, on the other, namely, its adaptation to the logic of relations (and notions of relations, "relatives" per se).

Thus, I shall first focus almost exclusively on one aspect of the theory, and construct it merely as an algebra, a calculus, which derives its laws necessarily from a small number of accurately formulated, fundamenta l stipulations [Festsetzungen]. Only after we have created a certain foundation and have amassed a considerable resource of absolutely established truthsmfacts of deduction---only then do I plan to come back in much later lectures to the foundat ion of the discipline, in order also to motivate heuristically the previous fundamental stipulations, to discuss it reflectively from the perspective of general logic, and, in particular, to prove it serviceable to the purposes of logic. Until then, logical interpretat ions of expressions or formulas of the calculus appear, at most, at the same time in the form of side glances, with the intent ion to awake the interest of the reader and to introduce him gradually to the art of interpretation which he will later acquire systematically.

For the sake of overall simplification, we ease the theory itself of all

2 1 0 S C H R O D E R ' S L E C T U R E I

additional details and summarize only later in a separate section what ought to be said to acknowledge the contributions of o ther researchers to its development , which is inseparable from a discussion of the history of the literature and from a critical discussion.

My notation follows very closely that which Peirce used in one (Peirce 1883) of his treatises; deviations will be described and justified later. For the sake of the numerous suffixes and also to leave space for the "exponents" or "powers," terms which are finding access in our discipline, we had to change from the vertical to the horizontal negation dash. There will be cases where both are used di f ferent lymal though not in proposit ions [Aussagen]myet for binary relatives (as well for those of a still h igher o r d e r ) m a point to which I will return later.

After having said all this, I immediately proceed to try: /3) First, to give a brief overview of the operat ional sphere [Operation-

skreis] of relative log ic~by compar ing it to the operat ional sphere of Page 3 arithmetical algebra. I thereby focus exclusively on the most impor tan t

part of the former, the

Algebra of Binary Relatives

(called "dual relatives" by Peirce), which constitutes the natural point of depar ture of the whole theory. It alone has so far exper ienced some upgrading, and perhaps the science may limit itself to it in order to deal with its most impor tant problems.

In the identity (system [Gebiete] or class) calculus, we had to get acquainted with three modesm"species"---of calculation [Rechnungsarten]: identity multiplication, identity addition, and negation. Of these, the first two are "tying' [kniipfende] operat ions which presuppose at least two operands (terms) as given for their execution; the latter, a "nontyin~' operat ion, can be carried out on one operand (term). The tying operations are here both associative and commutative.

We come across the same three identity calculus species in the logic of relatives where they constitute indeed the first main stage [erste Haupts- tufe] of e lementary operations. But three more species have to be added as a second main stage [zweite Hauptstufe]: the three "relative" elementary operations, that is relative multiplication (or composit ion) , relative addition, and conversion; the first two are tying and associative, but (in general) not commutat ive operations; the latter is a nontying operat ion which can be carried out on one operand.

Thus, in contrast to general arithmetic with its seven algebraic operations, the logic of relatives has advantages, using only six species. However, it can be argued in favor of arithmetic that it has been possible to reduce its seven species to four, namely, addition, multiplication, exponentiation, and logarithmication, by extending the domain of numbers [ Zahlengebiet] to the field of common complex numbers--by condensing subtraction as an addition of the opposite number,


division as a multiplication, and radication as exponentiation with the reciprocal number, three out of the four inverse operations went into the direct operations.

On the other hand, we have to stress that the six species of the relative logic also can be reduced to four (and this from the start) by reducing, through negation, the two additions to the corresponding multiplications (or conversely);

Page 4 by renouncing symmetry, these operations can be made superfluous. With respect to the definite number of essential, basic operations, the two disciplines thus stand on the same line.

In its thoroughgoing symmetry, the algebra of relatives has an aesthetic advantage over the algebra of numbers. It has two principles for multiplication, a doubling of its propositions, and each of its general propositions occurs mostly with three others, coupled as a tetrad, a quadruple , a "group"[Gespann] of propositions (or formulas); by means of contraposit ion, negation on both sides, it provides a dual relative proposition, and the pair by means of conversion on both sides provides a second "conjugated" pair of propositions whose validity it condit ions and guarantees.

7) Therefore , if the identity calculus appears as a mere part of relative log icBthe most e l e m e n t a r y u t h e latter being thus an extension [Er- weiterung] (special mode of application and cont inuat ion) of the former, there are obviously two possibilities for founding the algebra of relatives.

The first: as a continuation of the present course [Lehrgang] in which we started from the notion of subsumption in order to reach at the end a scientific definition of the individual. The other: as the possibility of an independent foundation, as the construction of the whole discipline on a tabula rasa.

Peirce has given us one such foundation, which takes its depar ture from the consideration of "elements" (or individuals), and compar ing the thereby created, totally distinctive foundat ion for the entire logic with o ther foundat ions can only be instructive. We therefore follow this course, especially since the "continuation" we men t ioned will be very easy and quickly accessible.

w 2. The Domains of Various Orders [Ordnung] and Their Individuals

As given, somehow conceptually de termined, we consider the following "elements" or individuals

A , B , C , D , E , . . . 1)

of a "common" manifold [Mannig[altigkeit] (cf. vol. 1, p. 342). These should always be considered as different from each other and from nothing (from 0). They have to be compatible (consistent) a m o n g themselves,

Page 5 so that to set out one of them does not prevent the possibility of another,

2 1 2 S C H R ( ~ D E R ' S L E C T U R E I

and they have to exclude each o t h e r (to be mutual ly disjoint), so that n o n e of the e lements could be in t e rp re t ed as a class which includes a n o t h e r of them.

I only mention this in advance, together with a few other things, to direct

the expectation of the reader in the proper direction, not because important conclusions could be drawn from this last remark. Even if somebody would consider this remark as insufficiently founded, he could not reject the formal logical necessity of this thought, thanks to which the fundamental stipulations

of the next section as its consequence are secured, and, consequently, the total construction of our theory.

We r ep resen t the totality of the e lements cons ide red by l inking their

names with a plus sign, as an "identity sum" (logical aggregate) and call

it the original or universe of discourse of the first order [Denkbereich der ersten Ordnung]: 11 (read: one to the power one) , so that

I I = A + B + C + D + ... " 2)

The universe of discourse 11 is to conta in more than one e lement . This

assumpt ion is necessary for the validity of a lmost all the propos i t ions of this theory. We will call the case where the universe conta ins but one e l emen t , the exceptional case.

For some formulas it will even be necessary to assume that the universe

conta ins more than two e lement s in o rde r to claim validity. Such formulas are to be m a r k e d by an asterisk ( . ) added to the n u m b e r of the formula .

In addi t ion, the set [Menge] of e lements which our universe comprises can be "finite" [endlich] (or b o u n d e d [begrenzte]), whereby the universe consists of an arbitrarily chosen "number" [Anzahl] of e lements . Or the system of e lements is "infinite" [unendlich] ( u n b o u n d e d [unbegrenzte]); he re one canno t talk of its "number ." In the lat ter case, the e l emen t s may be "discrete," such as when fo rming a so-called "simply infinite" system, or not; that is, they may also be t h o u g h t of as "concrete ," filling, for example , a "con t inuum" as the points of a line, a surface, a solid,

in par t icular of a s traight fine, a plane, or a space.

These remarks also anticipate later insights. It is one of the most important

tasks of the theory first to establish the notion of"finite" for a system of elements;

this constitutes a prerequisite for gaining the highly important notion of"num-

Page 6 ber," as well as for defining and differentiating the various types of"infinites."

In order not to slip into fallacy, the reader is advised not to lose sight of the possibility of the assumptions just mentioned.

For illustration we may "prefer' a universe which consists of all the poznts on a straight line (viz., a straight line open at both ends) (to which, as is general ly known, the real n u m b e r s c o r r e s p o n d ) = - o r consist ing only

of a par t of it, such as its half: that ray to which the positive real n u m b e r s

FROM PE IR C E TO SKOLEM 2x3

c o r r e s p o n d - - o r perhaps simply of the integers consist ing of equ id i s tan t

points on ou r s traight line, or their positive half. However, this will always

have to r ema in un impor t an t . In our theory, we may not p resuppose that the e l emen t s are in a

def ini te order [Reihenfolge] to begin with or can be b r o u g h t into such.

One may never speak a priori of "nearby" elements, of the predecessors or

successors of an elementmas there is no immediately preceding nor any im-

mediately succeeding point to another point on the straight line (even if it would run, for example, from left to right.) As one can make judgments about

all points, or about each point, of a surface (for example)rain order from this

judgment to derive other judgments logicallymwithout ascribing some kind of

order to these points, likewise one has to proceed in the theory with the elements

of our universe.

The "e lements" need not exist s imultaneously; they may r ep re sen t

"events" f rom the past, present , and future (coexis tent or s imul taneous , as well as successive). It is sufficient that they can be thought together.

For the purposes of exempli fying a logical system, it is r e c o m m e n d e d

to i n t e rp re t the e l ement s as "persons" of the h u m a n society or m a n k i n d in general .

Page 7

In addi t ion to uppercase r o m a n letters which have to s tand for definite e lements , in case we wish to call special a t t en t ion to some of them, we

also need a ca tegory of signs to r ep re sen t or as names for indef ini te or general e lements .

This need appears already at the first and most s imple ac tmwhich we

are not going to deal w i t h m t h e act of pu t t ing any two e l emen t s in relation [Beziehung] to each o t h e r or to cons ider t hem f rom such a perspective.

The e l emen t s be tween which, let us say, "a re la t ionship" [Verhiiltnis] is

said to be established, can be e i ther different, or they can also be the same or "equal."

The perspective, the "fundamentum relationis," can, for example , be

the love of one person for another .

W h e n person A loves person B, the pair of e l emen t s is cons ide red as

"A : B" f rom this perspective. If, at the same time, person B does not

love person A, the pair of e l emen t s "B: A" (which the re fo re is to be

d i f fe ren t ia ted f rom the previous pair) canno t be cons idered .

If pe rson A loves himself, the "pair of e l ement s ... . A : A " has to be

cons idered .

These two cases canno t be deal t with t o g e t h e r D w i t h i n the l imitat ions

of the p re sen t stock of s i gnsDbecause the assumpt ion B - A , which would subsume the second case to the first, is in con t rad ic t ion with the condi t ion A 4: B, which was i n t roduced and has to be firmly ma in ta ined . With this p rocedu re , one will not go beyond sticking to examples .

214 SCHRt~DER'S LECTURE I

We need new s i g n s m a n d these we select, for the t ime being, exclusively f rom the following sequence:

i j , h , k , l, m, n , p , q 3)

r a i n o r d e r to r ep resen t any one of the e l ement s A, B, C, D . . . . of our universe 1 l.

If we now take the pair of e l ement s i : j again to state that person i loves person j, both the assumpt ion j = i and the assumpt ion j g: i re-

main valid; all advantages of general i ty for our examina t ion are assured,

in addi t ion to the advantages which the in t roduc t ion of a min imal sign l anguage [ Zeichenschrift] guarantees .

Thus, while two of the letters A, B, C . . . . always have to r ep re sen t two d i f fe ren t e lements , two di f ferent letters f rom the sequence i, j, h . . . .

are not subject to this restriction. The reason for this d i f ference lies in the fact that while A, B, C . . . . r ep re sen t to us definite, that is to say,

"specifiear' e lements , the symbols i, j . . . . are used as representa t ives of Page 8 some, of any, unspecif ied or "generalelements." In ar i thmet ic , the f o r m e r

c o r r e s p o n d to numbers , the lat ter to literals or variables. As the latter, they are essential and allow us to realize analogous advantages.

The genera l e l e m e n t symbols i, j, h, k . . . . can also be used as "indices,"

"running indices," "variables of summation" or "of productation"; we will mostly cons ider them as such for the t ime being.

Indeed , now we can write the equat ion , which r e p r e s e n t e d the universe of discourse ll , in a m u c h m o r e concise form, as

ll = E i i . 4)

For a cor rec t in te rpre ta t ion , for an "analysis" or "evaluation" of the right- h a n d sum, it is only necessary to enjoin i as the s u m m a t i o n variable, to take each e l e m e n t A, B, C . . . . as its mean ing , o r E a s one may s a y E t o "run th rough" the totality of individuals of the universe 11.

This process, which in our theory is to be thought through with each ex-

pression of the form E i or Ej, E h . . . . of a f(i,j, h . . . . ), is to be separated from

the process of interpretation of such expressions as Ef(u), as often occurred in

the first and second volumes (and will also soon play a part here, although with

modified meaning), namely where the summation variable u had to run through

not only all the elements, but all possible systems or classes, that is, all sums of elements from the present universe of discourse.

The re fo re , every summat ion sign re la ted to a symbol of s equence 3) has to be unde r s tood with the "extension" [Erstreckung] descr ibed before (unless otherwise st ipulated).

T h e universe of discourse 11 forms a manifold [Mannigfaltigkeit] for which the ent i re "identity calculus" could be used without problems. But

we shall not use this fact until we get to a new founda t ion of the latter.


Each ( ident i ty) sum of elements of this universe 1' will s imply be cal led

an "absolute term," la ter on, a "system" (Gebiet) or a "class" (class t e rm) .

Now we can fo rmu la t e in gene ra l what was shown with the above

example : If, in o u r un iverse 1', any two elements i a n d j are set off in a definite

order a n d if m a i n t a i n e d in it as a " p a i r " - - n o m a t t e r wha t the reason , no

Page 9 m a t t e r what the p e r s p e c t i v e - - t h e resul t of the j u x t a p o s i t i o n may be

r e p r e s e n t e d by m e a n s of a co lon

i : j 5)

- - t o be said: i to j.

This is unproblematic even though the colon was employed provisionally in

w 23 of volume 1 to represent identity division; however, it was definitely proved

there to be dispensable, and thus it became free for other uses.

When an pair of elements is set off from the perspective of a definite relation between i and j and is represented by i: j, as one represents a (geometrical)

ratio in arithmetic, this is confirmed by its use in common language when the

words "relation" and "ratio" are almost synonymous. Furthermore, the arith-

metical equation (i : h) x (h :j) = i: j will reveal analogies with the composition

of relatives, which further justify our procedure to follow Peirce.

In itself, the colonmas a sign of division, as well as the minus sign--has the

disadvantage of representing an asymmetrical operation by a symmetrical sign,

as it looks the same on the left and right side. But the circumstance consoles

us that in arithmetic we are already used to not considering the operation as

commutative. Moreover, we remark that this notation can be discarded later, after we have

used it to achieve a certain stage of development in the algebra of relatives.

We call i : j a "pair of elements," tha t is to say, i is the a n t e c e d e n t ( the

first t e rm) or the relate, j is the c o n s e q u e n t ( fol lowing t e rm) or the

cor re la te of the pair. After what has b e e n said, j : i appea r s as another pair o f e l e m e n t s t han

i: j , as soon as j is d i f f e ren t f rom i. O r we may s t ipula te

(i = j ) = (i : j = j : i), (i e: j ) - (i : j e: j : i) 6)

as valid for all i and j.

Within these two propositional equivalences, four propositional subsumptions

are contained, of which these two

(i=j) =(c-- (i : j = j : i), (i r j) =gc- (i : j ~ j : i)

are to be considered as self-evident, or given, based on what has just been agreed

upon.

Page 10

Page 11

216 SCHRODER'S LECTURE I

From these the two others result, the reverse propositional subsumptions (crosswise or) alternately, by means of contraposition.

We want to stress that this remark , too, serves only for the p resen t in t roduct ion; 6) will be proved as a proposi t ion later f rom the fundamen ta l st ipulations of the next section.

The same is true for the s t a t emen t

i : j , 0 7)

by which we express that each pair of e lements is different from nothing. T h e pair of e lements j : i is called "conversg' of the pair of e lements

i : j . Because of the general validity of this st ipulation, it will be permissible to exchange the terms i and j; therefore also the pair of elements i : j has to be converse to j : i. The re la t ionship be tween converse pairs of e lements is reciprocative [gegenseitig].

All conceivable pairs of e lements , for the fo rmat ion of which the universe of discourse 1 ~ provides the e lements , may be arrayed or ar- r anged in a "block" or tab/e:

A : A , A ' B , A ' C , A ' D . . . .

B : A , B B, B C, B D,

C : A , C B, C C, C D,

D : A , D B, D C, D D,

8)

We want to r emark that these "specified" or special pairs of e lements will have to be set forth or d e n o t e d as

"individual binary relatives"

of which any one can be general ly r ep re sen ted by i:j . T h e totality of these individual binary relatives or pairs of e lements

forms a new, a un ique universe of discourse which we des ignate as the "universe of discourse of the second ordd' [Denkenbereich der zweiten Ordnung], r ep re sen t ed by 12 (to be said: one to the power two), so that we have

12 - ( A : A ) + ( A : B ) + (A: C) + ...

+ (B :A) + (B: B) + (B: C) + ...

+ (C:A) + (C:B) + (C: C) + . . . . . . . . , ~ ~

9)

- - w h e r e the parentheses a round the pairs of e lements can also be left out-- -or the abbreviat ion, which is possible because of the s u m m a t i o n sign,

12 - E~F~,(i'j) = E, Ei( i ' j ) = ~o( i ' j ) . 10)

Page 12


The universe of discourse 12 is thus formed from the total "variations to the second class with repetitions" of the elements of the universe of discourse l~Das mathematicians would say; it is the second class of the mentioned variations. It contains the elements of 11 in pairs linked in all possible connections [Verbin- dungen] (combinations) and arrangements (permutations).

It appears as self-evident (but we will not make essential use of it) that this universe of discourse, too, presents a mani fo ld on which the identi ty calculus can be applied. The investigation of the theory in this vo lume will deal with this universe almost exclusively. The re fo re , we will

d e n o t e it for short with a I only---dropping most of the t ime the e x p o n e n t 2 in all formulas (and m o r e rarely in the text).

To summar ize , we repea t equat ions 9) and 10) in thei r mos t s imple m o d e of express ion

l = E o i ' j =A" A + A . B + A . C +

+ B . A + B . B + B . C + . . .

+ C . A + C ' B + C . C + . . . . . . . . . .

11)

An individual binary relative i ' j in this table always stands in the row

marked i and in the co lumn marked j---if the e l emen t s i, j were natural numbers , we could say in an abbrevia ted way: in the ith row and in the

j th co lumn.

Al though, as already expla ined, the supposi t ion of a def ini te o r d e r or a r r a n g e m e n t of the e l ement s of the first universe may not be the

basis for the conclusions of the theory, not less also for the pairs of

e l emen t s of the second universe; we shall never theless accept the previous modes of express ion for the sake of conven ience and for overall clarity:

If we speak of such individual relatives i ' j , i ' h , i" k . . . . . which all agree in the relate, we will state f requent ly that they c o m e f rom the same horizontal row or line, and "the row be long ing to i" of the table 12 will simply m e a n the totality (identity sum) of all pairs of e l emen t s

f rom our second universe which have i for a relate. Also all pairs of

e l emen t s i ' j , h ' j , k ' j . . . . . which agree in the corre la te , will be attrib-

u ted to the same vertical row or co lumn. And we the re fo re dis t inguish

in our table 12 "rows" of pairs of e l ement s as parallel or n o r m a l to each

other, as well as horizontal or vertical.

If we look at the individual pairs of e lements of our universe 12, we

see two kinds, d e p e n d i n g on whe the r i = j or i ~ j is in i ' j . In the first case, we have an pair of e l ement s of the form i" i. It should

be called an individual (binary) "'self-relative." In the case opposed to it of i g= j, we call i ' j an individual (binary)

"'aliorelative."

Page 13

2 18 SCHR()DER'S LECTURE I

I hereby simply translate the names given by Peirce: "self-relative" and

"aliorelative."

It could be considered offensive that "self" or "Selbst-" does not originate

from Latin, as other compound words used for these terms. If one does not

want to say "ipsirelative," one could also substitute "idemrelative" for our "sell:

relative." Alternatives would be to say:

Autorelative and Heterorelative

or

Idio(homo-?)relative and Allorelative

the first half coming from Greek, the second from Latin. This corresponds to

the following terms where the first half comes from German:

Selbstrelative and Anderrelative

Eigenrelative and Fremdrelative.

A colleague also suggested to use "relation" for relative.

After careful consideration, I could not accommodate any of these sugges-

tions. In order to cover the need of our science for new terms, the dead, or

classical languages deserve to be preferred as a source for new, international

expressions. The word "other-relative" gives rise to awkward associations in case

we have to talk of other relatives or even of o ther other-relatives. The other

expressions seem less apt, and cover the idea less accurately.

Al though "self-relative" does not quite do justice to the international consid-

erations ment ioned above, I will keep it and leave it to the romance cultures

to coin a word to their taste; my term seems to me the best and most accurate,

at least for the Germanic languages including English.

In o u r table 12, the ind iv idua l self-relatives A : A , B: B, etc. , all s t and

in a s t r a igh t l ine which cuts across this table f r o m the u p p e r left to the

l ower r i g h t side, a n d is ca l led the "main diagonal" of the t a b l e - - a c c o r d i n g

to a c o m m o n exerc i se in the t h e o r y o f d e t e r m i n a n t s a n d mat r i ces . Un-

d e r the m a i n d i a g o n a l o f 1 2 - - t a k e n analytically, if e x p r e s s e d i n d e p e n -

d e n t o f g e o m e t r i c a l i l lus t ra t ions which s e e m to p r e s u p p o s e a de f i n i t e

a r r a n g e m e n t of pairs o f e l e m e n t s on a s u r f a c e - - w e thus u n d e r s t a n d

n o t h i n g else t h a n the total i ty ( iden t i ty s u m ) o f all i nd iv idua l self-relat ives

f r o m o u r s e c o n d universe .

T h e ind iv idua l al io-relat ives lie ou t s ide , s t and at t he s ide of, above

a n d be low the m a i n d i agona l .

Each individual self-relative is the converse of itself. O n the o t h e r h a n d ,

i nd iv idua l al io-relatives, conve r se to e ach o the r , s t and "symmetrically" to

the m a i n d i a g o n a l so that , if o n e looks at the l a t t e r as a m i r r o r l ine,

any o n e o f the two m u s t be the m i r r o r i m a g e o f t he o ther .

S ince we now k n o w a b o u t the "individual b ina ry relat ives,

t ion arises: wha t do we actual ly m e a n by a "binary relativg'? " t he ques-

Page 14

FROM PEIRCE TO SKOLEM 2x9

Al though this quest ion is not to be t reated systematically until the next section, we would like to ant icipate the answer: it is an identity sum (an aggregate) of any individual binary relatives.

We can pick any pairs of e lements f rom our universe 12 and c o m b i n e them by means of identity a d d i t i o n m b e it collectively to a "system of pairs of e lements ," be it general ly to a "class of pairs of e lements ." T h e result is to be called simply a binary relative.

T h e perspective f rom which we pe r fo rm such selections of pairs of e lements is pr imari ly that we jo in all those individual relatives i : j in a class or identity sum of pairs of e lements in which the relate i stands to the corre la te j in a "relation" of a definite kind, a relat ion charac te r ized by a cer tain "fundamentum relationis," on which our in teres t is mainly concen t ra ted .

Jus t as we did not set any limits to the (wider) ex tens ional logic of class format ion , and the individuals of a class were not he ld toge ther to const i tute a regular "concept ," co r r e spond ing to the r e q u i r e m e n t s of the (narrower) intensional logic, likewise the possibilities for select ing pairs of e lements which are to be j o i n e d in a binary relative should not be l imited in any way; this perspective, as the one m e n t i o n e d above, may be decisive as a reason for their selection f rom the universe 12;

however, its exis tence is not an essential condi t ion for the selections which can be carr ied out arbitrarily (from the start) (and be ma in t a ined f rom there on).

A combina t ion of any three e lements i, j, and h f rom our original universe 1~, if they are written in this defini te order, may be called an "triple of elements" or an "individual ternary relative" and be r e p r e s e n t e d by

i : j : h. 12)

T h e totality, the identity sum of all possible triples of e lements , constitutes a new universe of discourse which we call the "the universe of discourse of third order" and deno te it by 13, so that we can say

1~= E, ,EiE, ( i ' j " h) =Eohi" j" h. 13)

Specifically, the triples of e lements can only clearly be r e p r e s e n t e d in the form of a block--filling, for example , a book--on its first page would s tand the pairs of e lements of 12 in 11) with " : A" added; on the second page (or better, f ront page of the second sheet) , the same pairs of e l ements with ": B" added; on the third page (or third shee t of its f ront page) , the same pairs of e lements with " : C" added; and so on. But we abstain here f rom their specific illustration.

Mathemat ical ly speaking, the universe 1 ~ consists of the "third class

Page 15

2 2 0 S C H R O D E R ' S LECTURE I

of variations (or p e r m u t e d combina t ions) with repet i t ions" of e l ement s

f rom the original universe 11. According to whe the r all th ree e lements are equal to each o t h e r (i.e.,

identical) in i : j : h , o r m w h i c h is possible in three ways---only two, or,

eventually, none of them cor responds to one of the other, i.e., all th ree

be ing different , we would have to dist inguish be tween five types of individual ternary relatives, for which the examples

A : A : A , B : A : A , A : B : A , A : A : B , A : B : C 14)

are illustrative. An identity sum of triples of e lements , somehow selected f rom the

universe 13, is now to be called a "ternary relative." Because all individual ternary relatives have to be cons ide red as dif-

fer.ent f rom each o the r and f rom nothing, the identi ty calculus (as

system, as well as class calculus) is applicable, at least for the t ime being, to the universe 13 and the ternary relatives possibly con ta ined in it.

And so on. It is clear how one can con t inue in this way and c o m b i n e all possible

quadruples , quintuples , sextuples . . . . of e lements of the universe 11 to the universe 14 , 15 , 16 . . . . . of the fourth, fifth, sixth . . . . order , and state

with it the not ion of quaternary, quinary, senary, ... relative. To conclude , we may r emark that also the "absolute terms," "(Gebiete

or) systems," virtually "classes," that ismas m e n t i o n e d m t h e sums which can be t hough t to be built f rom e lements of the original universe 11, can also be considered, r ep resen ted , or d e n o t e d as "uninary* relatives" (called "simple relatives" by Pe i r ce ) - - a s indeed there is no p r o b l e m to call the e lements i of the universe 11 "individual uninary relatives."

We then have the u p p e r m o s t reason for the classification of all imag- inable relatives: the "order" [Ordnung] of them. We have to dist inguish be tween relatives of the first, second, third, etc., order. A relative of a

def ini te o r d e r is no th ing m o r e than the (identity) sum of any "individ-

ual" relatives of the same order ; by "individual relatives" of a def ini te

o r d e r we have to unde r s t and the "variations (with repet i t ions) in the same class" f rom e lements of the first universe of discourse. The lat ter

are r ep re sen t ed symbolical ly--for the h igher orders ( the "variations in

the first class" are, as is well known, the e lements t hemse lves ) - -by con-

nec t ing with a colon the e lements which en t e r into t hem in a def ini te

o r d e r [Reihenf0lge].

*I have dared to propose this neologism elsewhere since the expression "semelary" (which would come from the sequence semel, b/s, ter, .... of the basic vocabulary of our terminology) seems too strange. The word "singular" taken from the sequence singuli, bini, terni ..... is already in use with too many additional levels of meaning. Although the ending -ar/us, and not -narius, is used in "binary" and therefore "unary, multary," may be considered more correct than "uninary, multinary," I prefer the latter for my neologism.

Page 16

FROM PEIRCE TO SKOLEM 2 2 1

Finally, we want to draw attention in advance to the fact that the theory of relatives makes it possible and offers the p rocedure to reinterpret expressions, as well as relations, formulas, or proposit ions of relatives of a definite order [Ordnung] from their c o m m o n universe of discourse into a universe of another order. Namely, ei ther to "preinterpret" them into a universe of higher order; then all the relatives (of a given order) consti tuting the expression, or enter ing the relation or the formula, respectively, are rewritten ( transformed) into those of the desired higher order. Or (as long as the universe to which the given relatives belong is not of the lowest or first order) also to in terpret them back into a universe of lower order.

Certain moments (elements) of our knowledge about the constitution of the relatives in question are lost when interpret ing back, that is to say, they are ignored, one abstracts from them, or, in o ther words, certain parts of our knowledge are dropped, relinquished, and will not be re- gained in the course of possible following reinterpretat ions, are not restituted, so that the loss of knowledge caused by this in terpreta t ion is p e r m a n e n t - - o f course, without de t r iment to the validity and justification of the whole process.

In the correct grasp of these processes, in the adequate interpretation and use of the formulas set up for one of the universes in ano ther of our universes, lies the main difficulties that the proper understanding of our theory may encounter ; they have to be overcome by aiming at making it unders tandable . The new interpretat ion from the second into the first u n i v e r s e u a n d converselyuwill illustrate this point.

Therefore , we want to deal with this question again only after we have a thorough knowledge of these two universes. We now proceed to an in-depth examinat ion of the second universe, 12; w e will focus on this problem almost exclusively for a long time, i.e., we limit our examinat ion to the algebra and logic, the theory of binary relatives.

Appendix 2: Schr6der's Lecture II


In this lecture, which presents the basic identities of the algebra of binary

relatives, Schr6der begins by introducing the two-element Boolean al-

gebra of 0, 1. He notes that if a 0 is a 0, 1-valued function of i, j running

through a domain, then a binary relative is de t e rmined as a =

~oao( i ' j ) , and every relative is so determined. If we think of 1 and 0

as the largest and smallest elements of the Boolean algebra, and think of sums as least upper bounds and products as greatest lower bounds,

this makes perfect sense. Thus Schr6der is here in t roducing the char-

acteristic function of a relative on a domain. He points out (his page

24) that the operat ions on relatives can be carried out by simple formulas of combinat ions of their characteristic functions, where 1 is the identically 1 characteristic function, 0 is the identically zero characteristic

function, 1' is the characteristic function of the identity relation, and

its complemen t 0' is the characteristic function of unequal . On his page

27 he says that these characteristic functions are to be regarded as

proposit ion functions of pairs. He goes on to introduce product and sum over the domain of the

proposit ion functions, which he knows to correspond to greatest lower

bounds and least upper bounds in the Boolean algebra of relations.

This int roduct ion of product and sum over a fixed domain is his way

of in t roducing quantifiers, as he points out on his page 38. He then

gives many quantifier rules, which he sees as algebraic rules for the

algebra of truth functions, least upper bound as sum, greatest lower bound as product , not based on ordinary algebra.

Schr6der prefaces this lecture by stating that less than or equal (subsumption) instead of equality can be taken as a basis for all o ther relations. This is in the sense of the first volume of the Algebra der Logik,

223

2 2 4 SCHRO.DER'S LECTURE II

where lattices can be defined either in terms of order or by equations only.

Finally, in w 4 of this lecture Schr6der indicates that each proposition can be thought of analytically, that is, by formal proof; geometrically, that is, by properties of the matrices of zeros and ones corresponding to relatives; or rhetorically, that is, based on ordinary language reading of the propositions.

S e c o n d Lec ture

The Formal Basis, Especially of the Algebra of Binary Relatives

w 3. The 29 to 3I Fundamental Stipulations. The Representation of a Relative as a Sum. Proposition Schemes.

Essential--If we ignore abbreviations achieved through the introduction Page 17 of the sum and product signs E, H, as well as agreements regarding the

omission of parentheses and other external and minor things-- the whole algebra of binary (and of uninary) relatives--yes, if you want: the entire logic--is based on only 29 conventional stipulations which can be easily and clearly summarized on half a page (without the necessary explanations).

As in both previous volumes, we consider the relation of inclusion [Einordnung], subsumption, expressed by the sign =(=, as fundamental, by means of which (or its negation =(~) all other relations have to find their definition. We therefore give preeminence to the definition of equality (which always means: sameness, identity) which is binding for all symbols a, b of our theory. We formulate it in the manner of the "propositional calculus" [Aussagenkalkuls], as previously d o n e - - a n d to be justified again directly afterwards:

(a :~- b)(b :(r a) = (a = b). (1)

The following 14 (we proceed rapidly!) fundamental stipulations are as follows:

0 =(= O, 0 :(: 1, 1 =(= 1, 1 ~ 0 , (2)

0 " 0 = O" 1 = 1 " 0 = 0 , 1 + 1 = 1 + 0 = 0 + 1 = 1 , (3)

1 �9 1 = 1 , 0 + 0 = 0 ,

i = O, b = 1. (4)

After having already postulated more than half of the formal foundations of

FROM P E I R C E T O SKOLEM ~25

Page 18 our theory, we would like to pause in this enumeration and look more closely

at what has been said so far.

For a domain of values [ Wertbereich] that consists of only the two symbols 0 ( identical ly zero) and 1 (identically one ) , we have fully s t a t e d - - a l t h o u g h only in nuce--the laws of inclusion and non inc lus ion , of equal i ty and inequality, in addi t ion to a calculus which has as basic m o d e s of calcu-

lat ion the " three identity species": multiplication, addition, and negation. T h e four conven t ions (2) establish which of the possible s u b s u m p t i o n s

are valid in the d o m a i n of values, and which are not. T h r e e have validity,

bu t no t the four th .

In virtue of (1), we can also derive that of the four conceivable equations,

these two: 0 = 0 and 1 = 1, and of the four conceivable inequalities, these two:

1 #: 0 and 0 #: 1, are valid.

T h e e ight conven t ions (3) r ep re sen t the "abacus," the multiplication table and the addition table, for the d o m a i n of values res t r ic ted to the symbols 0 and 1. For this d o m a i n of values, they comple t e ly def ine the

product a" b or ab and the sum a + b of two values a and b - -howeve r these lat ter "general" values are d e t e r m i n e d , assumed, or t h o u g h t of

within that d o m a i n of values.

T h e mul t ip l ica t ion table co r r e sponds exactly to the numer i ca l table, as it would be expressed for the dyadic n u m b e r system or as par t of the

global dec imal mul t ip l ica t ion table known to everyone .

T h e addi t ion table shows only one deviat ion f rom the numer i ca l ad-

d i t ion table which is he re stated as 1 + 1 = 1. To mot ivate this deviat ion,

the following r e m a r k may no t be wasted: because in o u r discipl ine only

what is identical , same, is admissible as "equal," a r e p e a t e d use o f"equa l s" is no t o therwise possible than in the form of a tautological and the re fo re mean ing less repetition---comparable to the act ion of the chi ld who gives his f r iend "the same" object repeatedly. It is exactly to this devia t ion that

ou r discipl ine mainly owes its wonder fu l symmetry. The two conven t ions (4) def ine in genera l the negation d ("a bar" or

"not -a ' ) for every value a of this domain .

With respect to the number of conventions, it is clear that our way of counting

Page 19 has something arbitrary. One could further reduce the number of independent

fundamental conventions by using letters as general value signs. Thus, the first

and third convention (2) could be combined in the formula a =(= a,--our pre-

vious principle I - -and the six stipulations of the first line of (3) could be replaced

by the four, a ' 0 - 0 = 0 " a , a + l = l = l + a .

By the same token, the first three conventions (2) can be reduced to two,

0 :(= a ~= 1, which we know well as "definition (2)" of volume 1. And---even bettermall eight conventions (3) can be combined in the four well-known laws: a ' 0 = 0 , a + l - - 1 , a" l = a = a + 0 .

The most effective procedure to reduce our system of conventionsmifindeed

226 SCHR(~DER'S LECTURE II

one wants a further reduct ion-- is the procedure which leads us essentially to

repeat the method of establishment of the identity calculus, as it was given in

volume 1 for a much larger manifold, but would be used here for our very

restricted domain of values 0, 1. In particular, the eight conventions (3) would

be replaced by "definition (3)" of volume 1, page 196 related to product and

sum--s ta ted with general signs a, b, cmand to prove the abacus from it--as in

volume 1, page 271, "theorems 21) and 22)."

Undoubtedly, we could show that one could use a smaller number of inde-

penden t stipulations for what has been said.

But one could also produce a larger number (instead of 15, a maximum of

26). Because: it was also arbitrary to count subsumptions and equations, without

differentiations, as "stipulations," whereas each equation includes a pair of sub-

sumptions by virtue of (1).

Therefore, I do not want to argue about the exact number of stipulations for

independen t conventions which would have to form the formal foundation for

our entire theory.

By enumerating them I only intend to create a useful point of depar ture and

prepare a clear overview. Indeed, the 15 data ment ioned previously constitute

the core and the specified contents of an equivalent, but more general system of

conventions which would aim at summarizing the data more conciselywin what-

ever way it would be formulated. This core appears here as mere artless enu-

merat ion and detailed explanation.

It b e c o m e s i m m e d i a t e l y clear, a n d will be c o n f i r m e d , tha t o u r system

o f c o n v e n t i o n s is without contradictions', they a p p e a r f r o m the b e g i n n i n g

as independent of each other. Both ins ights s t em f r o m the o b s e r v a t i o n tha t

Page 20 e a c h of the s t ipu la t ions (1) to (4) de f ines "a new symbol ," so to speak,

wh ich was n o t b e e n m e n t i o n e d in any of the p rev ious s t ipu la t ions , so

tha t these c o u l d n o t affect the m e a n i n g of the new symbol .

So- - to begin with the end: even if we first agree on what we unders tand by

1, it remains open what we understand by 6. In whichever way we want to define

0 (because nothing has been agreed upon, we are not bound by anything), the

agreement will not be contained in it or any previous agreeme,~-ts, nor be able

to contradict them.

With the abacus (3 ) - - the products, respectively the sums, of the first line

have to be read not as equal to each other, but as equal to the last symbol 0,

respectively 1--each (thus separate) equation contains also a new operat ion

otherwise not existing between 0 and 1. That, for example, the equation 1 +

1 = 1 cannot follow from any other can be seen from the example of the nu-

merical addition table, where the other equations are valid, but it (alone) is

not. It is self-evident that it cannot be in contradiction with the others because

it mentions for the first and only time the expression 1 + 1 thus far undefined;

hence, anything in opposition to its meaning cannot possibly have been

established.


The conventions (2) finally contain, independent of each other, the stipulations about 0, respectively 1, as subject (or as predicate) to 0 or 1.

If we had condensed the last 14 conventions concisely, that is, into a smaller number of--consequently more general--formal stipulations (by using letters),

the conviction of the independence and noncontradictoriness of the fundamental conventions would have been gained in a less comfortable and easy

way--which, not in the least, induced us to prefer the above form of stating them.

Page 21

As already m e n t i o n e d , the previous 15 st ipulat ions form the basis and sufficient founda t ion for a calculation with letters, a "calculus," in which we have to suppose of each "general value symbol" or l e t t e r n a s a, b, c,

. . . m t h a t it represents the two values 0 and 1. The formal laws, proposi t ions, and formulas of this le t ter calculus are

no o the r than those of the "propositional calculus" (which is even r icher

in formulas) . The propos i t ional calculus is a subcase of the "identity calculus," which we m e t in volumes 1 and 2.

We now d e m a n d f rom the r eade r that he convinces h imse l f thoroughly of that, that is, at least that he checks that the def ini t ions and

pr inciples taken as the founda t ion for the calculus in the place cited result f rom our 15 st ipulat ions whereby their collective consequences

are also gua ran teed .

This can be d o n e in an unsophis t ica ted way by shee r verification of the founda t ions f rom the abacus, f rom our st ipulations.

For each le t ter we only have to dist inguish two cases: w h e t h e r it means

0 or w h e t h e r 1; in a fo rmula in which only 1, 2, or 3 let ters appear , only 2, 2 2= 4, or 2 3= 8, respectively, substi tut ions (of values 0 or 1 for the

letters) are to be carr ied out in o r d e r to prove it for all conceivable cases.

With our 15 stipulations, for example, we can prove with ease "principle II"

of volume 1: (a=~-b)(b=~c)=~--(a=g~--c), the definitions (3) of the same: (c=(=

a)(c=(v-b) -(c=g~--ab), etc., the associative law, and the full distributive law a(b+ c) - ab + ac.

There never occurred more than three letters in the previously mentioned

"definitions" and "principles" taken as formal basis for the propositional calculus.

In a similar way one could, of course, verify each complicated theorem, each

corollary of the propositional calculus which we deduced from those formal

foundations of the theory, and verify it, if necessary, based on our 15 stipulations.

At one stroke we may p resuppose that the r eade r is fully acqua in ted

with the comple t e formal ism of the propos i t ional calculus (implicitly

a long with it also the identity calculus).

The i m p o r t a n t fact is thus assured: that we are at any po in t free to i n t e rp re t 1 as a "true" proposition, 0 as a "false" propositionmwhere all t rue propos i t ions are equal to a n o t h e r ("equivalent") , are valid, and likewise

2 2 8 SCHR()DER'S LECTURE II

all false p ropos i t i ons - - i f we in t e rp re t at the same t ime the subsumpt ion be tween proposi t ions, the proposi t ional negat ion, the propos i t ional p roduc t , and the proposi t ional sum in the usual way.

These remarks do not at all ant ic ipate the appl icat ions which we in t end to show with the same value symbols 0 and 1 within the d o m a i n of values of the relatives.

The s tuden t has to r emember , and always pay close a t ten t ion to this fact that if we now draw conclusions f rom addi t ional st ipulations, that these inferences always has to follow the laws of this propos i t iona l cal-

Page 22 culus, whose foundat ions are f o r m e d by the previous st ipulat ions and which are no o the r than those of general logic--also of t radi t ional logic, in its most succinct and strictest version, however.

The four th convent ion (2) formula tes the opposi t ion "true" and "false" for proposi t ions, and expresses it in a concise way. For relatives,

it will only play an essential role when par t icular j u d g m e n t s are taken into cons idera t ion , and we will the re fo re do without it for the t ime being.

A fu r the r g roup o f - - s e v e n - - f u n d a m e n t a l st ipulations serves to def ine the general "binary relativg' and certain special relatives of this ( the second) o r d e r [Ordnung].

It is only with these conventions that we enter into the algebra "of relatives," since the previous ones belonged to the elementary branches of our discipline of exact logic.

We now want to formulate the definitions, which are also to be given verbally,

with the help of equations [Ansatzes yon Gleichungen]. The equation involves two subsumptions and presupposes that one knows the meaning of a subsumption between two binary relatives in order to understand its consequences fully according to the ideal stated in convention (1). In turn, this cannot be well expressed unless one knows what a binary relative is. We therefore have to postpone

the question of what a subsumption between relatives a and b means to the end

of our exposition, and take the notion of equality, identity--as it is customary with "definitions"--as the primary one. I would say "for didactic reasons"; not

everyone may agree with this, but it, incidentally, is a question of little weight.

We call "binary relative" a sum of pairs of elements set off f rom the universe 12--and, that is to say, of none, of some, or also of all.

The genera l form of any binary relative a can be set down in the following expression:

a=Eoao(i" j) (5)

Page 23 - - w h e r e the indices i and j in the sum E 0 have to run th rough all the e l emen t s of the universe 12 (as their m e a n i n g or "value"), i n d e p e n d e n t


of each o t h e r m 0 n condition that one restricts the "coefficients" a 0 (said: a sub ij), with which the pairs of e l emen t s i : j (as the accompany ing "constituents") are re la ted or "multiplied," to the d o m a i n of the two values 1 and 0, which the following formula would express:

(a 0 =1) + (a 0=0) =1. (5c~)

Its first coeff icient value would guaran tee , by means of the s t ipulat ion

l ( i : j ) = 1 �9 (i: j) = i : j , (5~)

the existence of i : j as a te rm of the sum; the lat ter coeff ic ient value would

guaran tee , by means of the st ipulat ion

0 ( i : j ) = 0" (i: j) = 0, (5./)

the absence of the pair of e l ement s i : j as a te rm of the sum.

The "relative coefficients," known from the suffix which is always a t t ached to t hem (as a rule in the form of a double index) , are the re fo re to be subject to the laws of the pure propositional calculus.

The opera t ions which will have to be p e r f o r m e d on, with, or between t hem are fully exp la ined by our first 15 st ipulat ions and are regu la ted accord ing to their laws. O t h e r opera t ions than the three identity species ( canno t and) will not be considered.

From now on, we will always express binary relatives with s imple letters f rom the lowercase latin a lphabet , that is, those without suffix, inc luding

those which we have already reserved for the r ep re sen ta t ion of indices, i rrespective of this use; this calls for an exp lana t ion to be given later (and gradually).

Equat ion (5) by itself would not yet be u n d e r s t a n d a b l e because the original obscurity of the coefficients a o and their workings has not yet been expla ined, unless one combines t hem with their verbal corol lar ies or addi t ional formulas which---cf. (5ol)massign each of the coefficients to one of the two values 0 and 1 and explain by means of (5/3) and

(53') the effect of factors 1 or 0 on a const i tuent . In a rat ional way one

can only set down the four s ta tements (5), (5c~), (5/3), and (53') as "one st ipulation," and to cons ider this (5) not only as a formula , but also to

look at it in connec t ion with the verbal text.

In particular, our explanation (5"),) (expressed in formula only for the sake of clarity) has to be thought of in its verbal form as the basis for the theory; in

Page 24 case a coefficient has the value 0, the absence of the accompanying constituent i : j as a term of the sum to be called a is simply required---otherwise we must initially postulate the proposition a + 0= a for relative a; to this small (and

actually not very important) circle, we would have to add that this postulate,

Page 25

23 ~ SCHRODER'S LECTURE II

temporarily stipulated as convention, will later be proved through added conventions.

It also has to be m e n t i o n e d that in mult iple indices, such as ij, ijh, .... be it in the suffix of the signs ~ or H, be it in the suffix of a relative symbol, such as a, d, a + b, etc., the e l e m e n t letters have to be t h o u g h t of as separated by a c o m m a - - w h i c h we omit only for the sake of saving space: in the suffix, i j is to be unde r s tood as representa t ive of i, j and not as the "product" of i and j (which, of course, would have to be r ep re sen t ed correctly as i j) .

Since the notat ion for the summat ion variables is ind i f fe rent to begin with, namely because we have any o the r te rm which has not been already al lot ted at our disposal, we can, of course, write (5) as

a = Ehk ahk (h: k),

and we would have to make such a change in the des ignat ion of the indices by using scheme (5) in cases where one of the terms i and j (perhaps to represen t a definite e lement ) has been al located otherwise and is thus no longer at our disposal.

It is possible to substitute b or c in (5) for a, and so on, that is to say, any symbol, be it simple or c o m p o u n d , w h i c h can rep resen t a binary relative or which we can consider as binary relative. The convent ion (5) ough t to be set forth as general and provide the "scheme" for all binary relatives.

Insofar as the left-hand symbols in the following equat ions refer to binary relatives, (5) implicitly states the following:

Corol lary to (5)

1 = E o l o ( i : J ) ' 0 = Z o O o ( i : j ) ,

i = [;hk ihk( h: k),

1'= E 01'0(i �9 j), 0 ' = E000' ( i ' j ) ,

i : j = Enk (i : j)hk(h : k),

ab = Zij (ab)o(i : j ) , a + b = ~0 (a + b)o(i : j ) , d = ~0 ( a)0(i : J) '

a ; b = Eo(a; b)ii(i : j ) , a j- b = ~o (a j- b)o(i : j ) , d = ~o( ~t)o(i : J)

---we stipulate this explicitly for an easier u n d e r s t a n d i n g of the f o l l o w i n g . -

If one knows for a definite universe of discourse for every possible suf f ix ij which value belongs to the coefficient a 0 of a binary relative a, namely, w h e t h e r it is --0 or - 1 (for this par t icular suffix ij), one also knows which pairs of e lements en te r the sum a exclusively, one knows how the binary relative a is composed of individual binary relatives of the universe 12. In o ther words, one knows the binary relative a itself.

By stating all its coefficients, namely, all their relevant values, a relative can be "de te rmined ," sufficiently described, be m a d e known. For the


determinat ion, complete specification of a binary relative, in o ther words, for the "definition" of a special binary relative, it suffices, and is even necessary in view of (5), to determine which values its coefficients will have. The description, specification of the relative is from now on reduced to the statement, specification of its coefficients.

It is now clear that through the following six stipulations

1;j = 1, 0o= O, (6)

1' = ( i= j ) ij

- -or , better distinguished

{1~= 1 , , 10=0 for i C j ,

and

! 0 0 : (i r j ) , (7)

' = 0 , 0 0 (7)

0 q = l for i c j ,

ihk = l'i; , , (8)

Page 26

(i " j)hk 1'i;,1' = ~, ( 9 )

- -which are to be thought as appointed for every suffix i, j , respectively--for (8) and (9)- -h , k-- the symbols 1, 0, 1', 0' and i, as well as i : j (also) have found their definition as "binary relatives."

These stipulations form, together with (5), the "second" group of fundamental conventions. Let us have a closer look at them.

The symbols 1 and 0 are to be called the two identity or "absolute" modules when interpreted as relatives.

By the first convention (6), the identity module 1 (one) is marked as binary relative which coincides with the universe of discourse 1 "~.

It is the universum, the fu l l sum, the total or the whole of the universe of discourse, the sum of all its individuals or pairs of elements.

The second convention (6) marks the identity module 0 (zero) as a completely empty relative, a relative which contains no pair of elements of our universe 12 (and also otherwise nothing).

Because of (6) and considering (5), (5/3), and (5y), we have

1 = E o i : j , 0 =

whereby the last equation has to be considered as a complete equation, al though it does not show anything on its right side. The right side here is a sum of which all terms "fall away," that is, literally "nothing." In order to avoid confusion with an incomplete equation of which the right side still has to be produced, the symbol 0 has to appear from now on in such cases where all terms on the one side fall away.

Page 27

232 SCHRODER'S LECTURE lI

T h e identi ty modu le s r ep re sen t the l imit ing cases, the two extremes, a m o n g the conceivable binary relatives. No relative (within 12) can contain m o r e individual binary relatives or pairs of e l emen t s than m o d u l e 1, n o n e can conta in less than m o d u l e 0; the re fo re one could pos tu la te 1 as "the maximal relative," 0 as "the min imal relative." T h e admissibility o f these l imit ing cases has been discussed u n d e r the genera l def in i t ion of a b inary relative.

In add i t ion to these two "identity" modules , two special (binary) relatives appea r in the theory which are de f ined by the two conven t ions (7), namely, the two "relative modules" 1' and 0 ' - - - p r o n o u n c e d as "one-ap" and "zero-ap" (as abbrevia t ion of "one-apostrophe" etc.).

F rom (5), we have the represen ta t ion :

1' - ~o (i =j)(i " j ) = ~ ( i " i), O' = Eq (i :1: j ) ( i " j) .

Namelymleft of cen te rmi f j g: i, then the propositional factor (i =j) equals 0 and the term i : j in the sum falls away. On the other hand, if j = i, then the propositional factor (i =j) has the value 1, and the term i: j appears in the sum. Then we may for j, j being equal to i, also use the term i; consequently, the existing terms can be represented in the form i : i , and these are now to be thought as formed for each i.

This means 1' is the sum, "the universum, the doma in" of all individual self-relatives of the universe of discourse 12, 0' is the sum of all individual aliorelatives of this universe and forms "the universe o f the aliorelatives" (cf. w 9).

O u r theory of binary relatives thus accepts four "modules," 1, 0, 1', 0 ~. T h e i r des igna t ion can soon be mot iva ted f u r t h e r . ~

Before en t e r i ng into the discussion of conven t ion (8), we would like to ant ic ipa te and m e n t i o n the following, a l t hough it is u n i m p o r t a n t for the algebra of relatives; it is, however, essential for the logic of relatives, for its i n t e rp re ta t ion and a p p l i c a t i o n ~ i t is thus mainly in the in teres t o f the appl icat ions that we i n t e n d e d to weave it into the topic of a lgebra for the pu rpose of illustration.

T h e relative coefficients which, as we m e n t i o n e d earlier, are subject to the propos i t iona l calculus, can be i n t e rp r e t e d any t ime as propositions. Thus, we can read:

a/j = (i is an a of j) .

T h e n a m e a of the binary relative can be recogn ized as a "relative name"

(cf. vol. 1, p. 76ff) equiva lent to "an a o f - , " such as "a lover o f - , a p ic ture o f - , an effect o f - , a fa ther o f - , " etc.; as a te rm which needs an addi t ional corre la te for its comple t ion . T he same names can be used as "absolute terms," since one can speak of "lovers, pictures, effects, fathers," e t c . ~ w i t h o u t the addi t ion of correlates.

T h e s t ipulat ion (8), which was essentially goes back to P e i r c e ~ o f

Page 28


which I only presented the most succinct vers ionmforms the basis for the transition of using names as relatives to using them as absolutes, and conversely.

It teaches us first of all: any individual or element i of the universe of discourse 1 ~ is to be considered and represented as a binary relative. It will later follow naturally how an "absolute te rm"--namely , a "system" or a "class" as the identity sum of elements, the individuals i of the universe l~mis to be represented at any time as a binary relative; or, conversely, how binary relatives are to be in terpre ted in the original universe of discourse, in o ther words, how they can be in te rpre ted back from 1 z into 1~.

It does not become me to declare the establ ishment of convention (8), insignificant as it may appear, as the highest and most consequential achievement in the whole theory. The s tudent will only gradually be able to unders tand the full range of its consequences, its p roper application, and its usages.

For the representat ion of i, we will have by virtue of (8) and (5):

i = ~hk l'0,(h " k) = ~h ~k (i = h)(h" k) = E,, (h = i) Zk (h" k) = ~k i" k,

that is, i is represented as the sum of all the pairs of elements which have i as their relate; the relative i comprises those terms that are marked with the e lement i in the table 12.

The algebra of binary relatives can be developed to a high stage without ever using convention (8). It may therefore be advisable for the reader to ignore this convention during the first and general part of the theory; otherwise, he may encounte r difficulties in unders tanding---confusions, objections may appear which he then would have to solve or elucidate alone without guidance, whereas here we will deal with it later in our theory and with easembecause we will be treating it systematically. However, for the sake of complet ing the enumera t ion of the formal foundations, we thought it necessary to ment ion this convention (8).

The convention (9) leads, according to the preceding: (7) left, (5), and (3) left, mainly to the acceptance of the equat ion

i : j = i : j .

It lets us recognize that the general coefficient

(i : j)hk = (i = h)(k =j)

of a binary relative denoted by i : j does not disappear only when the two equations h - / a n d k - j s i m u l t a n e o u s l y have the truth value 1---consequently, in the sixth double sum of our "corollary to (5)" only that pair of elements h: k will not fall away, will remain, for which h - i and k=j.

Page 29

234 SCHRODER'S LECTURE II

In o the r words, convent ion (9) guarantees the admissibility of the pair of e lements i : j as a (0ne-term, monomia l ) s u m of pairs of e lements ; it a r ranges in a row the pairs of e lements formally u n d e r the "binary relative" and gives us subsequen t and expressive indemni ty for having previously taken the liberty to presen t or to call this pair of e lements an "individual binary relative."

In view of the long-standing custom in arithmetical analysis with respect to sums, polynomials, aggregates, the definition of the binary relative, as given verbally by (5), may meet such wide acceptance that one may be inclined to stipulate for (9)--similar to (6)--that this convention need not be expressed but taken as self-evident from the rest. I do not want to argue this point with anyone. In any case, it is recommended for clear and easy reference to proceed liberally, generously when coding the fundamental conventions and to quote rather one too many than one too few.

Also the theory is in a position not to need to use convention (9) for a long time.

A third g roup o f - - s i x - - f u n d a m e n t a l st ipulations defines those binary relatives which are deducible , by means of the six species or basic modes of calculat ion of w 1; it explains the results of these six opera t ions (on or with binary relatives) as again binary relatives. These are

(ab)q - aob 0 , (a + b)ii - a 0 + b 0 , (10)

d 0 or (d)aj= (aii), (11)

(a; b)o = F, haihbhj , (a~ b) o = IIh(a~h + bhj), (12)

di~ or ( d ) q = a t , , (13)

and should be unde r s tood as general, as an a g r e e m e n t m a d e f o r every

suffix ij, which, for each of these conven t ionsms imi la r to (6 ) - (9 ) of the previous g r o u p - - w o u l d have to be expressed by a sign II o, p r eced ing the proposi t ion, to be placed in braces {}, by which the convent ion appears s t a t ed - - fo r the last, for example , by means of II0{~ 0 = aji}.

In view of what has been said by (5), the first three (10) and (11) of the above stipulations define the identity product a " b or ab, further, the identi ty s u m a + b of two relatives a and b, as well as, finally, the negat ive

("the negat ion") d (read a bar) of a relative a. Because according to the known theorems of the proposi t ional cal-

c u l u s - a l s o compare the abacus ( 3 ) - - t h e (ab) O above can be equal to

Page 30

Page 31


1 only if a 0 and b O both have the value 1, whereas (a + b)o will already be equal to 1 if a 0 or b 0 have the value 1; then one sees that the identity product ab will be that relative which contains exclusively the pairs of e lements common to the factor relatives a and b, whereas the identity sum a + b contains all pairs of elements, and those alone, which either belong to a or b, or to both summands.

The negation d or "not-a o f - " of a binary relative a will, by virtue of convention (4), unite those pairs of elements of the universe 12 which

are not represented, in the negation of a. The first 25 stipulations (1) to (11), together with the still missing

stipulation (14), which forms the conclusion of our list because it will finally define the "inclusion" [Einordnung] between binary rela- t i v e s - t h e s e 26 conventions can be considered as sufficient formal basis for the fact that the binary relatives are subject to the "identity calculus,"

whereas the "specific" laws of the propositional calculus do not at all need to apply.

In view of what has been said by (5), the last three (12) and (13) of the stipulations above define

The Relative Product a; b (said: a of b)

The Relative Sum

act b (I say: a piu b)

of two binary relatives a,b, and finally:

The Converse ("the conversion") ~ - - sa id : a converse--

of a binary relative a. "Relative multiplication," which from two binary relatives a and b

derives a third binary relative "a;b," may be called composition; if one wants to denote the "relative factors" a and b as the "components," the term "Kompos(i) t" or "Kompot" for "relative product" would not seem acceptable, because of fatal ins inuat ions--whereas the English term "compound" is satisfactory.

Stipulations (12) and (13) willulatermbe motivated from the need for linguistic expression. Concerning, for example, the first of these stipulations, linguistic expression shows the frequent juxtaposition of new relatives, such as "lover of a benefactor o f -" from given relatives such as "lover o f - " and "benefactor of- ."

Incidentally, we have to ment ion tha t - -because of the noncommu- tativity of relative opera t ions - - the two relative factors en ter the notion of the relative product in very different ways; they have thus to be dist inguished as "first relative factor," "relative prefactor" or "multiplicant" and "second relative factor, .... relative postfactor" or "multiplicator." When a ; b is formed, we will have to say that one can "relatively premultiply" b with a, or "relatively postmult iply" a with b. Likewise, when forming a relative sum ac~ b, the "first (relative) suxnmand," the "first relative term"


a is "preadded" to the "second' b, or the latter postadded to the former. When we speak of the relative addition or "summation" (respectively multiplication) of given terms, one has always to consider the order in

which they were given: one then joins the terms in the order in which they were expressed.

While the converse d of a relative a is easily described in words as

that binary relative which exclusively unites all individual binary relatives or pairs of elements which are "converse" to those contained in a (cf. p. 10) - - the formation of a;b and a j-b is complicated, and we reserve

the right to examine it in detail later. At this point we only want to

emphasize that the two relative operations, consisting of the two given

relatives a, b, are defined in (12) by the manne r in which their coefficients can be derived from those of the two terms a and b. For that purpose,

these last coefficients have to be taken in every possible way from the

rows of a and from the columns of b and be jo ined together* according

to the prescription of formula (11 ), by means of "identity" multiplication (respectively addition) that is to say, by modes of calculation which belong to the operational sphere of the propositional calculus. For the definit ion and unders tanding of the two relative operations, only the knowledge of the propositional calculus is necessary.

Because of the abacus (3) - - in addition to (4) - - these operat ions of

the proposit ional calculus are always applicable, and in every case of their application they give a "single-valued' [eindeutig] or fully determined result. The identity product and identity sum of any two values from the domain

of values 0, 1 is in each case again a fully determined value from exactly

this domain of va lues- -ne i ther more nor less than the negation of one such. The expressions are "totally single-valued" [voUkommen eindeutige], that is, they are "never undefined" [ hie undeutig] and "never multivalued"

Page 32 [nie mehrdeutig]. And this property can obviously be transferred to our six species for

obtaining the coefficients, for which an easily accessible me thod is pre-

scribed to produce them, namely, a definite p rocedure consisting of the

types men t ioned before--with the exception (if you will) of the con-

verse; there a mere exchange of the two indices has to be made which

causes the replacement of a coefficient of the operand a with another

of its known coefficients. In short, we can say so much:

The six species of our discipl ine-- the identity as well as the relative

modes of calculation--are "totally single-valued" operations. They are unconditionally applicable in our universe; if a, b mean given binary relatives, then the symbols

* In a way that reminds the mathematician of the (row-by-column wise) "multiplication of determinants."

FROM P E I R C E TO SKOLEM

ab, a+ b, d, a;b, aj-b, 5,

237

which m a r k the results of these species as such, are never m e a n i n g l e s s o r undefined signs, also never multivalued names , tha t is, o n e a n d only o n e value be longs to t h e m - - t o t a l l y d e t e r m i n e d - - i n the class o f b ina ry relatives.

Although this remark may be valuable for didactic purposes to the newcomer

to the theory, we only want to state it as a general philosophical perspective to make the correct understanding of our theory possible. One of the theory's important tasks will be to grasp the essence of "single-valuedness," "single-valued assignment" [eindeutigen Zuordnung], and to formulate the notion exactly, as well

as to deduce its laws. As long as it is still surrounded by the nimbus of linguistic uncertainty, no conclusions may be based on a notion of such abstract philosophical color.

Page 33

T h e last of o u r f u n d a m e n t a l s t ipulat ions, which we add to the th i rd g r o u p , is the def in i t ion of inclusion [Einordnung], s u b s u m p t i o n between binary relatives. This is as follows:

(a :(= b) = II;)(a0:~:b;) ) (14)

a n d leads back to the known n o t i o n of inc lus ion b e t w e e n c o r r e s p o n d i n g coeff ic ients of this relative, which has b e e n d e f i n e d by m e a n s o f stip- u la t ion (2). For two b inary relatives a and b, a is said to be included in b, a@b, if and only if, for every suffix ij, a;):~-b;). T h e r e f o r e , a@b tells us that all pairs of e l e m e n t s of a are to be f o u n d a m o n g those of b. We

can t h e n also say: a is part ( e i the r a p r o p e r pa r t o r also the whole) of b, is contained in b.

Because of (1), as can easily be seen, it follows:

Coro l la ry to (14)

(a = b) = Hi)(a 0 = b0)

- - a c c o r d i n g l y , t hen two relatives can be cal led equa l to each o t h e r if a n d only if they ag ree on the i r c o r r e s p o n d i n g coeff ic ients , tha t is, iden-

tically involve the same pairs of e l e m e n t s exclusively.

Thus , o u r verbal cons ide ra t i ons above, tha t a b ina ry relat ive be de-

t e r m i n e d by its coeff icients , finds its m a t h e m a t i c a l c o n f i r m a t i o n , and

the s t ipula t ions given in equa t i ons (5) to (13) are fully s e c u r e d by (14)

a n d (1).

Although one has to speak occasionally about the relations of subordination [Unterordnung] like a C b (where a is called a "proper" part of b), perhaps the secant a ~ b, etc., we can still consider as well as the relation a ~ b (a not equal to b), a:4~ b (a not included in b), defined on the basis of (14) (as per volume 2).

23 8 S C H R O D E R ' S LECTURE II

To c o n c l u d e , a w o r d o f j u s t i f i c a t i o n a b o u t the d e v i a t i o n s o f m y sys tem

o f n o t a t i o n f r o m Pe i rce ' s , r e spec t ive ly the o n e w h i c h c o m e s c loses t to

o u r s "

Because of the noncommutat ive character of relative addition, I have formed

the 'piu'-sign asymmetrically, whereas Peirce (1883) used the formal cross we

commonly find in obituary notices. For similar reasons, I chose for relative

mult iplication the semicolon as an asymmetrical connect ion sign; while I also

express symmetrically the identity multiplication as a commutat ive connect ion,

be it by means of the period as the sign for multiplication, be i t --as in most

cases--by simple juxtaposi t ion of factors (without any part icular sign of connec-

tion). In this respect, I deviate considerably from Peirce.

Peirce denotes the identity product by "a, b." This comma, as a sign of mul-

tiplication, seems less suitable for a commutat ive connect ion because of the

asymmetry to the left and right; fur thermore , I must declare the hyphen, the

often used dividing line in punctuat ion, to be completely unacceptable because of

the confusion it will create first and foremost in the text, as well as in formulas

where the functions of several a rguments have to be considered which are also

separa ted by commas (cf. vol. 1, p. 193ff).

Peirce expresses relative multiplication virtually symmetrically by means of the

simple juxtaposi t ion of factors. If for noth ing else, I could not accept this pro-

Page 34 cedure because it is used elsewhere.

However, two circumstances suppor t Peirce's procedure . The first one is not

significant: if a relative a is in terpre ted as "an a o f - " and, as I suggested, the

semicolon is read as "of" then "a; b" has to be read as "an a of of b"; of course,

I reject the tautological repeti t ion of "of." I diffuse any criticism by saying a can

be in te rpre ted as an absolute term, and as a relative; in the latter case, one

interprets not so much a, as actually "a;," that is, "a o f - " (of) any assumed and

following correlate.

The second weightier circumstance is the following: Unde r the not ion of

binary relative falls also--as we will s ee - - t he notion of mathematical substitution,

not less than the notion offuncti0n. However, one does not write '~f;x" for a

function of x, '~f(x)," sometimes also '~fx." Moreover, the relative multiplication

of the substitutions will be no o ther than its actual multiplication which the

theory of substitution has been expressing for a long time without connec t ing

signs, by mere juxtaposi t ion of the factor symbols. I do not want to deprive the

substi tution theory of the advantages of such a simple type of no ta t ion- -as long

as the theory has (as up to now) to deal only with the one opera t ion of common

( therefore "relative") multiplication. Simplifying deviations from systematic no-

tation for use in a special field are always admissible, however such practices for

a discipline so very general are not applicable.

The following circumstance is against Peirce: the simple juxtaposi t ion of terms

can represent the product of coefficients as well as of propositions for him (as well

as for me); but confusion can arise with his notation, because the opera t ion ab


is not subject to the same laws, depending on whether a and b refer to relatives or propositions. Furthermore, the coefficients can also be represented as (binary, so-called "distinguished") relatives although they are propositions! (See end of

w 25.) The "relative module" l'---corresponding to the "identical substitution" 1 of

the substitution theorymis simply denoted by 1 (without my apostrophe) by

Peircemwhich is only admissible because Peirce replaces my "identity" or "absolute module" 1 by the symbol 0e (infinite)mhowever, not without keeping my

(i.e., Boole's) 1 for the coefficients and propositions. Against the usage of this 00 I think I have said enough in volume 1, page 274ff. I may only add that we will need the ~ for very different~more mathematical~purposes, and that the beautiful analogies between the absolute and the relative modules are obscure in Peirce's notation, but clear in mine. Peirce expresses the relative module 0' as a gothic it (not as the latin n), corresponding to the first letter of"naught"

Page 35 or "nought" (nothing).

Finally, we could cons ider as f undamen ta l s t ipulat ions the rules (which we omi t t ed initially) that def ine and regula te the use of the product and sum signs:.

H and E.

By the "running index" (i.e., the "product variable," respectively "summation variable") u, we u n d e r s t a n d a relative symbol to which all values

of a def ini te (to be cons ide red as somehow given) d o m a i n of values shall be assigned. This doma in of values is called the "extension" [Er- streckung] of the "product H," respectively the "sum ~," "taken with respect to u" and will, in the most genera l case, be a well-defined "class" of (binary) relatives.

By the "general term" (factor or s u m m a n d , respectively) of the p r o d u c t H or sum ~ - - w h i c h can always be caught sight of after this s ign - -we

u u

u n d e r s t a n d a "function of u," f(u), i.e., an expression which is cons t ruc ted in some given way by means of opera t ions f rom the g r o u p of six species in our discipline, f rom u itself, and any o the r relatives a, b, c . . . . . x, y,

.... of which the m e a n i n g (value) has to r ema in constant, even if the

m e a n i n g of u (within that ex tens ion) changes. These lat ter relatives are

c a l l e d m i n contras t to the "argument" u n t h e "parameters" of the funct ion

f(u), and can be u n d e r s t o o d both as general relatives, as well as having

special values; in particular, they, or some of them, can be r ep laced by modules.

F u r t h e r m o r e , the funct ion f(u) is itself a binary relative whose value

for each assigned value of u and the fixed values of the genera l le t ter pa r ame te r s has to be fully d e t e r m i n e d - - t h e reason be ing that the results

of the opera t ions or species, constituting the express ion f(u) and there in

Page 36

2 4 0 SCHRODER'S LECTURE II

prescribed, are defined to be single-valued, by courtesy of our stipulations of them as binary relatives. Indeed, the general coefficient of the suffix ij of this relative f(u) can be easily represented by means of combined application of our six schemes (10) to (13) as a proposit ion function [Aussagenfunktion] of the general coefficient of both the a rgument u and all of its parameters according to a totally de termined, prescribed procedure . With f(u), we also know for each ij its relative coefficient

If(u)} O. We now have to define the symbols

H f(u) and El(u)

as binary relatives. This definition has, as always, to be done by means of a general s ta tement about their coefficients, which is provided by the two following stipulations:

{Hf(u)}# = H {f(u)}0, {El(u)} 0 = E {f(u)} 0 , (15)

which are to be "assumed" for every suffix ij. We will be able to give a simpler version of these stipulations in w 6:

(H a)0 = H a~i, (E a)0 = E a 0 . (15)

If we include these, we will have in total 29 + 2 = 31 fundamental stipulations.

In fact, there can be no doubts about the meaning and value of the coefficient (i.e., propositional) products or sums on the right side by which our coefficient on the left is to be made explicit.

In view of the goals pursued in our ninth lecture (w167 23 and 24), it is impor tant to discuss this point in detail and, in particular, to gain the conviction that it is not at all necessary for the evaluation of such propositional-II and -E to assume the notion of proposit ional product II (respectively propositional sum E) as established by explaining it as was done, for example, in Appendix 3 of volume 1--namely, "inductively"--by extending the associative laws of propositional multiplication and addition from three to any (even an unlimited) n u m b e r of terms by means of the "inference from n to n + 1." Rather, it is sufficient for the establishment of the notion and the explanation of the most impor tant propositions referring to it to only use the right to make a general consideration, namely, to think in universal and existential terms.

We therefore want to discuss in more detail the role of II and E in the propositional calculus.

Both notions are to be considered as established in their own right ( independently, not recursively or inductively), as follows. If A,, repre-


sents any p ropos i t ion re fer r ing to an object of t h o u g h t u, i.e., a osi t ion about u," then of the two symbols

2 4 1

"prop-

HA,, a n d EA,, u u

Page 37

- - - ex t ended over any given d o m a i n of "values" as the m e a n i n g s that symbol u shou ld take o n - - t h e first of these has to represen t : the proposi t ion that A,, is t rue for every one of these objects u (within the "extens ion") ; the s econd tells us: the p ropos i t ion that A,, is t rue for certain u (within this ex tens ion) , that there is at least one u in the ex tens ion [Erstreckungsbereiche] for which A,, is true.

T h e r e f o r e , the t ruth value 1 is appl icable to the p ropos i t i on I I A , if u

and only i f f0 r each of the aforesaid u, A,, = 1; on the o t h e r hand , the t ru th value 0 is appl icable if the re is at least one u for which A u is no t true; the re fore , where A,, = O.

T h e t ruth value 1 will apply to the p ropos i t ion E Au if the re is any u at all in the ex tens ion of u for which A u = 1; on ~he o t h e r hand , the t ru th value 0 applies if and only if the re is no such u, i.e., if for each u in its ex tens ion A,, is no t true, A, = 0.

If v represen ts a value, arbitrarily p r o d u c e d for u f rom the ex tens ion of u, we mus t have

(HA.= 1)=(= (A,, = 1) =(=(EA,. = 1), (EA,,=O)={c--(A,~ = 0) = ( = ( I I A . = 0) u u u u

or, shorter : m

HA,, a@ A,, =r r, A,, , E A,, =t-=A,, =C-HA,,, o~) u u u u

whereby it is given:

HA,, = A, , I IA, , , EA,, = A,, + EA, , . 13) u u u u

T h e last f o rmu la shows that for each value (v or u) f rom the ex tens ion , the so-called "genera l factor" A,, of the proposi t ional - I I can also be c o n s i d e r e d and p r e s e n t e d as a real ("actual") "factor" of the proposi - t ional "p roduc t" in the most na r row sense, a lready exp l a ined as a "binary" (two-factor) p roduc t , wi thout the H-sign; and also that the so-called "genera l te rm" of a proposi t ional-[ ; is an actual (or "real") summand of a binomial propos i t iona l sum, i.e., o f a p ropos i t iona l sum in the nar rowes t sense, as which ou r proposi t ional -E has to be always capable o f be ing

u n d e r s t o o d . Based on what has been said above, it mus t f u r t h e r m o r e be clear that

the negation of ou r proposi t ional- I I and E has to be "carr ied out" acco rd ing to the fol lowing scheme:

242 SCHRODER'S LECTURE II u

H A , = EA,, , EA u = I I A , . 3/) u u u u

In the "dictum de omni et de nuUo," appl ied here , by which we have ga ined all the p r e c e d i n g formulas (of which we i n d e e d d e m a n d accep-

Page 38 tance) , a real "axiom" is not to be seen; the d ic tum has only the cha rac t e r o f a "pr inciple" (in the sense of vo lume 1); it acts as a subst i tute f o r m a n d is in this sense no th ing m o r e t h a n - - a definition of the no t ions

"every" u, respectively, "some" or "certain"* u ("in genera l one" u, shor t e r "one u," a u)

- - t h e def in i t ion of which could probably no t be given "formally," as a " s tandard def in i t ion " Cf. pp. 67ff.

If the ex tens ion of u conta ins only one object v, it is easy to see that the m e a n i n g of bo th HA and EA is the p ropos i t ion A re fe r r ing to this o n e v, namely that

I I A , , = A v = E A , . u u

T h e II and E consist he re of only one term; they are "monomia l . " If the ex tens ion of u conta ins the two objects v and w, we recognize

easily that the m e a n i n g of

I IA , ,=Av A , , , E A , , = A v + A,o u 11

is de f ined by the binary produc t , respectively the binary (= b inomia l ) sum of individual propos i t ions re fe r r ing to v and w, es tabl ished t h r o u g h abacus (3) (without the II- and E-signs).

If the ex tens ion of u conta ins exactly th ree o b j e c t s m t o express it now mere ly for I I - - r e p r e s e n t e d at the m o m e n t by the letters u, v, w, then we cou ld see that the m e a n i n g of HA is de f ined by the te rnary ( three- factor) p r o d u c t A,A,,Aw as the c o r r e s p o n d i n g value of the two binary p ropos i t iona l p roduc ts A,,(A,,A,,,) and (A,,A,,)A~,---established on the basis o f the associative law for p ropos i t iona l mul t ip l icat ion. And so on.

In ou r theory, we can omit expla in ing that the propos i t ional -H (with a b o u n d e d extens ion restr icted to an arbi t rary "number , " a "finite set" of objects u) can be der ived t h r o u g h successive, binary multiplication of its factor proposi t ions , we can omi t stat ing this and mak ing essential use of it. At least until the n in th lecture, in which the " in fe rence f rom n to n + 1" will be r igorously proved. However, until this happens , the ma t t e r at h a n d will have to be cons ide red as sett led by A p p e n d i x 3 of vo lume 1.

[We have to take even less advance not ice of a d o m a i n of values consisting of an " u n b o u n d e d , " namely, "simply infinite" series o f discrete

* The German term "irgend ein" is not really suited here because of its double usage" it can replace the English "some" or "ally." The latter, meaning "any arbitrary one," has to be excluded (because it would be synonymo'us to "each").


Page 39 objects u, our IIA could be derived as a "limit" t h rough binary multiplication of the factor proposi t ions con t inued indefinitely [fortzusetzenden] ! ] '

If A, is i n d e p e n d e n t of, constant with respect to u, that is, if there is no men t ion of u in the proposi t ion which figures here as the general term, then we may omit (also in the formulas) the suffix u in the proposition A,, as un impor tan t , and represent it merely with A. T h e n the result certainly is

H A = A and E A = A . 6)

Likewise, if B,, is a proposi t ion with respect to u and B is a constant propos i t ion with respect to u, we have fur ther the scheme:

rI(A~-B,,)=(A=~c--HBu), H(A,,~-B)=(EA,~--B), e)

both of which can be combined into the general scheme:

II or IIII(A,,~(=B,,)=(E,A,,~IIBv), u , 1 ; u 11 1~

t')

where the extension of v is arbitrary [posssibly different f rom the extension of u].

Analogous to Peirce's formulas, my two schemes are also valid:

E(A,,~--B) = (rl A,, ~(=B), E(A =(=Bu)=(A ~(= E B,), 7/

which can be combined to the more general

~2 or F,r,(A,,=~-B~)=(H,A,,~-I2B~). u , ~ u IJ 1~

0)

If one specifies in e) and r/) A = 1 (by taking A hereaf ter for the remain ing B) or B = 0, then the following schemes result:

{ II ( a , = 1) = (HA,, = 1), II (a,, = 0 ) = (EA, = 0), u u t)

Z, (a,, = 0) = (II a , = 0), I~, ( a , = 1) = (E au = 1),

of which the first and last do not express anything [nichtssagend], in view Page 40 of the "specific principle" of the proposi t ional calculus:

( A = I ) = A ;

the o ther two, however, are very often used. Finally, we have to men t ion the proposi t ional scheme:

I Text (other than German terms) enclosed in brackets th roughout the appendices is Schr6der 's unless otherwise noted.

Page 41

244

{ (H, A,,::~-. H B,,) (A,,~B,,) ~ (IA,,~-EB,,)

14 U

SCHRODER'S LECTURE II

E (A,, :~--B,,) K) u

because it is most often used. Of the two middle subsumptions, one standing on top of the other, only one (or the other) need be taken as thesis (assertion, conclusion) or hypothesis (assumption, condition). This mainly permits sliding over products and sums from valid general subsumptions, etc., for the extension of u.

We have thus recapitulated and clearly compiled, for the benefit of the student, the most important schemes or theorems of the propositional calculus, as they refer to propositional-I/ and Emat least those that will suffice for the time being (and a few more which will directly follow).

They appear explicitly or in nuce in volume 2, although in scattered form (ibid. pp. 40, 180, 194, 258, 261, etc.). One can recognize in o~) the theorems 6) of volumes 1 and 2; in/3) an obvious corollary thereto by the authority of R. Grassmann's theorem 20); in ql) De Morgan's theorem 36); in 6) the laws of tautology 14); in e) the "definition" of Peirce, from volumes 1 and 2, numbered (3) there; in r/) my counterpart delivered for that purpose in volume 2, page 258 (only valid for propositions); in t) theorem 24) along with the counterpart delivered for that purpose in volume 2, page 261 (only valid for propositions); in K), finally, the extensions of theorem 17).

Not mentioned are the distributive laws for the propositional II and I2:

A E B . = r , AB,,, A + IIB, .= II (A + B,,), X) u u t t u

as well as their extensions to a multiplication rule for (propositional-) polynomials and their dual counterparts"

(~ A,,)E By = E A uB~, II A,, + I~ B~ = I~ (A + B~) Ix) ~) u , v u Ip u , 1~ lz �9

The counterpart to X)"

AI/B,, = II AB,., A + E B,. = ~ (A + B.), 1,) u u u u

is understood from 6), according to the identities:

(H A,,) II Bv = II A,,Bv = II A.B. , v u , 1~ u

EA u + EB,, = E (Au + B,,) = E (A. + B.), ~j)

of which the last is only valid if u and v have the same extension [ Erstrecku ng] .


We could fur ther add schemes for multiple sums and products. The most remarkable new theorem among these is this one, based

on Peirce: If A,,.~ represents a proposition referring to two objects u and v, which

are to be thought of as variable in their own respective extensions, then

E II A,,.v ~-- II E A,.,, o)

Page 42

-,---wherein the symbol A ..... can, of course, be replaced by Av. u. If there is at least one such u so constituted that for this u and every

v the proposit ion A is valid, then there is also for every v at least one u (i.e., the one we just ment ioned) such that the proposi t ion A is true of both. This conclusion is obviously not convertible.

We shall use the assumed schemes predominantly, if not exclusively, for such objects u, v, ... which are not only general relatives but also mere "elements" i ,j , ..., or "individuals of the first universe of discourse." In such cases, we add the running indices to I2 and II (as done so far)

as suffixes instead of writing them under ~ or II. In order not to be r edundan t and not repeat ourselves too much, we

want to consider the relevant or still unexamined schemes of the remarkable schemes only in w 7 and with the above restriction.

Already with the restriction of the indices to elements, we may point out and emphasize that the domain of values of our propositional-II and I2 may also always be a "continuum," as, for example, are the real numbers or the points forming a straight line. In such cases, the signs II and I] are definitely indispensable and it would never do to represent the proposit ional product explicitly, as an "actual" product with all its factors, which is represented "symbolically" by, for example, II.

In our theory, it will always b e m i f not mere proposit ional products, respectively propositional sums, then at least "identity" products II and sums I2, which these signs help us represent. In o ther words, the signs lI, I2 as such are only used by us for the first main stage [ erste Hauptstufe] to indicate an identity multiplication, respectively addition, of (in most

cases infinitely many) relatives. If ever these symbols should also be needed for the abbreviation of

relative products and sums, we shall write them (similar to the modules) with an apostrophe as II', 52'. Conceivably such usage will need in-depth consideration, perhaps in relation to the de te rmina t ion of general

coefficients. The s tudent has to think seriously about the above proposit ional

schemes a)...0) [as we have carried out for illustration unde r 0)] and try to absorb them succum et sanguinem.


The practice of the rest, which our theory permits, will fur ther con- 2 tribute to a better unders tanding ....

w 4. Threefold Analytical-Geometrical-Rhetorical Evidence

To grasp our theory correctly, it is of utmost importance that the reader, Page 64 before beginning with it, observes the following.

All propositions in the algebra of relatives may lay claim to a "threefold evidence."

We can talk about "an evidence" in three different senses--after what has already been said so farmwe can find it iUuminatingin three different ways, depending on whether we take as a starting point: the fundamental stipulations asserted in w 3; or, the geometric representat ion of relatives and the operations to be per formed with them, as we connected it to the consideration of the matrix; or, finally, the verbal interpretat ion of relative symbols, writing them with relative names from ordinary language for the purpose of applied logicmas we have just now hinted at in anticipation.

In short, al though it is not completely exhaustive, I shall call the three different evidences: the analytic, the geometric, and the rhetorical evidence. The purity of the method will require that we do not mix them, that we prefer one of t hem- - fo r algebra, the first o n e m a n d let it govern all

Page 65 essential inferences. The first, the "analytic" evidence, will be obtained through a rigorously

deductive "proof' of the formulas or propositions of our theory from the formal foundations given in w 3. It will show of each formula that it is already contained, as a consequence, in those conventions, and that it is therefore necessarily given by them. Such a proof is to be given by calculation [rechnerisch], in which at each step we realize by which scheme of the propositional calculus we perform that step, that is to say, by which law of general logic this step is legitimized. The following lectures will provide ample illustrations of the nature of this evidence and the way to obtain it. Another unambiguous name for it could be "coefficient evidence" because, for the fundamental conventions, the results of the six species as one relative each are only explained by means of the definition of its general coefficientmthus, we have to revert indi- rectly or directly to these coefficients for all proofs.

We shall consider this analytic evidence as exclusively valid in our theory, and a proposition in the algebra of relatives cannot be accepted as certain unless it has been "proved" in this way.

But we can also, as a second way, accompany or pursue the relations

Schr6der's exposition of the matrix representation of the algebra of" binary relatives (pp. 42-64) has been omitted.


and operat ions between relatives from a "geometric" point of viewmfor example, by means of a comparison through mental superposi t ion, making parts coincide, connect ing and separating their spatial figures, as well as by means of a lawful interlacing of their sequences of "filled circles" (o) [Augenreihen] in their matrices.

We immediately see, for example, for Figures 4 to 7, 3 that we have 1 ' ' 0 ' = 0 and 1' + 0 '= 1, that the two relative modules are disjoint and c o m p l e m e n t each o ther for the entire universe of discourse 12, in o ther words, that they are negations of each other.

O r m t o illustrate the "geometric evidence" with one more e x a m p l e m i t will make immedia te geometric sense that

( a = 0 ) = ( a ; l = 0 ) and (a~: 0 ) = ( a ; 1 g: 0)

after we have recognized that the relative a ;1 is always obta ined from a relative a when the rows of the latter occupied with (one or more)

Page 66 filled circles are changed into fully occupied rows or full rows, that is, a and a ;1 are mutually vanishing or nonvanishing.

Likewise it is clear that we have to have a~--a; 1, and m o r e m a s we have already recognized in volume 1 the proposit ions of the identity calculus for the figures or point systems which our relatives represent geometrically as directly evident.

Although it may not be essentially used (here as well as there) for the construct ion of the theory, the geometric evidence ought not to be rejected because it is a convenient and fruitful means of discovering propositions; it also gives us a most welcome control and makes it easy to r e m e m b e r many propositions at a time. We will thus pay special at tent ion to it in the theory; yes, the theory will even justify the tendency to gradually "replacg' the coefficient evidence by the geometr ic evidence in an analytically well grounded way.

The third type of evidence, the rhetorical evidence, is effective for ordinary thinking. We notice it, we become aware of it, as soon as we exempl i fy~wi th Peirce~rela t ives of a special nature, such as l = lover (of), b - benefactor (of), s -- servant (of), with those general relatives that appear as letters in our formulas.

Anybody will find the following proposit ion immediately i l luminating: The lover of a benefactor, who is a servant (of somebody) , is a lover of a benefactor and at the same time also a lover of a servant (of this somebody)mas the following formula expresses [somehow shorter and also more generally]:

l; (bs) ~ (/; b)(/; s).

Nobody will refrain from extending the proposit ion of the given rel-

~ See page 50 of Schr6der ' s second lecture.

24 8 S C H R O D E R ' S LECTURE II

atives of a special nature to any others, be they relative, or be they absolute terms, and recognize in it a principle which governs our whole thinking as a matter of course. The picture of a dead friend is certainly the picture of a dead person but also the picture of a friend; the buyer of an expensive horse is a buyer of something expensive and a buyer of a horse, etc.

We thus must have generally

a; (bc) ~ (a; b)(a; c).

As soon as we have become better acquainted with the translation of sign language into ordinary language, we cannot but recognize a high

Page 67 degree of direct intuitiveness in the more simple formulas of our theory. The logic, if one wants to develop the methods and schemes of de-

ductions and inferencesmfor relative as well as for absolute no- t ions - -cannot avoid to strive for the most complete possible registration and schematization of: the a priori, self-evident, identical, analytic, or irreducible judgmen t s or "truths" as those to which we can always appeal, when drawing any kind of conclusion.

However, if we would rely on such evidence when constructuring our theory, we would soon find ourselves obliged to recognize an excessive n u m b e r of strange "principles," and we would seem to justify Mr. Venn's complaint (1880, p. 400ff): that instead of one "simple and uniform set of rules," the very simple system of principles of the old logic, we "are in t roduced into a most perplexing variety of them" when enter ing the logic of relatives.

In addit ion to the fundamental conventions compiled in w 3, there is no other "principle" in the theory. And if somebody raises questions about the axiomatic foundat ion of our discipline of the algebra and logic of relatives, I can only agree with Mr. Peirce who comments on this question at the conclusion of Peirce (1883). The foundations are of the same level [Range], are no other, than the known "principles" of general logic. Contrary to geometry, logic and arithmetic need no p roper "axioms."

In order not to be misunderstood, I have to add: of course, geometry can also be considered just from the formal perspective of the consistency [Folgerichtigkeit] of its theory. Its so-called axioms can be presented as mere assumptions, perhaps very arbitrary assumptions, about whose validity, truth in any universe of discourse we are not at all concerned; we refrain from asserting anything. The geometric propositions then lay claim to only a relative truth and will only be accepted if those preconditions apply. In general, this does not happen; rightly so, in my opinion. Geometric axioms are taught, presented, and accepted, as having a claim on validity, truth, in reality, be it for our subjective intuition of space [Raumanschauung, be it for the reality that is objectively thought to be

Page 68


at the basis of it. These axioms are not at all analytic or tautological [nichtssagenden] judgments ; even if we call them "psychologicaUy necessary" in view of the nature of our spatial intuition, we cannot call them necessary in the logical sense, and thus geometry is more than a mere branch of logic; it is the most e lementary part in a large n u m b e r of physical sciences. Arithmetic is otherwise.

In volume 1, I have consciously shied away from using the name "axiom" for the "principles" appear ing in the theoretical analysis. Those "principles" are only h idden def ini t ionsm"are mere substitutes for definitions of the universal logical relations." In so far as the universal logical relations can be def inedmPei rce has a right to saymwe can do without any "principles" (all axioms may be dispensed with). This idea, I think, will find fur ther confirmation in the following lectures.

In particular, we will find it instructive that and how the proofs of the complete distributive law are conducted.

To come back to our three types of evidence, we cannot revert too much to the two latter ones in our theory; they can be used at most for the illustration of propositions.

For simple propositions, the second and third evidences easily over- take, readily speed ahead of, the first evidence; for relatively complex propositions, the third evidence in particular remains far behind---of ten without h o p e ~ a n d we are eventually painfully forced to give evidence by means of complicated deductive proofs, and the application of the o ther two types of evidence requires for the un t ra ined s tudent a considerable amoun t of thinking and "headache."

Appendix 3" Schr6der's Lecture III

Introduction

The first page of Schr6der 's third lecture in volume 3 is r ep roduced in translation here, followed by another part of the third lecture, beginning on page 97 to end of section 6. This is Schr6der 's algebraic t rea tment of quantifiers as arising from least upper bounds (sums) and greatest lower bounds (products).

Schr6der appears to think that by reducing each identity of relative calculus to an assertion about characteristic functions, he can reduce every relative identity to an equivalent propositional one. But when one actually does this reduction, in the front there is a series of universal quantifiers over all e lements of the domain: (v i) (u For a single finite domain, of course, de te rmining whether an identity holds reduces to proposit ional calculus. Schr6der apparently did not realize that an identity holding in all finite domains does not necessarily hold in all domains whatsoever. In volume 3, page .551, Schr6der tries to rewrite all first- order statements about binary relations as equations in the relational calculus. As was proved by Korselt (L6wenheim 1915), this cannot be done. But in 1941 Tarski announced that if there is a decision method for telling whether a relational equat ion is an identity, then there is a decision me thod for telling whether a first-order s ta tement about binary relations is true, contradict ing Church 's solution to the Entsheidung- sproblem. It was later proved that no finite set of true identities implies all true identities (Monk 1964).

That having been said, Schr6der, following Peirce, does derive a large n u m b e r of relational identities. However, it was beyond his world view to see that there is no finite axiomatization of the theory of identities in relation algebra.

251

25"2

T h i r d L e c t u r e

SCHRODER'S LECTURE III

The Propositions of the Most General Nature in the Algebra of Binary Relatives

w 6. Laws of the Species, If Only General Relatives Enter Their Expression. Duality and Conjugation.

The most impor tant laws of the six basic types of calculation have been Page 76 discussed with considerable completeness by Peirce.

The lowercase latin letters will always represent general binary relatives and in addition denote the "elements" ment ioned in 3) on page 7.

Of course, the laws of operat ion [Kniipfunggesetze] which have already come to light in the simplest operations will also play a role in the more complicated operations; they will, in all expectation, be the basis of more complex laws applying to the propositions with more complicated operations. As the simplest operation, we may stipulate one in which only 1, 2, 3 (at most 4) letters enter as symbols for as many i ndependen t and arbitrary relatives. Thus, we can roughly limit the field to the rules of inference [Folgesiitze] or the "laws" which can be called fundamenta l for our discipline. In order to discover heuristically conceived, fundamental laws, we need only write all conceivable expressions which we can construct from a very small n u m b e r of letters by means of our species with combinatorial completeness. For each of these expressions, we have to form the general coefficient, according to stipulations 10) through 13) on page 29, an exercise which we r e c o m m e n d for the beginner; and, finally, we have to find out which relations (of inclusion [Einord- nung] or equality) can be justified between these coefficients based on the theorems of the propositional calculus [Aussagenkalkuls]. Then we will also have gained the conviction that our compilat ion of propositions is complete, or, at least, gained the knowledge that noth ing impor tant has been overlooked . . . .

w 6. The II and ~ of Relatives

Finally, we have to stipulate the most impor tant of those propositions or formulas which are valid for the (identity) products II and sums ]2 of

Page 97 binary relatives, whether we consider these symbols as independent ly defined, as we have described it at the end of w 3, page 36, and, for example, as unavoidable for "continuous" extensions [Erstreckungsberev che] of II and E, or whether we only use them as abbreviations for the

Schr6der's derivations of relational identities not involving the quantifiers (pp. 77-97) have been omitted here.


results of binary identity operat ion species between arbitrarily many terms which it would be cumbersome to write in full.

That our formulas have to be the same for both interpretations (because the second interpretation is subordinated to the first) can only be considered as proved rigorously in the ninth lecture.

We have to remark in general that tile important propositions will be used only late in our theory, at a relatively advanced stage, so that the student may skip them for the time being.

Most of the propositions are known and valid from the identity calculus. They form counterparts [Gegenstiick], pendants , possibly also heavily modif ied (that is to say, weakened or defective) analogues to the schemes of the identity calculus which we have collected unde r c~) to ~) at the end of w 3---they are analogies because with regard to stipulation (14) they are basically consequences of those. But the analogy is not without exception; some of the propositional schemes will remain without a counterpar t (for relatives), and, from the point of view of our newly founded basis, the identity calculus will prove to be the one which allows fewer conclusions compared to the propositional calculus which is r icher in formulasmwe find that the last reason for this lies in the circumstance that the fundamenta l stipulation (14), constructed with YI 0, lacks a "propositionally dual" counterpar t constructed with E0' and remains definitely inadmissible there (cf. volume 2, pp. 43ff).

In order not to overload the formula with signs, we will omit the indices where there is only one general relative and the sign ~ or II is

Page 98 used. If a is constant, we have the law of tautology

I Ia = a = Ea 17)

however the extension [Erstreckung] of II or X] may be given---cf. 6) of w 3, page 39.

If a is variable, we then have

Ha :(= a :(= ~:a 18)

if the free a standing in the middle (of II and Z) represents only any

one of, an arbitrary one, of the changing terms (a), over which the II and ~ have to extend consistently [i ibereinst immend]---cf . o~) of w 3.

One such Ha, = II a, ought to be replaced by the following expression which is more general from a formal point of view or seemingly more

comprehensive:

1-I 4>(a). (z

This expression contains in fact the previous one, because we can, at any time, specialize 4)(a) = a as long as 4)(a) is of indefinite generality.

254 SCHRODER'S LECTURE III

However, instead of specifying the d o m a i n of values as the ex tens ion of H, which we have to add mental ly to the a r g u m e n t a of the genera l te rm qS(a), in o the r words, which a itself has "to run t h r o u g h " and t h r o u g h which the c o r r r e s p o n d i n g values of 4~(a) are un ique ly [ eindeutig] d e t e r m i n e d , we can immedia te ly cons ide r the d o m a i n of the lat ter values as given. This d o m a i n forms a new ex tens ion [Erstreckungsbereiche], which is in genera l d i f ferent f rom that of a, and, if we use it, ins tead of the previous one , as a basis, ou r express ion II 4~(a) will now be rep laced by

a

II ~b(a). ~(a)

Page 99

However, the two terms may not be posed as formally equal despi te their identi ty because in such an equality, on accoun t of the d i f fe rence of the ex tens ions on bo th sides, the sign II would a p p e a r as one used with a "doub le mean ing" [~b(a) does no t have to take or run t h r o u g h the value of a ! ] D t h e r e f o r e : because of the passage over the equal sign, we would have a change in the d e n o t a t i o n pr incip le [Bezeichnungsprinzipien] e n t e r i n g in the t e rminology or n o m e n c l a t u r e , which is no t permissible!

T h e r e are no m o r e obstacles to i n t roduc ing a shor t e r des igna t ion , in the form of the let ter c, for the formal te rm 4~(a), and let 4~(a) = c; thus we get the express ion

I-It, r

which has the same form as the previous II a in which only the ex tens ion a

is to be cons ide red as different , that is to say, consist ing of the values o f ~(a), ins tead of the values of a.

If we c o n d u c t a new or i n d e p e n d e n t examina t i on in c o m p l e t e generality, we may use f rom the b e g i n n i n g the te rm a ins tead of the let ter c, and we then get back to our previous express ion as one which is only apparently less general"

By the appropriate choice or modification of the extension, every expression of the form 11 do(a) can be transformed into one of the simpler form II a.

a a

We can assume a similar simplif icat ion for the express ions E4~(a),

~b(b), ~ ~(b), which we can replace by the s impler ~ a, H b, ~ b, a~ then the converse remains valid.

It is r e c o m m e n d e d to do this because of the gain in the clarity of the fo rmulas and the ease which we achieve.

O n this assumpt ion , we have, as c o u n t e r p a r t ~:o -y) o f w 3:

IIa = Ed , Ea = IId , 19)

nex t to which we immedia te ly posit:

Page 100


l i a - l iS , ~ a = ZS,

I ; a - I I a ,

255

20)

fu r the rmore , as coun t e rpa r t to ~), ~) of w 3:

~h (a :(= b) = (E a :(= H b) 21)

- - w h i c h can also be r ep resen ted by

II(a ~ b) = (Ea =(= lib)

also sufficiently expressive and not equivocal. The schemes r/) and 0) of w 3 lack an exact analogy in our theory

and br ing no formulas of the same scheme for our relatives, except the weakened:

E(a =(= b) =~: (Ha =~- Eb). 22)

C o r r e s p o n d i n g to schemes t) are:

H(a = 1) = (Ha = 1), H(a = 0) = (Ea = 0), 23)

E(a = 0) :(= (Ha = 0), E(a = 1) :(= (Za = 1), 24)

whereby the formulas of the second line seem weakened c o m p a r e d to the schemes in t).

C o r r e s p o n d i n g to K) of w 3, we have in the first par t

(Ha =(= lib), l i (a =(= b) =~: (Ea @ Eb), 25)

whereas the last par t or the end of that scheme remains wi thout coun te rpar t .

As c o u n t e r p a r t to X) and g) of w 3, we have the distributive laws:

{ a~ = ab a + li b = l i (a + b) t, ~ ' h t, ' 26)

(Ea) E b= E ab I Ia + l i b= li (a + b) a , b ' a b a , b ' b

and to y) and () of w 3, we have the proposi t ions

a l i b = ab a + E = Y (a + b), b ~ ~ b b

(Ha) H b = li ab r a + ~ b = r (a + b) a , b ' a 'J a . b '

27)

- - f o r the latter, in case the extens ion of both l i (respectively, E) is the same, the doub le p roduc t (respectively, the double sum) can also be con t rac ted into a s imple one [cf. 5) of w 7]:

256 S C H R O D E R ' S L E C T U R E I I I

II H ck(a)~,(b) = II ~(a)~(a), E E {4)(a) + ~(b)} = E {4)(a) + ~(a)} a ~ a a IJ a

nf ina l ly , we have the proposi t ion, as coun t e rpa r t to o) of w 3:

E Ha :(= II Ea. 28)

For the relative operations, we only have to add the following extens ion of proposi t ions 5) and 6) of this section:

a;~b=~a;b , a# IIb=II(a#b), b b b b

" (~ a ) ; b = E a ; b, II a j- b = II (a c~ b), 2 9 ) a a a

(~ a)" ~ b : F, a" b II a~ O b = II (a# b) ' a , b ' ' a ~ a , b '

Page 101

I(~ ;b:~-' a J b

a ~b=~---~a;b, E(a#b)~- ~ b , a) a; b, E (a# b) ~= E a~ b, 30)

(IIa);IIb IIa 'b E (a#b)~ .~a# Eb. \ - r

b a , b ' ' a , b a b

We can read the signs

II = I I I I = I I I I , I2 = I~I2 = E~; 31) a , b a b b a a , b a b b a

arbitrarily.

Appendix 4: Schr6der's Lecture V


The theory of solvability of finite sets of algebraic equations in n variables over the complex and other fields had been worked out in 1882 by Leopold Kronecker in his Grundziige einer arithmetischen Theorie der algebraischen Gr6ssen. Kronecker 's theory was based on generalizations of the already old theory of resultants. In the case of the complex field, a logic-oriented phrasing of his result is that we can associate with any finite set P of algebraic equations in x l, ... ,x,, a finite disjunction of conjunctions Q~ of equations and inequations in xz . . . . . x, such that for given values a2, . . . ,a,,, there is a value of al such that a 1, ... ,a,, satisfy P if and only if there is an i such that a 2 . . . . . a , satisfy P~.~ This is part of Kronecker 's el imination theory, a project for giving condit ions for solvability of sets of many-variable algebraic equations over finitely genera ted integral domains. In the context of the complex numbers as g round field, the set of all solutions to a set of equations being called an algebraic set, and the difference of two algebraic sets being called a Zariski open basic set, this theorem states that the projection of a Zariski open set is Zariski open. These are also called constructible sets in algebraic geometry (see Har tshorne ' s s tandard graduate textbook Algebraic Ge- ometry), and this is the theorem that they are closed unde r projection.

In Schr6der 's fifth lecture, he replaces a finite set of algebraic equations and inequations by a finite set of relational equat ions and inequations between terms built up from constant binary relations and variable binary relations by the six relational algebra operations. If R l . . . . , R,, are the unknown relations in the terms, the others being known relations that act as coefficients do in algebraic equations, he wants to know when, for given relations R2 . . . . . R~, as values for

i This is also the ma in thrus t of Tarski 's decis ion m e t h o d for a lgebraical ly closed fields. His m a t h e m a t i c s is a special case of Kronecker ' s , which also appl ies to m a n y o t h e r fields.

257

258 SCHRODER'S LECTURE V

Re . . . . . Rn, we can find a value R'~ for R, such that R'l . . . . , R~, satisfy the original equation. Schr6der shows that, unlike the case of algebraic equations, every inequality can be replaced by equalities, and every finite set of equalities by a single equality. (For our discussion, by the relational class corresponding to a relational equation with variables R~ . . . . . R. we mean the class of all n-tuples of relations (R' 1 . . . . . R',) that satisfy the equation when R 1 .. . . , R,, are valued R' 1 . . . . , R',). The relational classes are a (class) Boolean algebra. Schr6der thus hopes to prove that they are closed under projection. We do not know if anybody ever answered this question, which is a reconstruction of his thought. What in fact he does is to show that this works in many complicated examples.

Schr6der also asks for the general form of all solutions R 1 for given R 2 . . . . . R,. This is where he introduces a precursor of Skolem functions, replacing existential quantifiers by function symbols that witness them. For this he introduces his f(u), a multivalued function (binary relation) whose independen t variable ranges over some index set, and whose values run through all solutions R 1. This is a relational precursor of Skolem functions and should properly be written f(u,Rz,. . . ,R~, ). Schr6der 's intent is to follow the Kronecker pattern by induction and show that solving the original problem for R'I . . . . , R~, is possible if and only if a certain relational equation involving only the constants in the original equation is satisfied. This was only a fond desire. He carried out only examples. The elimination theory he was after would be characterized nowadays as an elimination of quantifiers for a first-order theory, where the model intended is the class of all binary relations, and the atomic formulas are the relation equations, which may all be written F = 0 .

Although the Schr6der elimination problem appears not to have been extensively studied again, the massive identifies Schr6der developed were axiomatized by Tarski in relation algebras. There are also theorems in Tarski-Givant's set theory without variables that are set-theoretic variants of Schr6der 's elimination problem.

L6wenheim's 1940 paper can be construed as saying that all of ordinary mathematics can be expressed by such equations, and all proofs in mathematics can be viewed as elimination problems.


Fifth Lecture

259

Page 151

The Solution Problem in the Algebra of Binary Relatives

w I I. Total Proposition of the Data of a Problem and General Solution

Page 150 The most remarkable aspect inheren t in the results demons t ra t ed in the previous lectures may be the f ac t~shown in formulas 5) in w 1 0 ~ t h a t in our algebra each inequality can be changed into an equality (of similar character) with the right side 0 or 1.

We would like to show the schemes once more, but not as dually cor responding schemes next to each other; we will begin with the scheme that has to be followed if one prefers the right side 0; below it, we will state the scheme which is to be followed if one should prefer to begin with the equations on the right (actually, it would be bet ter to say, left) side 1:

(a:/: 0) - ( 0 ~ d ~ 0 = 0 ) (a:/: 1) = ( 0 0 ~ a : t 0 = 0 ) 1)

= (1 ; a ; 1 = 1), = (1 ; d ; 1 = 1).

This fact is of great impor tance and gives an advantage to the algebra of relatives over the identity calculus in which, as we have seen previously (volume 2, pp. 91ff and pp. 180ff), inequalities can never take on the form of equalities, and the distinction of "particular" and "universal" j udgmen t s is final.

As we have proved in volume 2, w 40, the most general proposit ion, mathematical ly "secondary" in Boole's sense, is constructed out of "primary" propositions.

A primary proposit ion has e/ther the form of a subsumption or an equation--which is really the same, because the one form can always be changed into the o the r - -o r the negation thereof, that is, a nonsubsump- tion or a inequation--which, again, can be t ransformed into each other.

In the identity calculus, equations or inequations were considered as stated between classes or systems--but here, in our algebra, they are to be thought as stated between (binary) relatives.

As every equation (and subsumption) can be b rought to 0 or 1--already in the identity ca lcu lus- -and also every inequat ion (and nonsub- sumption) , and since the latter propositions can be rewritten in equations, after preparat ions as per 1), it is clear that inequations and nonsubsumptions can be converted into equations in our algebra without any restrictions.

Without loss of generality, we can from now on assume that the total proposit ion of the data of a problem is built from only (primary) equations by rules from the propositional calculus.

After "executing" the negations which may be prescr ibed in a prop-

260 S C H R O D E R ' S LECTURE V

osition function (and converting inequations which may have been in- t roduced therewith into equations), only (identity calculus) products and sums of such equations come into consideration.

The identity calculus could unify these into one single equation, but it could not further reduce it (except, of course, where all existing letters represent "propositions," that is, were limited to the domain of values 0, 1).

A further advantage of our algebra is based on the following circumstance: that not only products but also sums, that is to say, alternatives of equations can be united in a single equation--likewise, not only sums, but also products of inequations (even when their polynomials, or both sides, represent arbitrary relatives). In addition to the previously ment ioned or general formulas 5) of w 10, and the long known:

(a =0)(b =0) - (a + b =0), (a = 1)(b = 1) = (ab = 1),

(a ~ 0) + (b ~ 0) = ( a + b ~ 0), ( a ~ 1) + ( b ~ 1) = ( a b ~ 1),

2)

this is also based on the following propositions:

Oj-aj-O+Oj-bj-O=Oj-aj .Oj-bj-O, 1 ; a ; l " 1 ; b ; l = l ; a ; 1 ; b ; 1 , 3)

Page 152

(0j-aj-0)(00~ba-0)=00~ab0~0, 1 ; a ; l + l ; / / ; 1 - 1 ; ( a + b ) ; 1 , 4)

which can be easily extended from two to any number of terms, and should be thought of as extended.

By 3), since, because of the commutativity of the identity calculus operations, both the terms which are interspersed between only relative summands 0 and the terms which are interspersed between only relative factors 1 have to be able to be permuted, their order is unimportant.

First, in order to prove these propositions, we have to note that the formulas 3) are nothing more than the application of a general proposition, which reads:

a ; l " 1 ; b = a ; 1 ; b , a ~ O + O ~ b = a ~ O ~ b . 5)

Direct proof of the first formula:

Lo = (a;1)O(1 ; b)o = Zh aih " Zkbkj = Zhkaihbkj = R o,

q.e.d. Because of the associativity of the relative operations, we now have

1 ; a ; l " l ; b ; l = ( 1 ; a ) ; 1 . 1 ; ( b ; 1 ) = ( 1 ; a ) ; 1 ; ( b ; 1 ) - l ; a ; 1 ; b ; 1

and therefore also 3), q.e.d.

The propositions 4), already given by Peirce, are best proved by 4) of w 6, for example, the one on the fight as follows:


L = (1 ;a) ;1 + (1 ;b) ;1 = (1 ; a + 1 ; b ) ; l = { 1 ; ( a + b ) } ; l = R .

By the proposi t ions 1) t h rough 4) above, we now reach ou r goal m e n t i o n e d previously of easily reduc ing the equa t ions (likewise the inequat ions) to the following schemes:

r ( a = 0 ) ( b = 0 ) ( c = 0 ) " " = ( a + b + c + ' " =0)

={1 ; ( a + b + c + " " ) ; 1 =0}

= (dl;~: "" = I )

: ( 0 o'- d / : g . . . o'- 0 : I ) ,

( a = l ) ( b = l ) ( c = l ) " " : ( d + / ~ + ~ ? + " " =0)

=11 ; ( d + / ~ + ~+ . . . ) ;1 =01

= (abc" �9 =1)

k = ( O ~ a b c ' " ~ 0 = 1),

6)

( a ~ 0) + ( b ~ 0) + ( c ~ 0 ) " " = ( 0 # d / ~ ? ' " # 0 = 0 )

= { 1 ; ( a + b + c + "") ;1 =11

(a ~ 1) + (b :/: 1) + (c:/: 1 ) ' " = ( O ~ a b c " ' ~ - O = O )

:11 ; ( d + / ~ + ~+ " " ) ; 1 : 1},

7)

( a = 0 ) + ( b = 0 ) + ( c = 0 ) " " = ( 1 ; a ; 1 ; b ; 1 ; c ; l ' " - 0 )

= (Oct ~o~Oct/~o~Oct ~ t O " = 1)

( a = l ) + ( b = l ) + ( c = l ) ' - " = ( 1 ; , ~ ; 1 ; / ~ ; 1 ; ? ; 1 " " = 0 )

= ( O j - a # O # b j - O # c # O ' " = l ) ,

8)

Page 153

(a r O)(b ~ O)(c r O) "" = (0 o" d o" 0 e/~o'- 0 j- (ct 0 . . . . O)

= ( 1 ; a ; 1 ; b ; 1 ; c ; l ' " = l )

(a ~ 1)(b #: 1)(c r 1) = ( O j - a ~ O j - b e O e c e O = 0 )

= (1 ; d ; 1 ;/~;1 ;~;1 " " = 1).

9)

To prove these schemes, for 6) and 7)-- the latter coming from 6) by con- trapositionmno further remark is necessary.

Scheme 8) is best deduced from 9) by contraposition, and to justify 9) we

have, for example, top:

L = (0o~ dot0 = 0)(0j-/~0~0 = 0) �9 �9 = (0ctd j-0 + 0 j-/~,t0 + " =0) = R

by 1), 6), and 3), q.e.d.

Now, if all alternatives or sums of equat ions in the propos i t ion funct ion which represents the total proposi t ion of the data of a p rob lem, as per

Page 154


scheme 8), can be eliminated--by uniting them in a single e q u a t i o n u w e eventually can only have a product, that is, a "system" of coexisting or simultaneous equations.

This can be completely reduced, according to the propositions of Boole, known for a long time--cf. 6); it is replaced by a single equation which we have called the united equation of the system.

Thus, we have gained the important proposition: in the algebra of binary relatives, every complex of propositions--in particular, the totality of the data of any problemmcan be united in a single equation, in which next to, or except, its one equality sign, other signs of "secondary relation" (such as =, :(=, ~ , etc.) no longer occur. Also the "secondary" propositions (in Boole's sense) can be reduced to one "primary" proposition.

The equation can be begun with the right side arbitrarily 0 or 1; its "polynomial" is then a "function in the sense of our algebra of relatives" of all those relatives to which the subpropositions referred, that is, an expression, which appears as constructed from these relatives and possibly also the modules of our theory by means of its six species.

If the equation is not brought to 0 or 1, the same is true for both sides of the equation: each must be a "function" in the sense of the previous argument.

A "function" so considered is itself a binary relative and may be called f for the time being. We then have an equation in the form of

f=0

as the form for the data of the most general problem conceivable in our theory.

I shall call it, here as well, the "united equation" or "total proposition of the data," and begin with the right side 0, as in the general observations of volume 1, about which we will speak later. If so desired, the equation can always be written as a subsumption, with the predicate 0:

f ~ 0 .

The corresponding dual of what we have just s a i d ~ t h a t we can also achieve anything for the subject 1 with f = 1 or 1 : (=fuwil l not be men- t ioned from now on because it is obvious.

The polynomial f of our united equation can be built from already (elsewhere) fully determined relatives, such as those specified from initially given or, in short, "known" relat ives~to which the four modules of our theory also have to be c o u n t e d ~ o r it can also contain undetermined (or letter-) relatives as terms (operational terms, arguments) .

For the first case, the equation f = 0 is either simply true (correct) or unt rue (false). Then the relative f can simply be "calculated t h r o u g h " ~ b e c a u s e of the unequivocal executability of all operations


p r e s c r i b e d in its e x p r e s s i o n - - ( b y o b t a i n i n g all its coeff icients , possibly

its gene ra l coef f ic ient in o n e s t roke) . If we i n d e e d ob ta in 0 as the value o f f (or o f each of its coeff ic ients) ,

t h e n the e q u a t i o n f = 0 was correct because it will give

0 = 0 .

It is t h e n admiss ible to assume or assert it, even t h o u g h the a s s u m p t i o n

can be cal led e m p t y or w i thou t c o n t e n t [nichtssagende] (self-evident) (its

se l f -evidence or validity may, however, be qui te a r d u o u s to prove) .

If, however , we f ind tha t the value of f is d i f f e ren t f r om 0 (when at

least one coeff ic ient of f is - 1), t hen the e q u a t i o n f = 0 was false a n d

r ema ins inadmiss ib le as a s sumpt ion as well as an asser t ion. It can only

be a d m i t t e d provisional ly as an a s sumpt ion or c o n d i t i o n in o r d e r to

p rove its final re jec t ion "apagogically" by d e d u c i n g absu rd

Page 155 c o n s e q u e n c e s .

We may call the e q u a t i o n f = 0 i t se l f "absurd" or nonsens ica l , a n d keep

it for o u r disc ipl ine ( the theo ry of relatives) as the or iginal , or the

r ep resen ta t ive for all absu rd equa t ions of the s c h e m e

1 = 0

known f rom the ident i ty calculus.

This is indeed true in a triple sense. With the assertion f = 0 , first 1 = 0 is

required for the coefficients o f f different from 0 that were mentioned. Second, because we g o t f ~ 0 when "calculating through," the validity of ( f :# 0) - 1 and

therefore of ( f=0) = 0 is secured. The assertion f--0, that is, ( f=0) = 1, would have to lead to the contradiction 1 =0 (understood as propositional

equivalence).

Third, i f f ~: 0, we can derive immediately from the equation f = 0 the equation 1 = 0 with facility, interpreted as an equation between binary relatives, namely, as an

equation of the absolute modules 1 and 0, by pre- and post-multiplying on both

sides with 1; therefore, from f = 0 to 1 ;f; 1 - 1 ;0 ; l nwh ich , according to 1), will

result in 1 = 0.

In order, for example, to get the form 1 = 0 from the assertion 1'= 0 or from

0 ~ = 0, it suffices to take the relative product with 1 on both sides of the equation.

We can thus transform 1~= 0 or 01= 0 into 1 = 0, just as, conversely, from 1 = 0,

1'= 0 and 0'= 0 would be given because of 1 = 1' + 0'. The equation 1'= 0 (for

example) and 1 = 0 are thus proved to be equivalent, and both have to be called

"absurd."

A problem ( w h e t h e r a so lu t ion p r o b l e m or an e l i m i n a t i o n p r o b l e m )

c a n n o t arise in this "first" case which we have e x a m i n e d so far.

It is d i f f e ren t in the "second" case, whe re u n d e t e r m i n e d relatives in

the exp res s ion of the po lynomia l f in the e q u a t i o n f = 0 occur.


Even by the sheer posing of the equa t ionmby pronounc ing or stating this equation "f= 0," be it as a (conditional or uncondi t ional) assertion, be it to make the equation an assumption for an invest igationwwe impose on the reader, we commit ourselves to the following: to think of or imagine a system of values under the letters which occur as names of u n d e t e r m i n e d relatives in the equation for which the equation is true. Each system of specified relatives which, substituted for those unde t e rmined relative symbols, "satisfies" the equation, "fulfills it," that is, makes it true, is called a system of "roots" of the equation, as is well known, and in as far as we deal with the discovery of a system of roots, those un-

Page 156 de te rmined relatives are also called "unknowns," are designated as "the unknowns." The determinat ion of a system of roots is called "a solution" of the equation, and the determinat ion (sometimes also simply the s tatement) of all possible systems of roots of the equation will be called the general (or complete) solution.

Thus, with the equation itself, the d e m a n d and obligation has been established to "solve" it for the unde te rmined relatives that occur in it as "unknowns"; the equation involves or states a problem.

Thereby two limiting cases can occur: First, the case in which there is no system of roots at all. In this case,

the equat ion posits a d e m a n d which it is impossible to fulfill; the problem remains unsolvable, and the equation is inadmissible (its " roots"wif we can still talk about roots here, where there are none, a l though they have names in the form of x, ...; they have obtained them prematurely, so to speakmwould have to be called "undefined," [undeutig], that is, they are incapable of being interpreted) . In these cases we say: "the resultant of the elimination" of all unknowns from the equation, or, also, any of them, is the "absurd" 0, that is, the equation 1 = 0; we may even call the equat ion f = 0 "absurd," and thus designate it as an "inconsistency" (in the broadest sense of the word).

We may also have the case in which every system of (an equal n u m b e r of) relatives (present as unknowns) is also a system of roots and satisfies the equation f = 0. In this case, we call the equation "universally valid," "analytical, .... self-evident," or also an "identity," or a "formuld' (in any o ther case, we call it "synthetic"); its roots remain undetermined, that is to say, arbitrary. We also say the "resultant" of the el imination of all unknowns (or any of them) from the equat ion is 1, or 0 = 0, or the equat ion f = 0 gives "no resultant." And of the equation f = 0 itself, we say that it is "tautological" [nichtssagend], that it will result in 0 = 0; indeed, it does not yield any knowledge, any information whatsoever, about the unde te rmined relatives which occur in it.

For both limiting cases which lead to the result 1 = 0 or 0 = 0, we can say that with every elimination all the unknowns "fall out of the equat ion

f= 0."

Page 157

Page 158


Also bo th these l imit ing cases, which are of ten no t easily recognizab le

and provable as such (in spite of the na tu re of self-evidence of o n e of

t h e m ) , offer no real solut ion p r o b l e m [Aufl6sung~oblem]. On the o t h e r hand , solut ion p rob lems arise in every o t h e r case; and

we now want to deal with t hem in dep th . Thus , ou r r e m a i n i n g observat ions are based on the c o n d i t i o n that in

the d o m a i n of b inary relatives there is at least o n e system (and possibly

many systems) of values which make the equa t i on true, u n d e r s t o o d with

respect to the "unknowns" x, y, z . . . . , a, b . . . . o ccu r r i ng in the equa t ion ;

bu t the value of these variables may no t always be a s sumed arbi trar i ly

if the equa t ion is to be satisfied, in o t h e r words, if the e q u a t i o n is to

be ne i t he r absurd, no r formal , bu t is r equ i r ed or s u p p o s e d to satisfy a

real "relation" be tween all unknowns .

"Relation between-" is understood here in the broadest sense possible. The

relation can also "split" into mostly simpler and, finally, no longer "splitable"

relations (relations in the narrow sense of the word) between only those un-

knowns which will enter into it. If only one unknown enters into such a "rela-

t ion"mfor example, if we had obtained x =(= 0 ' - - then the expression "relation

between x,y,z . . . . " would seem false and ought to be thought as replaced by

"relation for x." The relation thus stated would then be a unary one. But it would

be too complicated if, in general, we would talk about "relations between the

unknowns and, possibly, for the unknown." Furthermore, the distinction seems

unimportant for our theory because degenerate cases fall under the general

case, at least externally, by means of the total proposition, that is to say, the

systems and alternatives of"split" relations can be expressed formally as a relation

between all unknowns.

If we emphas i ze a m o n g the unknowns any par t icu la r o n e - - l e t us call

it x - - t h e n only one of the two following cases can apply with respec t to the o t h e r unknowns y, z . . . . . a, b . . . . :

E i ther the lat ter could be assumed arbitrarily, whereby the re is a value

or values of x for each system of values that may be a d d e d to it, which

t o g e t h e r form a "system of roots" of the equa t ion f = 0. Or this is no t

true.

In the first (sub-)case, we say that the e l imina t ion of the u n k n o w n x

f rom the equa t i on f---0 gives "no" resul tant , or "the resu l tan t of this

e l imina t ion" is the equa t i on 0 = 0; f rom the e q u a t i o n all u n k n o w n s 'fell out" t o g e t h e r with x, and there is no re la t ion be tween the r e m a i n i n g

unknowns , these be ing unrestricted genera l relatives ("parameters"). In

fact, we may then a t t r ibute to the lat ter an arbi t rary value system, and it will only ma t t e r to obta in the app rop r i a t e values of x which t o g e t h e r

with it fo rm a system of roots, in o t h e r words, to express the u n k n o w n x in terms of the o t h e r unknowns . I f this case can be proved to be correct, we have a "pure" solut ion p rob lem, the p r o b l e m of the so lu t ion of equa-


tion f = 0 of the one u n k n o w n x; then we have ident i f ied this equa t i on

as an "unconditionally" solvable one , and we can p r o c e e d to its solut ion.

For the second (sub-)case there are some value systems (at least one)

to which may not be given to the unknowns y, z . . . . . a, b . . . . because the re

is no value of x t oge the r with which they could r ep r e sen t a system of

roots. Each propos i t ion which truly states that a system of values

y, z . . . . . a, b . . . . is inadmissible, "excludes it," can be cons ide r ed a "rela-

t ion" be tween these o the r unknowns , which, if we so desire, can again

be expressed in the form of an equa t ion , and it may be called "a resultant of the e l imina t ion of x f rom the equa t ion f = 0" because it d e p e n d s on

the equa t i on f = 0 (necessary for its fulf i l lment) , but the n a m e of the

u n k n o w n x does no t appea r in it.

T h e un i t ed equa t ion , total p ropos i t ion of all resul tants (of e l imina t ion

of x f rom f = 0), mus t not only be a necessary but also a sufficient condit ion for the solut ion of the equa t ion f = 0 for the u n k n o w n x; we call it "the" complete or "full resul tant" of the aforesaid e l iminat ion . It excepts all

inadmiss ible systems of values of the o t h e r unknowns y, z . . . . . a, b . . . . .

Each system of values of this u n k n o w n that is sufficient is an "admissible"

one , w h i c h yields, t oge the r with some values of x, a system of roots of

the equa t i on f = 0; it can also be charac te r ized as a re la t ion be tween

the u n k n o w n s wi thout x, which d e p e n d s on the equa t ion f = 0, the

satisfaction of which guaran tees the solubility "for x" of the equa t i on

f = 0, that is, it gua ran tees the exis tence of at least one roo t value x

which satisfies this equat ion; it states the "validity condi t ion" for x.

Since we can only deal with the solut ion of the equa t i on f = 0 for the

Page 159 u n k n o w n x in the cases in which the equa t ion is solvable, in which there

are values of x that satisfy it, the solut ion for x has to be preceded by the

d e t e r m i n a t i o n and the fu~llment of its (full) resultant (of the e l imina t ion

of x).

T h e f o r m e r has to be called an elimination problem. T h e latter, i.e., the p r o b l e m of first satisfying ou r resultants , is again

a solut ion p r o b l e m which, however, has (at least) one u n k n o w n (x) less.

O u r original solut ion p r o b l e m has changed , and in its place we now

have a s impler one. For the latter, the same pr inciples apply as those

es tabl ished or those that we will establish for the original p rob lem.

We will try to determine the resultant, for example, by a suitable determination

of any one of the remaining unknowns (which did not fall out together with x

in the elimination of x from f=0 ) , by seeking to represent it as a function of

the others. Thereby, we may again obtain a resultant from its elimination, which then has to be dealt with in the same way as the others. Etc.

The result which we have ob t a ined in the second subcase can now

be f o r m u l a t e d as follows and, at the same time, can be e x t e n d e d to the

Page 160


first subcase, as well as to the two l imit ing cases previously discussed;

thus we can call it ent i rely general : In the algebra of relatives, every solution problem is inseparably connected with

the elimination problem (just as in the ident i ty calculus); the so lu t ion of an equa t i on for one (or for a system of) unknowns c a n n o t reasonably be b e g u n before we have ob t a ined the full resu l tan t of the e l imina t ion

of this u n k n o w n , which, in turn, is solved in terms of all o t h e r (or the

the re in occur r ing) unknowns ; it requires , first, the c o m p l e t i o n of this

e l imina t ion as a fo rego ing or preliminary task. The results of the o t h e r cases are s u b s u m e d u n d e r this one , insofar

as the p r o o f of the absence of a resul tant can be es tabl i shed a n d was

establ ished, to get a resul tant 0 - 0 ( that is, to achieve the p r o o f that

"the resul tant" leads only to 0 = 0), and, further , the p r o o f of the im- possibility of the solut ion, or the absurdi ty of the e q u a t i o n f = 0 was r e g a r d e d as a p r o o f that the resul tant leads to 1 - 0 .

The problems mentioned before, which we occasionally characterized as dif-

ficult problems, will have to be dealt with under elimination problems.

If we assume, for the t ime being, that the set of u n d e t e r m i n e d relatives

or "unknowns" which occur in the equa t ion f = 0 is limited (its n u m b e r be ing "finite"), t hen these two main p rob lems remain:

First, to eliminate one u n k n o w n f rom one equa t ion .

Second, "to calculate" an u n k n o w n f rom the equa t ion , tha t is, to solve the genera l equa t ion for it, if "the" resul tant of the e l imina t ion is

fulfilled. Let us suppose that we can hand le bo th these p rob lems , the p r o b l e m

of the e l imina t ion of one u n k n o w n and the solut ion for one u n k n o w n in every case; then we can satisfy every r e q u i r e m e n t f = 0, which is no t

absurd, in general : We e l imina te one u n k n o w n after the o t h e r in any sequence , unti l

they have all "fallen out" and we have r eached the resu l tan t 0 = 0. This

will h a p p e n , at the latest, with the e l imina t ion of the last unknown .

Along with the elimination of a particular unknown, it is possible that several

other unknowns (which we did not intend to eliminate) will also fall outmas

we saw in the two limiting cases, where all fell out. "The" resultant of the elim-

ination of an x does certainly not "mention" or "contain" this eliminant as a

term, operational term, or argument; but there can be other unknowns unre-

presented in it, or missing in it, which were represented in the equation with

which we began the elimination.

We thus get a series of resultants of which surely one , and p e r h a p s even

more , conta ins fewer unknowns than its predecessor . The or iginal equa-

t ion f = 0 itself may be charac te r ized as "zer0th resul tant" whereas the ident i ty 0 = 0 is to be seen, as we have already said, as its "last."

The satisfaction of any R' of these resul tants is a necessary and suf-

Page 161


ficient cond i t ion for the solvability of any immedia te ly p r e c e d i n g R for any one of the unknowns which fell ou t with the e l imina t ion ; that is, which are no longe r in R', bu t could be in R. Whereas the o t h e r "supe r f luous" unknowns can be assumed arbitrarily, we n e e d for the satisfaction of R, as soon as R' is satisfied, to solve R only for one of them, in o t h e r words, to express one of these supe r f luous u n k n o w n s in terms o f the o thers (and already d e t e r m i n e d f rom R'), the first o f which will t hen r ema in u n d e t e r m i n e d .

We now have the rule: to satisfy the resultants in o r d e r [Reihenfo lge] ,

in the reverse o r d e r f rom that in which they were ob t a ine d t h r o u g h successive e l iminat ion.

We thus in the first place satisfy the second but last resu l tan t by means of its solut ion for any one of the r e m a i n i n g u n k n o w n s in it, leaving u n d e t e r m i n e d the others , and o t h e r u n k n o w n s fallen ou t f rom its predecessors; it has to be solved uncondi t ional ly , because the last resu l tan t 0 - 0 is certainly satisfied. We then substi tute the system of values o f roots, thus ga ined for the u n k n o w n s u n d e r cons idera t ion , into all previous resultants (to and inc lud ing the equa t ion f = 0 ) , and then deal likewise with the previous resultant , and so on, until we have solved the e q u a t i o n f = 0 for the first u n k n o w n e l iminated .

This observat ion seems to justify the fact that we will occupy ourselves f rom now on only With the apparen t ly very m u c h m o r e special ized problem of e l imina t ion and solut ion which has only one relative as its eli- m i n a n t or unknown.

Finally, a few remarks. Examples will justify the theory. Please excuse the repetition, in principle, of several of the observations in

volume 1, w 22, which we made or hinted at, analogous to the identity calculus and its fewer consequences.

As we can see, there is no basic difference between the relatives thought to be generally "given" and those "sought." Also, those which are given as "parameters" of the problem (for example, polynomial coefficients a, b, c . . . . . etc.), if they are not "specified" (i.e., as special relatives), have to be considered as "un-

knowns" to begin with and have to be dealt with jus t as the x, y . . . . . .

The astute reader will easily understand why our discipline differs in this

respect from arithmetical analysis.

Page 162


w 12. General and Rigorous Solutions

Let

269

F(x) = 0 1)

be an e q u a t i o n to be solved for an u n k n o w n x, which is "solvable," tha t

is, has at least o n e roo t x.

If other undetermined relatives occur in the equation besides x, and if the

equation involves a relation between the latter (which we call "the resultant" of

the elimination of x from it), we assume that this relation would be satisfied via

suitable determination of the other letter relatives which has turned our equation

into one that can be solved. Equation 1) will be considered as unconditionally solvable; its solvability for x must not be tied to the condition of its satisfaction

of a resultant (free of x), or the elimination of x from it may not give us any

resultant.

We u n d e r s t a n d by the "comple te" so lu t ion of e q u a t i o n 1) for x (p.

156) the specif icat ion of all relatives which, w h e n subs t i t u t ed for x, satisfy

the e q u a t i o n a c c o r d i n g to the laws of relative a lgebra , s epa ra te f rom all relatives which do no t satisfy it.

T h o s e r e l a t i ve s - - t he "roots" of the e q u a t i o n n c a n be co l l ec ted in

t heo ry as well as in pract ice into o n e united expression which inc ludes all

o f t hem, bu t only t hem, a n d is thus cal led the "general root (or solution)" of the equa t ion .

We now aim to establish some f u n d a m e n t a l p ropos i t i ons a b o u t t hem,

which impress u p o n o u r whole discipl ine a c h a r a c t e r of its own.

First, I claim: The general root of equation I) can always be given in the form

x = f(u), 2)

where u represents an undetermined relative which we call arbitrary if the

u n k n o w n x has no o t h e r d e t e r m i n a t i o n s than to satisfy e q u a t i o n 1),

where , fur ther , f signifies a some "function in the sense of the a lgebra

o f b inary relatives."

This function f iswlet us say this right away--more or less determined by the

given F, which forms the polynomial of the equation to be solved; to express it

more succinctly: it is, in general, "not fully" or "only incompletely" determined,

that is to say, we can choose in an infinite universe of discourse from infinitely

many functions f, which do not only "formally" appear different, according to

the outer arrangement of the expression, but are "essentially" different because

they often "yield," that is, represent by their functional values, very different roots

x of equation 1) for the same value of u. We can speak of an expression for the

general root - -or "the" general solution---of equation 1) in many different ways, and only the totality of all meanings off(u), this function being formed, can be

27 ~ SCHRODER'S LECTURE V

considered for all possible values of u; it will be the same in all cases, it will be P a g e 163 coincide with the class of all roots x, the totality of all roots x, which equation

1) allows.

Second, I claim: that every general solution f(u) of equation 1) is sufficiently characterized by the propositional equivalence

{ F(x) = 0 } = I2 { x = f(u) }, 3) 1l

where the sum on the right has to ex tend over all possible relatives u within .12.

And, third, I claim: that we can impose certain o the r r equ i rements , which I call "adventivg' because they are not already inc luded in the not ion of a general solution, on a funct ion f which, accord ing to 3) (and also, fu r the rmore , according to 2)), is capable of r ep resen t ing "a genera l so lu t i on"mtha t is, all roots exclusively--of equa t ion 1), and is d e t e r m i n e d by it. In particular, we can state in theory, as well in practice, the general solution f always in such a form that it satisfies the following "(first) adventive" requirement

IF(x) = 0} = If(x) = x}, 4)

which for practical purposes is to be prefer red , forces itself upon us as imminen t ly useful.

Page 164

We want to begin the p roof of our claims by showing that the equiv- a lence 3) expresses the not ion o f f ( u ) as the genera l root of equa t ion 1).

If an expression 2) is to r ep resen t this general root, it must possess two propert ies .

First, it has to yield for every value of u a correct root x of our equa t ion F(x) = 0, such that

Flf(u)} =0 5)

is an identity; in o ther words, this equa t ion is valid as genera l fo rmula for any arbitrarily composed relative u. This means that our express ion 2) may yield only roots f rom our equa t ion 1).

T h e p roo f that this is indeed true for a defini te funct ion f(u) will be called "Proof I"; this funct ion represents the general solution of equa t ion 1).

More complete ly than by 5), this r e q u i r e m e n t will be expressed by

II [{x =f(u)} =(= IF(x) = 0} ], 6) u

which means: for every u, if the value o f f (u ) is x, then F(x) = O.

If we use the designation f(u) consistently in 6) for x, introduced by the


condition, the hypothesis, the conditional statement of the propositional sub- sumpfion in brackets in 6), then the condition is satisfied as an identity, and receives the propositional value {f(u) =f(u)} = 1. By the "specific principle" of the propositional calculus (1 :~-A)=A, as we have called it in volume 2, the assertion, thesis, corollary of this propositional subsumption will be now clearly satisfied, that is, formula 6) is reduced to

II[Flf(u)} =0], 7) u

which is nothing other than formula 3), written in a more expressive waymthe expression that is connected with the above, auxiliary proposition.

Conversely, 6) follows from 7), on other hand, if we introduce x for flu), so that both propositions 6) and 7) [i.e., also 5), understood as a universally valid formula] are to be considered as equivalent and equipollent.

But because the p red ica te of the p ropos i t iona l s u b s u m p t i o n in the brackets [ ] in 6) is i n d e p e n d e n t of, cons tan t in view of the p r o d u c t variable u, 6) can be rewri t ten equipol len t ly acco rd ing to p ropos i t ions

a l ready known (namely, acco rd ing to Th. 3+)) as

E {x =f(u)] ~ IF(x) = 01. 8) u

If this requirement is satisfied alone, without the one we are about to mention, we then say: x =f(u) represents a "particular" solution of the equation F(x) =0, even when this solution is still of great generality and may yield infinitely many roots.

Second , ou r express ion f(u) also has to yield every roo t x o f ou r equa- t ion 1); that is, if x represen ts any one given relative in such a way that it satisfies the equa t ion F(x) = 0, then there also has to be a relative u for which ou r f(u) will be equal to this x.

This r e q u i r e m e n t is correct ly expressed by

IF(x) = 0} :(= E If(u) = x}. 9)

T h e proof , for a def ini te func t ion flu), that it satisfies this r e q u i r e m e n t 9), will be called "Proof 2"; this f(u) represen ts the gene ra l so lu t ion of equa t i on 1).

T h e two r e q u i r e m e n t s 8) and 9) tell us that the express ion f(u) Page 165 "yields," or comprises , only roots and also every roo t o f e q u a t i o n 1); they

are now necessary and sufficient condi t ions , so that we can call f(u) the genera l solut ion of equa t ion 1); they charac ter ize f(u) as " the genera l

root" o f 1). T h e two p ropos i t iona l subsumpt ions 8) and 9) can be pu l led t o g e t h e r

as forward- and backward-equ ipo l l en t to the p ropos i t iona l equation 3),

272 SCHRI~DER'S LECTURE V

which formula tes n o t h i n g m o r e than the notion of 2) as the "genera l solut ion" of 1), as was c la imed ( u n d e r the "second" claim), q.e.d.

Page 166

To c o n t i n u e with the just i f icat ion of ou r claims, the main task is now

the following:

We assume equa t ion 1) to be solvable; it has thus at least one root.

Let the relative a be one such root; then we h a v e m n o t j u s t as an equa t i on which has to be fulfilled, but one which is t r u e m

F(a) =0 10)

- - w h e r e a s for an arbitrari ly chosen x, we have in genera l F(x) :/: O, and the equa t i on F(x) = 0 has to be cons ide red no t as satisfied, bu t r a the r

as "a formula" which has to be satisfied by app rop r i a t e d e t e r m i n a t i o n of x.

If we now form the express ion

f ( u ) = a " 1; F(u) ;1 + u " {O ~ F(u) ~ O}, 11)

then , indeed ,

x =f(u)

mus t be a form of the genera l solut ion of 1), satisfying r e q u i r e m e n t 3). Proof. Cons ide r ing that, accord ing to 1) of w 11, the relative

1 if F(u) :/: O, 1 ; F ( u ) ; 1 - 0 i fF (u ) =0,

and, conversely, the relative

0 if F(u) :/: O, O ~ F(u) o ~ 0 - 1 i fF (u ) =0,

we see immedia te ly that we have f ( u ) = a" 1 + u" 0 = a, when u is not

a roo t of the equa t ion F(x) = 0, but, on the o t h e r hand , we get f ( u ) =

a �9 0 + u �9 1 = u, when u satisfies the equa t ion F(u) = 0, and can now be a s sumed as a root x of it.

In particular, for the assumpt ion u = a, bo th of these results coincide.

If we assume that the relative u is chosen arbitrarily, then it is e i the r

no t a roo t of equa t ion 1) or it is one: u--- x. In bo th casesmas we have

jus t s e e n m f ( u ) yields a root; in the fo rmer case, it is always the one we

a l ready know, root a, exist ing by assumpt ion; in the lat ter case, it is the

luckily guessed root x. In either case, the above express ion f ( u ) yields only

roots of the equa t ion F(x) = 0, and it yields all roots of this equa t i on

because it already p roduces any des i red x of these roots by the as- s u m p t i o n u -- x.

O u r f ( u ) is thus not only sufficient for the r e q u i r e m e n t s which are

Page 167


inc luded in the not ion of the genera l root of 1), but also for the first adventive r e q u i r e m e n t 4)---cf. p. 171.

If we e x p r e s s - - t o facilitate p r i n t i n g - - t h e nega t ion of F(u) m o r e convenient ly by F(u), instead of by F(u), then we have the t heo rem:

{F(a) = 0} :(= ({F(x) = 0} = ~ [x = a" 1 ;F(u) ; 1 + u{0 ~ F(u) j- 0}]) 12) u

- -wh ich , since it is valid for every a, may be p r e c e d e d by the symbol II.

a

In explanation of this: if a is not a root, and thus {F(a) =0}- 0, then 12) is of course valid as a propositional subsumption of the form 0 :~-R, although it is meaningless [nightssagend]. The condition that the equation F(x) --0 be solvable can be expressed in the form

IF(a) = 0} = 1 a

and guarantees that there is some a for which the premiss of our theorem satisfies {F(a) ---0}, is = 1. For each such a, the right side R of theorem 12) must be valid because of (1 :(=R)= (R= 1)= R; and this expresses correctly, according to

scheme 3), that the expression 11) given above for f(u) is the general root, which we proved a short while ago.

We may fu r the r r emark that the genera l solut ion x =f(u) , which we found by 11) and gave in 12), corresponds to itself according to the principle of duality (by cont rapos i t ion) .

Namely, the expression of our x =f(u) formed by dual correspondence from 11) would be

x = (a + 0 ~ F j- 0)(u + 1 �9 F; 1),

which, if we multiply it out, results in x= au + f(u), and where, because f(u) is either = a or = u, the term au is absorbed and only x--f(u) is reproduced.

Thus, each solut ion i s - -a f te r a is once chosen for the known root or par t icular solut ion of 1 ) - - c o m p l e t e l y d e t e r m i n e d in its form. The expression is only d e p e n d e n t on the choice of a.

It will soon be clear why we call it "the rigorous solution" (be long ing to root a) of equa t ion F(x) = O.

The "rigorous" solut ion is one of the forms of the "general" solution.

The p roo f of the exis tence of the genera l form of the solut ion (by establ ishing it explicitly) now also yields the missing a r g u m e n t s for what

we c la imed u n d e r the "first" and "third" c l a ims - - a t least as far as we

in tended : that it be "theoretically" possible that the genera l root x of

1) can only be r ep re sen t ed in the form of 2), where u is arbi t rary and f also fulfills the adventive r e q u i r e m e n t 4).

But there is more . We may be sure of the exis tence of a genera l

solut ion of such a character , not only in theory, but also we can in


prac t i ce es tabl ish i t - - a t least as a " r igorous" so lu t ion . T h e p r o b l e m o f a c o m p l e t e so lu t ion of a solvable e q u a t i o n 1) for o n e u n k n o w n is reduced to the discovery of one single particular solution, or very special r oo t a, of this equation.

A n d we can always d iscover o n e such r o o t - - t h u s especial ly in all p rob-

lems with which o u r t h eo ry has to deal . In gene ra l , I have to fo rgo

p r o v i n g these claims he re , which r e g a r d "pract ice ." But I wou ld like at

least to give s o m e hints.

Often it suffices, as an experiment, to substitute the four modules for x in

F(x) as trial solutions of the roots in question in order to recognize one or more

of them as real solutions or roots. If, for example, we are dealing with the solution

of the equation x; x--x for x, we immediately have 0, 1, and 1' as particular

solutions. Likewise, 1' is known a priori as a root of that equation which has a relative

x to be defined as an invertible [gegenseitig eindeutige] mapping. In other cases, there occur parameter values or certain simple functional

expressions built out of them, which are easily recognizable as particular solu-

tions. Thus, if we had to solve the equation a;x--x;a, we would immediately

have, in addition to 0 and 1', the particular solution x = a.

The resultant (of the elimination of x), mostly (previous to the solution for

x) to be satisfied generally, apart from all other unknowns, gives these remarks

Page 168 still more weight.

If, for example, we ask for the solution of the equation x; b = a, the resultant

will require (see the next but one lecture) that a itself is of the form c; b, and

thus we know a particular solution x = c (= a,r And more of this kind.

A l t h o u g h we re joice in hav ing a c q u i r e d the very gene ra l l y o b t a i n e d

so lu t ion 12), this joy is cons ide rab ly t o n e d down, we even t u rn quie t ,

w h e n we look at it m o r e closely a n d l ea rn m o r e a b o u t the n a t u r e o f

such " r igorous" solut ions.

It does n o t g u a r a n t e e an i m m e d i a t e a b u n d a n c e of d e s i r e d pa r t i cu l a r

so lu t ions o r roots for a series of arbi t rar i ly a s s u m e d values o f its indef-

ini te a r g u m e n t s u, b u t - - i f . w e are n o t lucky e n o u g h to have hi t a r oo t

o f e q u a t i o n 1) with the a s s u m e d v a l u e m i t only po in t s aga in a n d aga in

to the long known, a n d t h e r e f o r e u n i n t e r e s t i n g - - n o t to say "bor-

i n g " m r o o t a. T h e des i re to d iscover all roots o f e q u a t i o n 1) with the

h e l p o f exp re s s i o n 11) for the r igo rous so lu t ion really leads us to prove for all poss ible relatives u w h e t h e r they may be satisfied for the e q u a t i o n .

T h e " r igorous" so lu t ion is t h e r e f o r e no t yet a satisfactory f o r m of the g e n e r a l so lu t ion; it solves the p r o b l e m only in an e m e r g e n c y - - / ~ la ri-

g u e u r ~ a n d that is why I gave it its n a m e because it was necessa ry to

d i s t ingu i sh it f rom o t h e r m o r e p r o m i s i n g fo rms of the g e n e r a l so lu t ion .

But we got s o m e hints a b o u t the fo rm in which we have to f ind the

g e n e r a l so lu t ion of an eq u a t i o n ; it also t augh t us tha t the c o m p l e t e

Page 169


s o l u t i o n of e q u a t i o n 1) F(x) = 0 exists in the f o r m 2) x = f (u ) . A n d this

r e m a i n s a last r e so r t if we c a n n o t f ind a "be t t e r " f o r m of t he g e n e r a l

r o o t for a g iven e q u a t i o n , for all cases w h e n we n e e d an e x p r e s s i o n for

this r o o t to c o n t i n u e the e x a m i n a t i o n .

It may n o t be easy to es tabl ish a n o t i o n o f wha t cons t i t u t e s "a satis-

factory" f o r m a n d of wha t may even tua l ly be ca l l ed " the best" f o r m of

a g e n e r a l so lu t ion ; m o s t likely, it will on ly e m e r g e g r a d u a l l y o u t o f the

p rac t i ce o f o u r sc ience.

At least, we can, in gene ra l , jus t i fy o r motivate "the first adventive requirement" for the g e n e r a l so lu t ion .

In ana logy with the exerc i se a l r eady avai lable f r o m a r i t h m e t i c a l anal-

ysis, we can call the " i n d e t e r m i n a t e a r g u m e n t " u o f the g e n e r a l so lu t i on

f(u) in 2) its ( i n d e p e n d e n t ) " p a r a m e t e r . "

However, we have thereby created a double meaning; the concept and ex-

pression are not to be confused with the one of the same name on page 158,

in which we talked about the "parameters" of equation 1) F(x)=0 or of its

polynomial F(x). The common characteristic of both kinds of parameters i s u t o

our just i f icat ion-- their unrestricted arbitrariness.

We have already emphasized that we can only consider u as an arbitrary relative

when the unknown x is de termined by the requirement to satisfy the equation

F(x) = 0, but that, of course, if there are other specifications available concerning

x, or if x is even fully determined, the previously absolute indeterminateness of

the parameter u will be subject to certain restrictions, and that it can even turn

out to be absolutely determined in individual cases; it can, for example, happen

that u =0 has to be taken to yield the root x- -0mas we have already learned

from the identity calculus and as many examples have shown. Since we have to

deal with the solution of an equation for an unknown, and not for a known, no

harm is done when we present the parameter u in general and irrespective of

the possibility just ment ioned as an undetermined parameter, no less when we

present it as arbitrary--as "the arbitrary parameter" of the solution.

A l t h o u g h the t heo re t i c a l r e q u i r e m e n t s on f(u), which a re i n c l u d e d

in n o t i o n o f the g e n e r a l so lu t ion , are e x h a u s t e d with 3), t h e r e is still

o n e r e q u i r e m e n t - - 4 ) - - i n r e l a t i on to this p a r a m e t e r u, which is re-

q u i r e d in practice. H o w can we know this u o r such a u, which yields a de f in i t e , p e r h a p s

a l r e ady known , a given or desired r o o t x?

Systematical ly, o n e such u c o u l d be g a i n e d by the s o l u t i o n o f e q u a t i o n

2)

f ( u ) = X

for the u n k n o w n u. But this so lu t i on p r o b l e m may n o t s e l d o m be m u c h

m o r e diff icul t t h a n the o n e u n d e r 1), the so lu t i on o f which was ex-

p r e s s e d in e q u a t i o n 2).

Page 170

Page 171


It is an unde r s t andab le wish to know for each unknown x that satisfies the equa t ion F(x) = 0 immedia te ly one u--at l eas t - -which would yield, if subst i tuted in f(u), this x. This d e m a n d canno t be satisfied in a s impler

and be t te r w a y m n o t even m n e m o n i c a l l y m t h a n when the genera l solut ion is so a r r anged that it itself yields this x for each u = x.

To posit such a r e q u i r e m e n t for a genera l solut ion is also just i f ied f rom a second and a third perspective.

The second is the check or p r o o f of the correctness of a solut ion that has been found (or also a root) for equa t ion 1). The "general solut ion" should also take care of this check for each special or par t icular value of the genera l root; it should spare us the p roo f by subst i tut ing the x,

found as x =f(u) , in the polynomial F(x) of the equa t ion to be solved,

and offer certain guaran tees of its correctness in itself; it should also show the following.

In o r d e r to get various or even all roots f rom 1), we have to assume,

as to be subst i tuted or as be ing subst i tuted in the express ion f (u) , others ,

in pr inciple all conceivable relatives for the u n d e t e r m i n e d p a r a m e t e r

u Wi thou t cont rad ic t ing the not ion 3) of the genera l solut ion of 1), f(u) can be so const i tu ted that, when we substi tute a root x~ for u, any

o t h e r root x 2 =f(x~) results; and if we substi tute x 2, again a d i f fe ren t root , x 3 - f (x2) , will come out, and so on. W h e t h e r a value taken for u is pe rhaps not a root of equa t ion F(x) = 0 canno t be seen in this case

wi thout mak ing a direct proof: subst i tut ing it into the polynomial F(x) of our equa t ion 1) to check whe the r it will vanish. O n e of the major goals of the genera l solution, namely to save us once and for all f rom giving this proof, thus evaporates into thin air. We migh t as well re- n o u n c e the genera l solut ion and be con ten t with separa t ing the relatives empirical ly into two classes by processing them individually into those u for which F(u) ~ 0 will result, and into those u that will be called x for which we obtain F(u) = O.

A parable will illustrate the point . Transpor ta t ion by railway would

not help us if the train passed the des i red station wi thout s topping or if the stations were not identified.

If we have correctly assumed or guessed a root x, which may be of par t icu lar interest to us, perhaps ob ta ined it f rom reflect ions of still

dubious value, then the genera l solut ion has to tell us that this is the cor rec t root.

We have now reached the desi red final des t inat ion, but we have to

know that we are already there; the train may not con t inue to a n o t h e r root.

In addi t ion to the pr imary or min imal r equ i r emen t s of the genera l solut ion, which are inc luded in the not ion of a genera l solut ion, we have a secondary or adventive r e q u i r e m e n t because of two reasons which

have already been men t ioned : that t he genera l solut ion thus yields every

Page 172


root that was happily guessed; and that its expression, when it is sub-

s t i tuted for the u in f (u ) , gives us this root itself, reproduces it. This means f (u) has to be so const i tuted, if it may be called a satis-

factory genera l solution, that

{F(x) =0} :(= {f(x) = x}. 13)

And, moreover , since the converse propos i t ional s u b s u m p t i o n - - a s

c la imed for u = x accord ing to 3 )mis valid, we may also give this subsumpt ion the form of equation 4).

This addi t ional or adventive r e q u i r e m e n t 4) of the genera l solut ion

suffices, in turn, if, for some funct ion, f is satisfied, bu t not yet to justify this funct ion as suitable to r ep re sen t the genera l solut ion of 1); it only

guaran tees that the funct ion f (u) encompasses all roots x and leaves open w h e t h e r it can assume or yield others as root values; it only tells

us that "Proof 2" has to be correct .

On the o t h e r hand, we have seen that 4) is no t at all the logical c o n s e q u e n c e of 3 ) ~ a s we will later prove rigorously, as if it were

necessary!

To character ize a "not initially unsatisfiable" genera l solut ion, we ough t to write the p roduc t of proposi t ions 3) and 4) or the i r nega t ion

as double equation

{F(x) = 0} = E Ix =f(u)} = {f(x) = x}. 14) u

For all solutions of special p rob lems which we st ipulate f rom now on,

we will have to be careful that these adventive r e q u i r e m e n t s are satisfied, and our specification of solutions should indeed satisfy t h e m (if not o therwise r emarked ) and claim t hem to be satisfied. It would be too t roub le some to accoun t for this fact with the explicit a d d e n d u m of f (x) = x---especially where f (u) has a compl ica ted e x p r e s s i o n m a n d we shall thus l imit ourselves to express ing the solut ion in the form of 3),

r e m e m b e r i n g that u = x is always an admissible value for u, able to yield the root x.

The beginner may think it strange that, disregarding the equivalence between

the two propositions E {x =f(u)} and f(x) = x, as 14) has established, the former u

suffices for the characterization off(u) as the general root of 1), but not the

latter. For the assumed f, which suffices for requirement 3) including the adventive requirement 4), both propositions indeed are valid when x is a root of

equation 1); both are not valid if x represents another relative (no root). They are equivalent. However, this does not make them equipollent. The propositions,

that 2 x 2 - 4 and that matter is indestructible, are also equivalent. This does

not mean that the first proposition can be used as a char~icterization or definition of matter, although perhaps the second one can.

The "rigorous" solut ion has shown that we can always satisfy the ad-


vent ive r e q u i r e m e n t 4) as well as 3) with the c o n s t r u c t i o n o f a su i table f u n c t i o n f (u) .

But we can satisfy the r e q u i r e m e n t 3 ) - - e v e n b o t h r e q u i r e m e n t s to-

g e t h e r m i n infinitely many ways. A n d this c i r c u m s t a n c e po in t s to a th i rd

pe rspec t ive , which mot iva tes the use of the advent ive r e q u i r e m e n t : tha t

we have to a im at f l a m i n g o u r p r o b l e m as a m o r e de f in i t e one , to m a k e

the n o t i o n o f "one" g en e ra l so lu t ion prec ise in such a way tha t we can

speak o f " the" gene ra l so lu t ion o f a specific p r o b l e m , o r o f "its" so lu t ion in an u n c h a n g i n g way.

Al ready in the ident i ty calculus we cou ld ind ica te the func t i ons tha t

a re su i table for all values. For e x a m p l e , cu + ~2 is o n e such func t ion ,

s i tua t ed s o m e w h e r e b e t w e e n c~ = 0 a n d c + ~ = l m s e e v o l u m e 1, p. 427.

If we take c to be i n d e t e r m i n a t e , we have an inf in i te set.

This m e a n s tha t we have to a d m i t tha t t he r e are also func t i ons in the

a l g e b r a o f relatives, which, w h e n d e s i g n a t e d ~(u) , a re c apab l e o f t ak ing

on the value o f any des i r ed relat ive b e t w e e n 0 a n d 1 ( inclus ive) , ac-

c o r d i n g to the value which we give to u. I n d e e d , t he r e is an inf in i te

n u m b e r o f func t ions ~(u) with this p roper ty : th(u) - ~ o r ~ o r u are f u r t h e r m o r e ( the s imples t ) e x a m p l e s of t h e m . . .

If ~(u) is thus sui table for all values ("varying w i t h o u t l imits") , tha t is

to say, x =f (u ) is a g e n e ra l so lu t ion o f 1) in the earl ier , " b r o a d e r " sense (only l imi ted by 3), t h e n obviously

x =j~O(u)} 15)

Page 173 b e c o m e s aga in "a g en e ra l so lu t ion" in this b r o a d e r sense.

For if a determinate value v of u yields a determinate root x with x - f (u), then f{O(u)} will yield this same root x when we take u to be such that we have

~(u) = v~which is always possible with the assumption that was made concerning ~; it goes without saying that each u in 15) gives us a correct root, because each w does this with f ( w ) . ~

Just as f(u) represents the general root of 1) correctly, so does, for example,

f(bu+[n2), f(~i), f((,), f(u), f(cu+ i,i), f(d~+ {tu).

If, furthermore, f(u) satisfies the adventive condition 4), then f(~), if desig-

nated by ~(u), will not satisfy it; less still ~F(x) = x, but certainly only 'F(~) = x

~ana logous ly 'F(~) = x only if we interpret ~F(u) as f (d) , etc. If fully exemplified

by the special function f, such considerations help us to obtain the (above-

ment ioned) rigorous proof that 4) cannot follow from 3), because we are now

in a position to specify funct ionsfwhich satisfy the requirement 3) without 4 ) . ~

Already from the first of the above examples, namely, the expression f(bu + /~7i), we obtain, with an infinite universe of discourse and by varying b, infinitely

many functions f(u) as correct general roots of 1), corresponding to the notions 3).

Page 174


We have thus proved that there exists in general an infinite number of functions f(u) which, according to 3), are able to represent the general root x of 1).

The indeterminateness of the notion of the general solution of 1) is somewhat control led by the adventive requ i rement 4 ) - -by which in fact the solutions, men t ioned as examples above, at least in general , are all excluded. They remain correct, but are impractical, if not almost useless forms of a general solution, and accordingly are to be rejected.

Tha t this requ i rement 4), with the adjunct 3) that is included the notion of a general solution, is not yet sufficient to de te rmine completely a function f(u) as general root of 1), and that there may be many different functions f(u) which satisfy the conditions of 3) and 4) and yet are essentially different, will be seen in special solution problems in the theory . - -

For the application of formula 12) it is useful to observe that we have to establish rigorous solutions to special problems, the expression of which can be considerably simplified in the cases where a = 0 or a = 1 should be a root of the equation F(x) = 0 to be solved.

We can easily obtain both subcases of the general proposition:

{F(0) =O}~({F(x) =O}=~[x=ulOj.-F(u) j.O}]), 16) u

{F(1) = 0}=~- [{F(x) = 0 } = E { x = u + 1 ;F(u);1]]. u

17)

So much for the solution problem in general. The elimination problem is, without fail, associated with it and culmi-

nates in the requ i rement to eliminate from every equat ion 1) or subsumption F(x) =~-0 one relative x. If we are able to el iminate any desired relative, then we are able to eliminate several relatives in any sequence and thus also any one system of re la t ivesmsimultaneous in their ef- f ec t - - ( a t least, certainly with a finite n u m b e r of eliminants).

If we change the notation a little, it means to learn how to el iminate from any one equation

f(u) = 0

a relative u It seems useful to tackle the problem in the formal, ra ther general

version; to form immediately the resultant of the el iminat ion of u from the equat ion of the form

f ( u ) = X.

Since the latter could be easily brought to the predicate 0, it is clear

Page 175


that, as soon as we have obtained (for any assumed x) the resultant of the elimination of u in the form of a subsumption

F(x) ~ O,

we have also have found in the form of

F(0) :(= 0

the resultant of the previous problem of el imination (in a more special form).

In this ex tended version, the el imination problem appears as the immedia te converse, inverse of a pure solution problem, and it has the advantage of letting us see that with every pure solution problem a certain elimination problem is also solved, and vice versa.

The former allowed us to proceed from the equat ion F(x) = 0 to its general solution x-f(u); the latter demands that the reverse path be taken!

If we have found the solution of 1) with 3), we have the subsump- t ionmfol lowing afor t ior i from 8 ) m

If(u) = x} =(= IF(x) =0} 18)

and the solution to the problem of the elimination of u from the left side of the equation, namely, from 2); and indeed the right side, that is, equat ion 1)mbecause of 9)mis the complete resultant.

We are no longer surprised that there are so many essentially different forms of the general root for the solution problem; this circumstance is now characterized as a consequence from the fact, plausible from the beginning, that very different initial equations will yield the same resultants, jus t as occasionally very different premises yield the same conclusions.

To obtain the latter is the goal of the el imination problem. And conversely, we could posit obtaining all general solutions to a given equat ion 1) (as the total proposit ion of a propositional system) as the answer to the question: "what premises yield a given conclusion?" 2

To give an answer to this question can hardly be more important than the solution of the first problem, that is to say, to answer the question which conclusion follows from a given premiss; Peirce (1885) has already emphasized this point on page 196---without paying any attention to the solution problem per s e . m

At the beginning of this section we assumed (what can always be added in theory) that the equation F(x) = 0, to be solved, if at all, is to be solved without conditions, that is to say, that the el imination of x would

Translator's note: English in text.

Page 176


not yield any "resultant." In practice, the opposite is usually the case. And if we have now found for that case the general scheme by which the solution always has to be approached, we still have to deal with this case. We have to tax the patience of the reader with the following question: how to modify our scheme if equation 1), to be solved, is to yield a resultant R = 0 (free of x)?

By R we have to imagine any function O(b, c . . . . . y, z . . . . ) of parameters thought to be given, such as polynomial coefficients, for example, possibly also of o ther unknowns.

The answer to this question is simple to give: that the resultant as a propositional factor of ~ is to be included, or prefixed, to the right side, so that the general sc~heme for the solution is

IF(x) = 01 = (R = 0) I: {x =f(u)1. 19)

Indeed, the resultant of the unde te rmined relatives which occur in 1), besides x, is ei ther not satisfied or it is satisfied.

In the first case, we have (R = 0) = 0, and the right side of our scheme will have the truth value 0. But then the left-hand equat ion cannot be solved, is {F(x)= 0}- 0, or the equation F(x)= 0, for every meaning which we may attach to x, is absurd. Our scheme the reupon proves to be the proposit ional equivalence 0 - 0.

In the second case, we have (R = 0) = 1. Then the condit ion is satisfied according to which we have justified scheme 3), that is to say, equat ion 1) is solvable. Our scheme 19) then passes into the same scheme 3). And it proves to be true for all cases.

If we introduce the abbreviations

A={F(x) =01, [ 3 = ( R = 0 ) , r = ~ { x = f ( u ) } ,

it is already certain, by what we stipulated earlier, that

A:(=[3 and B : ~ : ( A = E ) ,

and it is easy in the propositional calculus to prove this pair of subsumptions to be equivalent to the equation:

A = B r .

To conclude this discussion, I would like to say a word about the methods for solving the two problems.

We can also represent these problems which are associated with equation 1) as the analogous problem for the coefficients of the unknowns, respectively, of the el iminant x, by calculating or expanding, for each suffix ij, the left side of the condit ion to be satisfied

{ F(x) }o = 0 20)

2 8 2 SCHRODER'S LECTURE V

accord ing to the st ipulat ions of w 3. T h e n we only have to calculate,

respectively, to e l iminate , the general coeff ic ient Xhk----or better, all Xhk m a s an u n k n o w n f rom the equa t ion ; then the p r o b l e m presents itself

Page 177 as one of the pure p ropos i t iona l calculus. Already in the ident i ty cal-

culus, even m o r e for this calculus, the m e t h o d s of solut ion and elimi-

na t ion were b r o u g h t to a cer ta in stage of per fec t ion . They were devel-

o p e d in extenso and cult ivated into m e t h o d s that could be satisfactorily

wielded. Nevertheless , it would be wrong to th ink that every p r o b l e m can be

solved in our d i s c i p l i n e m a n d this for the reason that .... because we have only focused on cer tain and limited sets of unknowns , respectively

e l iminants , and because in fact this m e t h o d appears sufficient or fairly

qual i f ied only for the calculat ion of or e l imina t ion of these sets.

As a rule, the universe of discourse is to be assumed to be infinite,

and here we almost always have an infinite or at least an i n d e t e r m i n a t e

set of unknowns and el iminants ; even if the universe of d iscourse 11

consists only of a few e l emen t s as i n d i v i d u a l s ~ l e t us say, th ree or

m o r e m t h e calculat ions to be car r ied ou t accord ing to known m e t h o d s will reveal themselves to be scarcely feasible in practice, with the n u m b e r

of u n k n o w n s increasing by the square. Finally, even if the p r o b l e m could be solved for the coefficients, the

reverse conc lus ion back to the relative itself, for which relat ion, re-

spectively, relations, we inqui red , is no t that easy to do. It is easy to acquire a mastery of the basic pr inciples of ou r discipline,

bu t its two f u n d a m e n t a l p rob lems have to be called difficult. So far, the re is no method to solve t hem in general.

In our next lecture we will propose such a solution for a group of 512 prob-

lems. Concerning elimination problems, we only have a study by Peirce (w 28), in which something like a "method" of some generality is hinted at, and, concerning solution problems, an extension of my procedure on "symmetric general

solutions" (volume 1, pp. 498,503 and volume 2, w 51) may serve well for certain

classes of problems, as we shall soon see.

Otherwise, for numerous problems in our theory we depend on a deepening

of particular aspects, and especially on special techniques, or simply luck. For

others, we can only hope to find the solution sometime in the far future in the

united work of many researchers.

Under these circumstances, it seems important to know very broad classes of

problems for which the solution can be achieved with the help of a uniform

Page 178 scheme; therefore, I would like to identify some of them.

Page 179


w 13. Continuation. Iteration. Limiting Values and Convergence. Power.

R a t h e r g e n e r a l are the two classes of so lu t ion p r o b l e m s , in which the p r o p o s i t i o n to be solved for x can be r e p r e s e n t e d equ iva len t ly in e i t h e r o f the two fo l lowing ways:

x ~ ( x ) , ~(x) 4= x.

T h e s e are cases in which the p o l y n o m i a l F(x) o f o u r e q u a t i o n F(x) ~c--O has the fac tor x, o r Y - - t h e n its cofactor , respec t ive ly the fac tor

itself, can be p u t on the o t h e r side, n e g a t e d (as an a d d e n d to 0).

A "satisfactory" g e n e r a l so lu t ion o f the p r o b l e m in b o t h cases can

always be given in the fo rm of the often infinitely iterated function of an

a rb i t r a ry relat ive u, n a m e l y as x=f=(u), w h e r e f(u) r e p r e s e n t s s o m e express ion . In fact, the two t h e o r e m s are valid:

{x =(= 4~(x)} = ~ {x =f=(u)}, {~b(x) :(=x} = E {x =f=(u)} , u 1)

w h e r e f (u) = uq~(u), w h e r e f(u) = u + ~b(u).

T h e e x p l a n a t i o n a n d p r o o f o f these two p r o p o s i t i o n s give rise to several i m p o r t a n t r emarks .

First, we will de f ine the "iteration" f~(u) for any given f u n c t i o n f (u)

( u n d e r s t o o d as relative function or " func t ion" in the usual sense o f the

a lgeb ra o f b ina ry re l a t ives ) - - f i r s t for all " e x p o n e n t s " r tha t a re (f inite) na tu ra l n u m b e r s .

This de f in i t i on has to be m a d e in the usual way "by i n d u c t i o n , " by

which we m e a n n a m e l y tha t

f~ =u, f ' (u ) =f(u) , f2 (u ) =j{f(u)} . . . . 2)

in g e n e r a l " f~+l(u) - f{ f (u)}

- - t o this we only have to r e m a r k that the symbols wh ich a p p e a r as

"exponents" in o u r theory, O, 1,2 . . . . . r, r + 1 . . . . , n eve r o u g h t to be un- d e r s t o o d as relatives, bu t always only as natural numbers.

Of course, one of the most important purposes of our theory is this: to provide the logical basis of the theory of numbers, that is to say, to dig to the very

foundation of the concept of number [Anzahl], namely, to justify the so-called

"definition by induction" as a definition, to demonstrate its efficacy as one which

really determines the object to be defined, likewise to prove as admissible the

inference of mathematical induction, and so on. In pursuing these goals, we

are not allowed to make any assumptions about these topics; this we will re-

member in the specific lectures of our book that are devoted to these goals.

But this does not hinder us from using for the time being, in lectures remote

from those particular goals, those numerical concepts, as well as the inductive definitions and inferences that were ment ioned-- jus t as it occasionally hap-

pened in the course of the previous volumes, and, incidentally, as it is customary


in the whole world of mathematics and science. So much the more, we may

proceed in this manner, since this procedure is justified at some place in our

book in a way which will satisfy the most rigorous demands that can be made

on us from the point of view of logic.

It is true that this anticipation documents a certain imperfection in our lec-

tures, which do not follow Euclid's ideal of an absolutely rigorous, step-by-step

construct ionDas, for example, Mr. Dedekind's monograph did.

But the specialmI would say: austere--beauty of such a rigorous, step-by-step

construction is bought at the price of certain disadvantages, too, which appear

especially in the field of pedagogy or didactics; it can only be realized at the

cost of the overall view of the whole and the highlighting of general perspectives. I hope to keep to the middle of the road and trust to find readers who know

how to read eclectically, choosing (and occasionally skipping certain passages in

order to come back to them later), readers who are also ready to descend a few

echelons for the purpose of questioning fundamental knowledge, and who can

leave behind, can leave unused, knowledge that they have already acquired

elsewhere.

And thus we wish to proceed speedily here and recognize the iteration of the

function f(u) for all iteration exponents as "defined" by "recursion" in equation

2) which fixes the meaning and significance of f '+ l (u) as soon as the meaning

and significance are fixed for f ' ( u ) ~ a s soon as fl(u), as f(u), or, if you would,

f~ as u, has found its definition with equation 2).

Likewise, we wish to admit as valid h e r e ~ a n d we will have to return to this

l a t e r ~ t h e propositions, which evidentially follow from the definition:

as well as in general

Page 180 m a n d if we wish also

f"+ 1 (u) =f'{f(u)}, 3)

f " {f" (u)} = f . . . . (u) =f"{f" (u)} 4)

( f ' ) " (u) = f ..... (u) = (f")"' (u) 5)

Das indeed will be immediately clear to those who already unders tand the

concept of number because, for example, the three expressions that are posited

as equal in equation 4) mean nothing more than the function f taken m + n

times on u, and so on.

F u r t h e r m o r e , "the infinite iteration" f~(u) is to be defined, provided that t he n a m e is capab le of a de f in i t i on which is based on the n o t i o n o f

f • a n d is m o t i v a t e d by the b e h a v i o r of this re la t ive for all and , in

par t icu la r , for inf ini te ly i n c r e a s i n g i t e r a t ion e x p o n e n t s X. T h e c o n d i t i o n

for this last case will be ca l led the convergence condition for fX(u), n a m e l y

for inf in i te ly i nc rea s ing ~.

In g e n e r a l , if the value of a re la t ive u• is determinate for the inf in i te

Page 181


series o f na tura l n u m b e r s 3, = 0, 1,2, 3 . . . . . for example , actually given for arbitrari ly many relatives of the "series"

UO, U 1, U 2, U3, . . . ,

conceptually fixed for the rest by a law or pr inciple , then we have to say for an infinitely increas ing index ;~ (as na tura l n u m b e r ) in gene ra l that u• "diverges" and that the symbol u~ has no mean ing ; bu t the re is also a class of cases in which we can say that the gene ra l t e rm u• o f ou r series "converges" because it "strives" toward a d e t e r m i n a t e , f ixed relative and thus toward a relative to be des igna ted u= as the "limit."

T h e lat ter case occurs if, and only if, for every position on the matr ix of 1 ~ m a r k e d by a suffix ij---or, in o t h e r words, the matr ix o f u• number n can be given or exists in such a way that the position in u• bears a "filled circle" (o), is occupied for every ~ > n or is empty, remains empty for every X > n.

A posi t ion ij of the matr ix of the relative u x with variable ;k shall be called a final [endgiiltig], "definitely" occupied posi t ion of this variable relative, if the re is such a value n of ;k such that for all ~ > n the posi t ion in u• turns ou t to be occupied ; it shall be called a final unoccupied or definitely empty position, if there is such a n u m b e r n that the posi t ion in u• remains u n o c c u p i e d for all X > n.

By using this form of express ion, we can say m o r e briefly: u• shall be called convergent with increas ing X if, for every posi t ion of

its matr ix, we can d e t e r m i n e if it is definitely to be o c c u p i e d or defini tely to r ema in empty.

By the limiting value [Grenzwert] (limes) of u• (for lim ;k = oc) then we u n d e r s t a n d that relative which in the "definitely occup ied" posi t ions of u• has "filled circles"; the "definitely u n o c c u p i e d " posi t ions of u• are empty. And this l imit ing value we des ignate u= for short .

In genera l , however, there are posi t ions ij which b e l o n g n e i t h e r to the defini tely occup ied no r to the definitely u n o c c u p i e d posi t ions of the u• d e p e n d i n g on ~; there , for each index n, however g rea t it is, the re is always a n u m b e r m > n such that, if the posi t ion is o c c u p i e d in u,,, it appears again as u n o c c u p i e d in u,,, and conversely.

Such posi t ions in u• oscillating with increas ing ;k ( a l t hough no t necessarily in regu la r variat ion) , which are somet imes o c c u p i e d and some- t imes u n o c c u p i e d , may be called "oscillatory occupied" (or, likewise, unoccup ied ) or (finally) oscillatory* posi t ions of u•

W h e t h e r the re is only one such posi t ion or w h e t h e r the re are several, no def ini te no t ion (such as that of a relative) can be associated with

* Not: "oscillating"; the posit ions themselves do not fluctuate, only their occupant , the "filled circle" oscillates (blinks, scintiUates) between presence and absence.


t h e s ign u0~ b e c a u s e t h e r e is a b s o l u t e l y n o r e a s o n to h e l p us d e c i d e

w h e t h e r s u c h p o s i t i o n s o u g h t to f igu re as o c c u p i e d o r as u n o c c u p i e d .

T h e s y m b o l u~ t h e n r e m a i n s m e a n i n g l e s s , a n d u• d iverges . In o u r

d i s c ip l i ne , d i v e r g e n c e can always on ly take p l a c e as "osc i l la tory ."

Al though this symbol is meaningless, we can nevertheless associate a definite

m e a n i n g to it with subsumpt ions in which it occurs as subject or as predi-

c a t e - a l t h o u g h I do not want to attach great impor tance to this point at this

time. But we can write

in o rder to express that:

a has "filled circles" only or, at most, at those positions which are a m o n g

those definitely occupied for some arbitrary u• of a sufficiently great index X;

a thus has to have all positions, continually oscillatory occupied at u x, and the

definitely unoccup ied positions as empty positions.

Likewise, b has empty positions only or, at most, at those positions which are

a m o n g those definitely unoccupied for some sufficiently great u• b thus has to

show as occupied, or, at least, with "filled circles," all the positions which are

definitely occupied at u• as well as the continually oscillatory occupied positions

Of u•

In such cases, there is also a most encompass ing or m a x i m u m value of a,

which still satisfies the r equ i r emen t a~6--Uoo (can possibly be = 0), which we may

Page 182 call the "lower limit" (limes inferior) of this not completely definable relative

symbol uoo; likewise for a m i n i m u m encompass ing or m i n i m u m value of b, which

still satisfies the r equ i r emen t uoo =(=b (which can be = 1) and is therefore called

"the u p p e r limit" (limes superior) for u=.

Many times we will be able to say that u• cont inuously oscillates between these

two limits, after a sufficiently large X, and can only pass th rough or assume

" in termedia te values" between them.

[It may be that too much has been said with the last corollary, a l though it

will often be t rue - - in part icular always with a finite n u m b e r of e lements in the

universe of discourse, at times also with a finite n u m b e r of positions in the

matrix.

If there is, for example, a n u m b e r n for every "definitely occupied" position

ij, for which, after X exceeds it, the position will no longer appear as an empty

position in u• then there is also a n u m b e r m i n the form of the greatest value

individually assigned to the positions of the s e t I f o r every finite set of such

positions ij; this n u m b e r has the proper ty that, after X has exceeded it, all named

positions have found their final occupat ion and can never again be empty po-

sitions in u x. But if the set of tile positions which we take into considerat ion

increases indefinitely, it is problematic , and future more subtle examinat ions

will have to decide whether or not this greatest of all (smallest) values n (which

be long to each position) is pushed far ther away in the n u m b e r sequence, and


the sequence of values n, itself of infinite growth, does not include any value

as the greatest one. Then we can indicate for every individual position a number

)~ = n, from where the position in u• has found a final occupation, but not for

the total of all positions to be occupied definitely. Etc.]

We have to s e p a r a t e c lear ly the p r o b l e m of c o n v e r g e n c e , o r diver-

g e n c e , of the series itself f rom the p r o b l e m of c o n v e r g e n c e o r d i v e r g e n c e

o f the general term u• of o u r series, if the t e rms a re t h o u g h t to be con-

n e c t e d by a j o i n i n g o p e r a t i o n (for e x a m p l e , o n e the six spec ies ) . If the

t e rms of the series a re j o i n e d to each o t h e r by an identity species, t h e n

we ge t an "infinite product" or an "infinite sum" (or "series" in the n a r r o w

sense) . T h e n , the resu l t o f c o m b i n i n g the first 3, + 1 t e r m s is

V k u 0 u 1 u 2 . . . u k , U x = u 0 -~- u 1 + u 2 - - 1 - ' ' ' ~- u x , 6)

tha t is, " the p r o d u c t i o n a l factor," respectively, the so-cal led " s u m m a t o r y

t e r m " of the series, is tha t g e n e r a l t e r m a b o u t whose c o n v e r g e n c e we

are c o n c e r n e d in the la t te r case.

H e r e the r e m a r k a b l e d o u b l e p r o p o s i t i o n is valid: every identity infinite product and every identity infinite sum is convergent. Even w h e n the g e n e r a l

t e r m u• is d ive rgen t : we have in o u r d i sc ip l ine c o n v e r g e n t p r o d u c t s

Page 183 f r o m d i v e r g e n t factors a n d c o n v e r g e n t sums (series) with d i v e r g e n t

t e r m s - - t h e s ame wou ld have to be e x p l a i n e d in an a r i t h m e t i c a l analysis

o f p a r a d o x e s ! In any case,

U~: = u 0 u I u 2 u 3 - ' - , U ~ - u 0 + u 1 + u 2 --[- u 3 - ~ - . . - (in infinitum) 7)

has an u n c o n d i t i o n a l m e a n i n g a n d c o m p l e t e l y d e t e r m i n e d va lue in the

d o m a i n o f b ina ry relatives.

This is relatively easy to see here. It rests upon the fact that every empty

position in the U• on the left remains definitely so, as many factors u• as may

be jo ined to the product with increasing 3,; each occupied position of the U x

on the right has to keep its "filled circle" permanently, as many terms u• as may

be added to the sum of the already assembled one. To be more precise: because

( r l - n (u , j ) -- n u , ) , u ,j - (u , ) ) - u , )

is def ined- - for every choice, for example, a series, of values u as extension of

H, of Z- - the re are only two possibilities for a determinate position ij, namely:

Left: Either there is a number n for which u• at ij has an empty position,

(u,,)O =0, or not. In the first case, U• also has an empty position at ij for each

3, > n, and it becomes the definitely unoccupied one. In the latter case, for )~,

every (u• has to be equal to 1, and the position remains definitely occupied.

Right: Either there is an n for which u x at ij has "a filled circle " (u,,) O, or it

does not apply. In the first case, U• also has a "filled circle" at the position /j

for each )~ > n, and it becomes a definitely occupied position. In the latter case,

for )~, all (u• are equal to 0, and the position remains empty.

Page 184


Left and right show that all positions of the matrix of U• are either definitely occupied or definitely unoccupied, and a third solution, oscillator 3, positions, is

not possible at all.---q.e.d. The attentive reader will immediately see that the numerical value of the

index plays only a minor role in these considerations. The considerations remain valid if, for example, in IIxu x the index ;k would

have to run through a "continuum" of numerical values. The proposition and proof are also valid for H u and Z u, in whatever way the

u u

extension was provided. Concerning IIu, for example, there has to be for any arbitrary suffix ij in the

extension either a u for which u 0 =0, or not. In the first case, IIu at ij has a

definitely empty space; in the latter case, where thus "for all u" u 0 -- 1, IIu will have a definitely occupied position at ij; a third possibility (an oscillatory occupied position) is unthinkable.

Etc. (that is, analogously for E u). u

For relative infinite products and sums, a similar propos i t ion of the same general i ty does not hold. We can easily illustrate this point , for

example , with the relative p roduc t in case of always equal factors.

a relative product o f factors that are all equal is called a "power."

We def ine the power (u;)• or (; u) • when X represents a natural number , simply as u • (u to the power ;k), e i ther "by induct ion" ("re-

cursion") by means of the stipulation:

U 1 U , U 2 U , U , , U ~ , + 1 U X = = �9 . - . = ; u , 8 )

or " independent ly" as

u • ; u ; u ; " " ;u. 9)

T h e r e follows the known proposi t ions for the powers in ar i thmet ic :

u x + ~ = u ; u x, u ~',u • ~+x=u • ~, (u~) x = u ~ ' • 2 1 5 ~. 10)

It is not necessary to characterize the "power" as a "relative" by adding an

adjective, because the law of tautology uu = u excludes the possibility of making

the mistake of taking the "power" as identity product (of equal factors).

Here again, numbers play a role with the exponents. If somebody feels un- comfortable with this, he should accept u x only as a "stenographic code," a

conventional sign for the purpose of simplification.

The dual coun te rpa r t to the power is the relative sum of s u m m a n d s

which are all equal:

uct uct u # " " # u = (u#)• or also ( # u) • 11)

I will occasionally call t hem "iterates" or "iterative sums." If we wish to

push the analogy with the nota t ion in ar i thmet ic construct ions , then

Page 185


t he a b b r e v i a t i o n s p rev ious ly i n t r o d u c e d - - w h i c h a re pa ra l l e l to wr i t i ng

u • for (u ; ) • have to be c h a n g e d , w h e r e t he i t e r a t i o n e x p o n e n t

~, has to be p u t as a " m u l t i p l i e r " b e h i n d the u. We w o u l d t h e n have to

d i s t i n g u i s h t h r e e m u l t i p l i c a t i o n s - - - c o u n t i n g the a r i t h m e t i c a l o n e ,

f o u r - - a n d the s a m e n u m b e r o f m u l t i p l i c a t i o n signs, w h i c h are de f in i t e ly

too m u c h ("All g o o d th ings a re t h r e e " ) .

T h e r e a d e r m a y wri te d o w n o r f igure o u t t he dua l c o u n t e r p a r t s to

t he above laws of power.

We can thus c la im: t he p o w e r u • d iverges in g e n e r a l .

A s i m p l e e x a m p l e shows this; as the fo l lowing a s s u m p t i o n b e l o n g s to

t he u n i v e r s e o f d i s c o u r s e 1~ of on ly t h r e e e l e m e n t s :

0 �9 0 0 �9

u = �9 o �9 for wh ich ~i= �9 o 0 �9 0 0 �9

and w e n o w have u; u = u 2 = u , u 2 ; u -- u 3 - u ; s o u x oscillates, fluctuates regularly, w i t h o u t e n d , f r o m o n e to the o t h e r o f t he two va lues u a n d

~i b e c a u s e we have

U 2~ = Z l , U 2 X + 1 __ U ,

a n d u • is d i v e r g e n t , t he symbol u,0 h e r e is meaningless. Of course, we could arbitrarily attribute to this meaningless name any chosen

meaning. In whatever way we would make this choice, rational considerations

cannot be found for it, the introduction of such a name would not grant any

advantage; on the contrary, it would do much damage. This name would not

fit into any rational system of notation, especially not into that created in this

book. It would even disturb the regularity and lawfulness of each such, if not

completely cancel them out; it would create artificial obstacles and turn into a

source of embarrassment , as it would necessitate all sorts of cumbersome ex-

ceptions which necessarily occur and which we have not encoun te red here.

Meaningless names in a discipline are, as it were, by-products of a certain (no-

tation) industry. At times, they present valuable raw material which can be proc-

essed in ano ther industry and thus finds a valuable usage in the overall econ-

o m y - t h e same is true in the field of natural numbers, with the meaningless

names of negative numbers, and the name ~/Z-i-, unusable as a rational n u m b e r

(and meaningless in the domain of real numbers) , etc., in the ex tended domains

of numbers.

But the circumstances are not always so favorable; much waste has to be

discarded.

Now we have

o 0

~ k U X = U "+" U 2 -t-" U 3 ~'- U 4 "- t -" '"

1

29 ~ S C H R O D E R ' S L E C T U R E V

as an obvious e x a m p l e of a c o n v e r g e n t s e q u e n c e with a d i v e r g e n t gene ra l

te rm. It has for o u r u above the sum u + ~i - 1.

For the c o n v e r g e n c e of the power u • o f a relative u---with an e x p o n e n t

X inc reas ing wi thou t e n d - - t h e necessary and sufficient cond i t i ons are no t

yet known. But some c i r cums tances can be p r o v e d as suff icient cond i t ions .

So x • has to conve rge for 3, = ~, if x has the p r o p e r t y tha t x;x:~--x, as well as if it has the p r o p e r t y x:~--x; x. In the first case it is easy to see

tha t x • =(=x• in the last, tha t x • :~x • has to ho ld for each X (however

grea t ) . In that case the empty positions will be finally c o n s e r v e d by con-

t inual relative mul t ip l i ca t ion with x, because w h e r e v e r x • has an e m p t y

pos i t ion , also x • mus t have one ; the power converges t h e n "decreasingly" Page 186 toward a f ixed limit. In this one , the same is valid for the "filled circles,"

a n d the power converges "increasingly" to a limit.

In part icular , if x is of the fo rm of 1' + a, t hen we have x; x = 1' +

a + a ; a, a n d thus in fact x ~ - x ; x ; c o n s e q u e n t l y the symbol (1' + a) = has

a m e a n i n g for each m e a n i n g of a.

Also, w h e n the base x of a power x • is of the fo rm a ; 6 ( tha t is to say, t aken for a, at the same t ime also of the fo rm d;a) , this p o w e r has

to c o n v e r g e - - a n d this based on the p ropos i t ion :

a ; ~ :(= a ; ~ ; a ; ~ , a ~ a ~ a~t a ~: a ~ ~,

which we will identify la ter as a special case (for b = 4) o f a gene ra l

p r o p o s i t i o n - - 2 1 ) of w 18 - - i n the m e a n t i m e , we prove it by the coeffi-

c i en t ev idence , with the r e m a r k tha t the te rms of L 0 = ~haihajh are all

p r e s e n t in R 0 = Ehktaihakhaktajl w h e n k = i, l = h. After this de tour , we now r e t u r n to the i te ra t ion of the func t ions a n d

t h e n to o u r T h e o r e m 1.

T h e r e are cases in which "the r times (rth) iteration of a func t i on f (u) ," namely, f r (u) for lim r = 00 converges , tha t is to say, "in general"for every argument u.

An e x a m p l e f rom the ident i ty calculus will con f i rm this point . W h e n

we assume

f (u) = au + b, for which fZ(u) = a(au + b) + b = au + b,

thus f e (u) =f lu ) , a n d t h e r e f o r e also

f3(u) = f{ fZ(u)} = f{ f(u)} =fZ(u) =f(u) ,

in genera l : f r (u) =f(u) a n d thus f~(u) - f ( u ) . A f u n c t i o n f with the p r o p e r t y that, in genera l , for every a r g u m e n t ,

its s e c o n d i te ra t ion is equa l to the first (or the func t ion itself) may be

cal led "invariant." All i tera t ions of such a func t ion , to beg in f rom the

ze ro th ab, are t hen equal to it, as can be easily seen: each invariantfunction

Page 187


remains unchanged by iteration, and also the func t ion which is i tera ted infinitely often is then no o the r than itself.

u itself is also an invariant funct ion of u. In general , the rth i terat ion of a somehow given func t ion f (u ) must

diverge with increasing r. This point , too, the identity calculus can illustrate. T h e mos t genera l

func t ion of u which can be f o r m e d with the species of this discipline is _

f (u ) = au + bfi. Thus we have f { f (u)} = a(au + bgt) + b(du + b~i); therefore fZ(u) = (a + b)u + abfi = (a + b)u + ab.

This is d i f ferent f rom f (u ) in gene ra lmas , for example , the assumpt ion b = d easily shows, where for f (u) = au + gt~i we now have f 2 (u ) = u =

f ~ etc. On the o the r hand , we have again in the above genera l case

f~ (u) - f ( u ) , f4(u) =fZ(u); in general ,

fzK+'(u) =f(u), f2K+Z(u) =fZ(u),

thus, fr(u) is d ivergent and f~176 is meaningless , excep t in the special cases m e n t i o n e d previously w h e r e f Z ( u ) = f ( u ) , and this funct ion is invariant.

If there is a pair of n u m b e r s m, n such that for every def ini te func t ion

f (u ) we have, for ever?, u,

fro+, (U) =fro(U) ,

we call the funct ion a "periodically (or oscillating) i terate, with an iteration period n" or, in short, "with the i terat ion per iod n" in case n is at the same t ime the smallest n u m b e r of the n a m e d proper ty (which such a pair of n u m b e r s m, n has).

The re fo re , the not ion of an invariant funct ion is s u b s u m e d u n d e r that of a periodical ly i terat ing funct ion of period I.

While the i terations of such funct ions are convergent , each periodically i terat ing funct ion, which has a per iod n > 1, has to be divergent- i terating.

To make it shorter, an example may illustrate this point. If we have in general--eliminating the argument (u), which has to be assumed--f s =fs, then we also have f:~ =f6, f ,0 =f7, f,~ =fs, fl2 =f~, f,:~ =fv, f,4 =fs, and so on. The iterations of f that have a period equal to 3 will eternally oscillate from one of the three values f~, f6, fv (which were assumed as different) in a ring (from the last again to the first), and f= is not capable of yielding any definite interpretation.

This is also the case if m = 0; therefore f"(u) = u is itself; here again, the values f0, f, f2, f:~ . . . . . f " - ' , [f" (or u), f, etc.] repeat themselves in stable sequence in infinite iteration.

Page 188

292 SCHRC)DER'S LECTURE V

Fo r the general function in the identity calculus we have s e e n tha t it has to be,

if it is n o t invar ian t , a periodically iterating function with the period 2. The example f(u) = a; u shows that the relative operations can also lead (and,

in general, will lead) to the formation of functions with divergent iterations; in this example, we had f (u) = a ~ ; u, where the power a ~ itself oscillates in general.

For the two meanings of f (u) ment ioned in our T h e o r e m 1, we now h a v e

f~+' (u) =f~(u)4~{f~(u)}, f~+l(u) =f~(u) + r 12)

By successive calculations of the iterations o f f , we have in addition to the already known expression o f f ' ( u ) only one factor: "4~ of all up to now," respectively one summand: "r of all up to now"; the law of format ion of the iterating function is easily overlooked, al though the expressions become rapidly more complicated with increasing exponents . We have, for example, on the left:

f (u) = u~(u) , f'~(u) = u~(u) 4,{u4~(u)}, f 3 (u ) -- uch(u) 4~{u4~(u)} 4~[u4~(u) 4~{u4)(u)], ...

and on the right

f (u ) = u + r f2(u) = u + r + r + r

f~ (u ) = u + r + r + r + r + r r + r " " .

Al though the names for "all up to now" increase in length, the difficulties or efforts in calculation do not increase at all. The formation o f f ~+l(u) from f~(u), already obtained, remains as easy as before, and requires no more work than the calculation of the function 4), respectively r itself for any given argument .

The indefinitely cont inued iteration of the function f (u ) appears in the form of an identity infinite product , respectively an identity infinite series of sums; therefore they are convergent (according to the general proposit ion proved above); f=(u) has a very definite value.

"Proof 1" for our Theo rem 1 requires us to show that it specifies a root of the solvable subsumption, whatever value of the arbitrary pa- ramete r u may be chosen.

"Proof 2" requires us to show that if we have from the beginning

x ~ ( x ) , r ~ x ,

then also f~ = x must hold. The latter is easy, in view of the fact that the assumptions can be

written as equivalences:

x = x ~ ( x ) , x + r = x,

thus:

Page 189


f ( x ) = X .

[This observation has led to the discovery of thef(u) which yields tile solution by iterating.]

But if tor any function f(u) and some value x of u, f(x) = x, "as previously mentioned, then we must have

f'~(x) =f{f(x)} =f(x) = x, etc.,

in general: f '(x) = xandf~(x) = x. Because oflf'(x)} 0 = xq not only for some value of r but in general, the left-hand coefficient has the value of the right-hand one, thus fr(x) has the "filled circles" of x as the definitely occupied positions of its matrix, and the empty spaces of x for the definitely unoccupied positions. There- tore, we had to determine the relative f=(x), so that the same is true for the latter.

P roof 2 is thus certain, and it also appears cer ta in that our solut ion 1) will yield all roots of the problem.

T h e first is not that simple, namely, to show "Proof 1" or br ing the p roo f that our solution 1) always yields only roots of the p r o b l e m (for every u).

If we accept the proposi t ion which every ma thema t i c i an knows, that if f =(u) has a mean ing , namely fr(u) converges for infinitely increas ing r, then we have f=+'(u), unde r s tood as f{f=(u)}, = f = ( u ) , and so the p roo f is easily obta ined, because we have f~(u) =f{ f~(u)}= f=(u)ch{f=(u)} ~-4~{f~(u)}, therefore f~(u) ~-4~{f~(u)}; ~b{f=(u)} =(= f'~(u) + ~{f=(u)} =f{f=(u)} =f=(u) , therefore ~{f~(u)} ~--f=(u), thus for x =f'~(u) P roof 1 is indeed true, namely, for every u, x@4)(x), respectively ~(x) ~ x .

But that "proposi t ion" itself is not that easy to conf i rm for our discipline. Before discussing it in its generality, I will l imit myself to the p resen t c a se - - fo r example , the left s i d e - - a n d say:

By the left side of scheme 12), taken for u n b o u n d e d l y increas ing i terat ion exponen t s r, f=(u) has as a factor 4~{fr(u)} for every r, however large; it thus has also 4~{f=(u)} as a factor and has to be con ta ined in it. In o the r words, when fo rming the infinite p roduc t f rom which we had to gain f=(u), in addi t ion to the sequence of factors wi thout end already m e n t i o n e d , we also have 4~ of everything m e n t i o n e d so far as a fu r the r factor; u n d e r "everything m e n t i o n e d so far" also figures ("finally"?) also the total f=(u) i tself--q.e.d. (?)

This ref lect ion is certainly unassailable as soon as our universe of discourse is finitely bounded. Because then the set of conceivable relatives is also finitely b o u n d e d ; the factors of our factor s equence c a n n o t con- t inue to be differ f rom each o the r endlessly; and finally the p r o d u c t has to be constant ; namely, it will r ep roduce itself tautologically when further factors are added. In this f=(u), 4~{f=(u)} really occurs as factor.

Page 190

294 SCHRt~DER'S LECTURE V

S c h e m e 1), accord ing to which we can cons t rue genera l solut ions satisfactorily for n u m e r o u s individual p rob lems , is t he re fo re jus t i f ied as a cor rec t s cheme of solut ion, at least for every finite universe of discourse--and with this, we have ga ined a lot!

But I main ta in it in g e n e r a l - - a l s o for the infinite universe o f discourse, a l t h o u g h I have to confess that what I will show to prove it is no t completely satisfactory. Whoeve r shares my scepticism n e e d do no m o r e than trust the special solut ions based on schemes only within the men- t ioned limitations.

Appendix 5: Schr6der's Lecture IX


At the end of his fifth lecture, Schr6der treats the relational identities that express the inductive definition of plus in terms of successor, and times in terms of plus and successor, as relational equations. The first must be solved for plus, given the constant relation successor. The second must be solved for times, given plus and successor as constants. Schr6der points out informally how these arise as the smallest set of o rdered triples satisfying conditions that can be read from such equations. In the ninth lecture, he does the same thing for inductive definitions in general.

Here he presents the definitions for primitive recursion, which of course includes those for plus and times. He compl iments Dedekind as the first person to see this method, and criticizes Frege and others for not appreciat ing what Dedekind had done. Recursion equations, defining a function in terms of simpler functions, are, as Dedekind realized, implicit conditions that must be proved to have a solution. Tha t is, the set of o rdered triples defining, for example, multiplication, must be defined in some direct way. Dedekind 's theory of chains was developed for this purpose. Schr6der realized that Dedekind 's key notion was the

transitive closure of a binary relation, the smallest transitive relation

containing the given relation, def ined as the intersection of all transitive

relations containing the given relation. At that time it was not realized

that the axiom of infinity was needed to show that there is at least one relation that is a set (not a class), and transitive, and conta ining the

given one. This me thod of a rgument was Dedekind 's , who implicitly used but did not ment ion the axiom of infinity. Without it, the integers could be a p roper class, and the transitive closure of a relation a p roper class. This actually happens in the ranked universe of level omega (a.k.a. the Ackermann model) , which contains only finite sets.

295

296 SCIIRODER'S LECTURE IX

However, Schr6der wished to re formula te these not ions and prove Dedek ind ' s results entirely within the calculus of relatives. Here inductive defini t ions of funct ions are t h o u g h t of as implicit solutions of the

relat ional equa t ion form of the recurs ion clauses, which then must be

solved. The solution is provided by transitive closures of appropr ia te

relations, and the device for hand l ing transitive closures is Schr6der ' s

version of Dedek ind chains. Since this was Schr6der ' s only e lucidat ion

and extens ion of Dedekind ' s work on justifying defini t ions by induc t ion ,

we r e p r o d u c e this lecture in its entirety. It can really only be read with

a copy of Dedek ind ' s m o n o g r a p h at hand. It seems likely that the pur-

pose of this lecture was to show that the most delicate piece of foun-

dat ions work thus far in the history of mathemat ics could be carr ied

ou t neatly in the calculus of relatives. O n e should note that D e d e k i n d used the word "system" where we use the word "set." O n e of the aims

of Sch r6de r appears to be to avoid ever m e n t i o n i n g e lements of sets,

replacing all set a rguments by the use of relational identities.

Lecture IX

The Theory of Chains

w 23. Dedekind's Chain Theory and the Proof by Induction. Its Simplification.

Page 346 The reader must have been wondering for some time what kind of precious goals can possibly be reached with the extensive resources of our theory? (When

I say "our theory," I mean the theory proposed in this book; its beginnings go

back to A. De Morgan, and its main contributor is Charles S. Peirce; the cir-

cumstance that it was my lot to expand on various aspects of this theory may

give me the right to join my name to those of these authors.)

Patience! The instrument in our hands is still in a very incomplete state! The

farther we go in its construction and the wider the sphere of its applications

becomes, the more powerful it will reveal itself to be.

However, in order not to delay the proof of its capacity any longer, I will now

proceed to integrate R. Dedekind's "theory of chains" into our discipline. The

advantages that this theory will gain, which I hope to make very clear, will serve

to document for the first time the worth of our discipline.

The "chain theory" is but a part, although a fundamental one, of Dedekind's

pioneering work "Was sind und was sollen die Zahlen ?" To incorporate it completely

into the structure of general logic, together with its other essential parts, con-

stitutes the most important goal of my own work.

I must therefore start with a discussion of this essay and will later repeatedly

refer to it. Thereby I will cite Dedekind's 167 propositions and definitions in


the abbreviated form of ~ 1 to ~ 167 and mark his direct s ta tements with quo-

tation marks, reserving the right to italicize impor tan t passages.

The mathematical world has general ly recognized the impor t ance of this work:

it went rapidly out of print and was then repr in ted again in u n c h a n g e d form;

however, some individual mathematic ians have profoundly unde res t ima ted it,

both with respect to its possibilities and to its me r i t smthe most blatant case

Page 347 being the negative review by the well-known edi tor of a mathemat ica l journa l ,

Mr. R. Hoppe , who only saw an "intellectual exercise" in Dedekind ' s essay and

to whom the goal that "the writer in tended to gain r ema ined obscure."

Similarly, the achievements of Mr. Dedekind have not been sufficiently rec-

ognized and apprecia ted by phi losophers or those leaning toward phi losophy

(Frege 1893; Husserl 1891). The au thor of this book therefore takes satisfaction

in put t ing them in their right and due light.

In o rde r to show the general t rend of Mr. Dedekind 's work, we let him speak

for himself:

Page 348

What can be proved, ought not to be believed without p roof in science. Al though this d e m a n d seems clear enough , it is not always fulfilled,

as I believe, in the proofs of the most simple science, that is to say, that part of logic which deals with the theory of numbers , as the most recent publications show.* By calling ari thmetic (algebra, analysis) only a part of logic, I show that I consider the concep t of numbers as totally i n d e p e n d e n t of any concept or any idea of space and time, and that I consider it as the immedia te result of the pure laws of

thought . My main response to the question asked in the title of this essay is: numbers are free creations of the h u m a n mind; they serve as a med ium to conceptual ize with ease and clarity the differences between things. T h r o u g h the purely logical s t ructure of the science of numbers and th rough the cont inuous realm of numbers ga ined by it, we are in a position to examine our ideas of space and time, by

compar ing them to the realm of numbers created in our mind.** If one pursues accurately what one does when countinga collection or n u m b e r of things, one is led to see the capacity of our mind, which is, to refer things to things, to let a thing correspond to another thing, or to represent a thing by a thing. Without this capacity, no thinking is possible. In my opinion, the whole science of numbers has to be based on this foundat ion alone, a basis which is essential in o ther respects as well, as I have said in an a n n o u n c e m e n t of this essay.*

* Reference to the author's Lehrbuch der Arithmetik und Algebra and to the essays of Kronecker and Helmholtz on the concept of numbers, and on counting and measuring, in the collection of philosophical essays addressed to E. Zeller, Leipzig 1887. "The publication of these essays induced me to come out and publish an opinion, similar in many respects, but considerably different in its proofs, which I have been forming for many years without any external influence from any side."

** Reference to w 3 of Dedekind's essay (1872) on continuity and irrational numbers. * Reference to Mr. Dedekind's Vorlesungen fiber Zahlentheorie yon Lejeune Dirichlet, 3rd

edition 1879, w 163, footnote on p. 470.

298 SCHRODER'S LECTURE IX

After giving a sho r t h is tor ica l b a c k g r o u n d of his essay, Mr. D e d e k i n d

stresses its m a i n points : the sharp distinction of the finite [Endlich] and infinite [ Unendlich] ( ,~64) , the concept of the "number" [Anzahl] of things ( ,~161) , the proof that the mode of proof known by the name of mathematical induction (or the i n f e r e n c e f r o m n to n + 1) /s really conclusive ( ~ 5 9 , 60,

80) , a n d t h e r e f o r e tha t the definition by induction (or recursion) is definite and free of contradictions ( ,~ 126):

Page 349

Anybody who possesses what is called common sense can unders tand my essay; philosophical or mathematical schooling is not at all a prerequisite. But I know very well that many people will hardly recognize in the shadowy figures which I will introduce the numbers that have accompanied them as true and faithful friends through life; they will be frightened by the many simple inferences which we have to draw according to the nature of our step-by-step intellect, and by the dry dissection of the series of thoughts on which the laws of numbers rest; they will also become impatient when they have to find proofs for truths which seem certain and clear from the start because of supposed inner perceptions. But I perceive in the possibility of tracing back such truths to other, simpler truths, the convincing proof that they are never given immediately by inner perception, but are always reached through a m o r e or less complete repetition of individual inferences, even if these inferences are long and seemingly artificial. I would like to compare this activity, difficult to follow because of its rapidity, with the action of a practiced reader; reading is always a more or less complete repetition of individual steps which the beginner executes with difficulty; but only a small intellectual effort is necessary for the practiced reader to recognize a right word even if only with a high degree of probability; as is well known, even the most expe- rienced editor overlooks a typographical error once in a while. That is, he misreads a word; this would be impossible if the chain of thoughts, necessary for spelling, would be repeated completely. From our birth, we are constantly and increasingly forced to relate things to other things and thus to exercise that capacity of our mind on which the creation of numbers rests. By this continuous and unin- tended exercise, which begins in early childhood, and the related formation of judgments and conclusions, we acquire a stock of arithmetical truths on which our first teachers later rely as something simple, self-evident, given in inner perception. Thus, it comes that many actually extremely complex concepts (such as the number of things) are erroneously considered to be simple. In this sense . . . . the following pages, at tempting to establish the science of numbers on a uniform basis, ought to encounter favorable acceptance, and I hope that other mathematicians will be stimulated to reduce the long series of inferences to a more modest and manageable quantity.

For the purpose of this essay, I limit my examination to the series of the so-called natural numbers.

Page 350


So much from the Preface. To begin, I would like to point out a n u m b e r of

those "provable" propositions which are generally accepted without proof;

among them are the propositions: that in every finite set of numbers there is a

greatest and least n u m b e r (~114) ; that if a part of a system is infinite, the whole

system has to be infinite ( , ~68 ) - - and so on.

Mr. Dedekind has thus successfully concerned himself with the filling of a

large and significant gap which has existed up to now in all representations, in all books on arithmetic and algebra (the one by this writer [Schr6der 1873] not

excepted). The gap is particularly large at that point where the science of ar-

i thmetic ought to originate from general logic, where it ought to be rooted in

order to branch off in different directions. We missed- -and actually are still

missing in pa r t - - t he connection of that discipline of ari thmetic to the algebra of relatives, which includes the theory of single-valued assignment [eindeutigen Zuord- hung] or mapping [Abbildung] as its particular or special branch.

I can hardly blame myself or any other expositor of ari thmetic for this gap

when I imagine, on the one hand, how advanced the deve lopment of logical

calculation had to be in order to produce the missing connect ion smoothly, and,

on the o ther hand, how great an intellect as Dedekind was necessary to fill the

gap (to which I only supply the glue). And I have to pay the greatest tribute to

the mind which created the missing link.

The final goal of the work is to reach a strictly logical definition of the relative concept "number o f -" ["Anzahl von-'] from which all proposit ions regarding

this concept can be derived purely deductively.

We may also ment ion that considerable "benefit" of this essay which consists

in destroying the nourishing ground for endless quarrels by pseudo-philosophers

about the nature of number. Since the concept of n u m b e r is only applicable to

finite sets (of "units" [Einheiten]), the determinat ion of the concept of finiteness

is anyway necessary to reach our goal. This in itself is not easy.

How true is the striking remark of Dedekind that the concept of n u m b e r

passes falsely for simple. Our book will prove this point.

Fur thermore , we want to stress the fine point that Dedekind 's essay makes

by int roducing ordinal numbers before cardinal numbers; and this considerably

earlier.

This, incidentally, corresponds to the historical deve lopment of a series of

areas of magni tude to which we gradually introduce and master the concept of

quantity.

As it is still the case, for example, in physics and its concept of the hardness

of substance (compare Moser's scale of hardness)rebut , since the int roduct ion

of the absolute zero point, no longer true for the concept of t e m p e r a t u r e - - o r

in physiology and its estimation of the intensity or degree of similar sense im-

pressions, or in the field of economics and its numerous value determinat ions,

or in aesthetics and its comparisonsmwe are able to range in a certain order,

grade, stage, sequence the objects falling under a concept, without being able

to measure them in absolute terms, long before we can represent them as being

3 0 0 SCHRIDDER'S LECTURE IX

equal to a "number" of a measur ing unit. Being "comparable," they are by far

not "measurable" quan t i t i e su"quant i t i e s" in the true sense of the word. If we

take, for example, two firm substances, it is always possible to de t e rmine which

one of them can be called "harder"; nevertheless, we cannot connec t any concept

to the assertion of double hardness, of "being twice as hard."

The same is true for the ordinal number , as the original and s impler concept ,

and the cardinal number , as the derived and less simple concept .

So much about Dedekind 's essay in general .

Enter ing into a revision of our theory from the s tandpoin t of that theory, I

must now distinguish three parts of Dedekind 's essay, which I also must keep

strictly apart in the discussion.

The "first part" consists of Dedekind 's w 1, subtit led "Systems of Elements," compris ing ~ 1 to ~ 2 0 , be it definitions or propositions.

This first part is essentially only an exposit ion of the most elementary (but for

the au thor essential) concepts and proposi t ions of the identity calculus as a cal-

Page 351 culus with ("classes" or) "Gebieten"; this last word- -which I mainly use in volume

l u i s replaced by "systems" in Dedekind; I also prefer it here in volume 3.

The d~nitions introduce: the "system" of "elements" (our "individuals" of the

first universe of discourse); fu r thermore , the relations of inclusion [Einordnung] or c o n t a i n m e n t [Enthaltensein] of the "part"[Teil] in the "whold'[Ganze], as well

as of equality, and the subordina t ion [Unterordnung] of the "proper" part [echter Teil] to the whole; finally, the identity p roduc t (called the "commonal i ty" [Ge- meinheit] in Dedekind) and the identity sum of two or more systems (Dedekind

calls the latter the "compound" system consisting of them) , c is called "common part" [ Gemeinteil] of a and b, if c=(c--a and c=Ec-b; likewise i is called "common" e l emen t

of a and b, if i=(=a and i=(c-:b ( ~ 17).

The propositions are the well-known and very close corollaries of those which

we are already enti t led to use not only for ( ~ ' s ) "systems," but also for binary relatives.

This first part can thus be omi t t ed - - fo r us; instead of drawing on its prop-

ositions, we can get by with the most familiar results of the identity calculus. We

may see in the omission of this part (and its 20 proposi t ions) a small

" s h o r t c u t ' m a t least as such~to Dedekind 's long series of inferences in a dis-

cipline which is not dry, but ra ther rich in its applications. Otherwise one canno t

say from my voluminous volume 1 that a shortcut has been achieved.

We have fur ther to ment ion that the scheme of the subsumpt ion inference,

s t ipulated as "principle II" in our volume 1, is stated as a "proposi t ion" with a

"proof ' in ~ 7 . Within the mean ing of the author, who generally uses "argu-

men ta t ion regarding individuals (elements) ," this is v a l i d E b u t we are also en-

titled to present the scheme from our point of depar tu re loc. cit. as unprovable .

A (real) duality has to run th rough the first part. But this is nowhere clearly

expressed, probably because of the circumstance m e n t i o n e d (that he argues

cont inual ly with regard to individuals); and the dually co r respond ing proposi-

tion to .~9, for example, is not explicitly stated. But Dedekind at least documen t s

F R O M P E I R C E T O S K O L E M 3 o l

the duality by stating propositions as neighbors; they follow each o ther closely.

Whenever the opportunity presents itself, I will stress this duality with greater

clarity by juxtaposing the corresponding propositions or formulas, as I have

done so far.

I will give an opinion at a later opportunity regarding the question of whether

the "element" of Dedekind, presented as a conceptual thing, is conceived too

broadly and needs to be limited conceptually.

Page 352 The "second part" consists of ~ 2 2 up to and including ~ 2 4 , subtitled as

"~ w 2 "Mapping of a System," and further the "Mapping of a System to Itself entitled

w 4: ~36- -~63 .

It culminates in the statement and the proof of the propositions ~ 5 9 , 60,

which constitute "the scientific basis of the proof by mathematical induction." This

part alone, containing the "theory of chains," will interest us first and exclusively.

The "third part" constitutes the rest of the essay, namely ~ 2 1 and "~25 of

~ w 2, ~ w 3 from ~26-35 , and finally ~ w 5-14 from ~64-167 . In addition

to its great importance and its essential function, which is also characteristic of

the second part, this third part constitutes the gist of the whole essay. We shall

nevertheless deal with it much later and not incorporate it to its full extent.

That part begins there and contains all the propositions where for Dedekind's

"mappings" single-valuedness [Eindeutigkeit] is required as an essential condition, in

other words, is truly indispensable for the validity of tile propositions.

It is especially remarkable that the whole "second part" of Dedekind 's essay is

independent of this (imposed) condition.

The propositions contained in this second part are not only valid for the relatives called "systems" by Dedekind and by us, they are also valid for relatives in general; they are not only valid for Dedekind's "single-valued" assignment, unders tood as

"mapping," but they remain fillly valid when one uses the term "mapping" in

the broadest sense of which it seems able, namely what is unders tood by an

assignment occasionally also "multivalued," not the least an assignment possibly also remaining undone (not occurring, "un-valued")--in which case the term is syn-

onymous with the general concept of a (binary) relative. The "second part" of Dedekind's essay has thus a far broader scope, a much

farther range, than the author himself attributed to it. This fact will be clearly

shown; it gives a credit to Dedekind's theory of chains because of its possible

incorporat ion in the algebra of relatives.

Even Dedekind's proofs of his relevant propositions can be maintained mostly

or at least in their main traits; they will have to be modified slightly once in a

while--because one has to eliminate argumentat ion regarding "elements"

(which would only prove a proposition for "systems").

Concerning the relationship of the terminology used here to that of Dedekind,

in this I try to follow the latter as closely as possible. I will therefore keep ~ ' s

expression "image of -" ["Bild von -"] (with respect to a given assignment rule

Page 353 a [Zuordnungsprinzip], which Dedekind almost never mentions), a l though it has

to be used with the broader meaning ment ioned above.


Nevertheless, some small deviations will be unavoidable. We shall never omit

the mapp ing rule [Abbildungsprinzip] in the formulas, and will suppress it less often in the text; thus we will have to say for Dedekind 's

b ' - "the image of b": a;b = "the a-image of b,"

which means the same as "(an) a of b," where a, like b, represents an arbitrarily chosen binary relative.

By analogy, we have to say for Dedekind 's

b 0 - "the chain of b": a0; b = "the a-chain of b."

By a 0 = "the a-chain" or "the chain of a" we unders tand someth ing essentially

different from what Dedekind would have to mean, namely a certain relative

(derived from a), which does not at all explicitly appear in Dedekind ' s chain

theorymsimilar ly by b0. And therefore we translate Dedekind 's

b 0 = "the image chain of b" by "a00" b = "the a-image chain of b,"

where a00 --- "the a-image chain-" for us = "the image chain of a-" has again an

i n d e p e n d e n t mean ing as a certain relative derived alone from a.

The reader now holds the key to translate one presenta t ion of the chain theory

to the other.

As can be seen, our me thod of notat ion is more expressive. This is general ly

only bough t at the expense of length or de t a i lmand often degenera tes into

pedantry. However, the p rocedure also has its advantages; in particular, we have

to admit that the small sacrifice to grea ter detail is outweighed by o the r advan-

tages; our presentat ion of the chain theory is second to none with respect to

c la r i tymnot even the originator 's , a master of precision and conciseness.

After these prel iminary remarks, necessary in research l i terature to elucidate

the cont inuous transition from one theory to another, we could now begin in medias res, if I did not think it necessary to raise one more point in advance.

While there is, as ment ioned , a real duality in the "first part" of Dedekind ' s

essay, there is also a duality in the second part; however, I would call it a pseudo

or mock duality. This is apparen t in the fact that the proposi t ions referr ing to

Page 354 images or chains of sums and of products appear in pairs with analogous wording and follow immediately in ~ . While one proposi t ion of such a pair states an

equation, the o ther conspicuously only states an inclusion and only passes into

an equat ion when the mapping, considered singled-valued [gegenseitig eindeu-

tig]by Dedekind, is taken as invertible [eindeutig]. I will separate such "pseudo-

dual" proposi t ions with a double strike.

I consider the elucidation of this mock duality as a fur ther gain which springs

out of our r ev i s ion~a l though it responds merely to an aesthetic r equ i r emen t

of the intellect (which cannot possibly be satisfied with the startling inaccuracy m e n t i o n e d before) .

Fu r the rmore , the merit of the real duality will be revealed, and this, toge ther

Page 355

F R O M P E I R C E T O S K O L E M 3o3

with the conjugation rule, quadruples with one stroke the recognition of the richness of Dedekind's theory.

I would like to begin simply by giving a overview in p r o p e r sequence of all definit ions and proposi t ions const i tut ing Dedek ind ' s chain theory, in the nota t ional language o f our algebra and wi thout commentary .

Only the last of these propositions, ~63, is omitted for the time being because Dedekind added it as an appendix without proof and without ever using it.

The letters of the lowercase latin a lphabet signify any binary relative, and all formulas have universal validity in the algebra. I g roup them according to opportuni ty:

~ ,~22. ~ ~ ~36. ,~37.

~ ,~38. ~ ,~39.

'~40. '~41.

"~42. ,~43.

,~44.

~ 4 5 . ~ 4 6 . ~ 4 7 . ,~48.

• , ~50 .

.~51.

(b:~--c) @ (a ; b:~--a ; c).

a ; ( b + c + ' " ) = a ; b + a ; c + ' " . [[ a ; b c ' " : ~ - - a ; b " a ; c ' " . Def. (a;b:~--b) = (a maps b to itself). Def. (a;b:~--b) = (b is a "chain" with respect to a). a; 1 :(= 1. (a; b:~--b) :(= (a ; a; b:~--a;b).

1 o)

(a; c + b @ c ) ~: (a ; b:~--c). {a; (b+ c) :(=c} :(= E (a; u:~--cu)(b:~--u) = E ( a ; u + b:~--u)(a; u:~--c).

u u

(a ; b :~--b)(a ; c @c) "" ~ {a ; (b + c + "" ") :(c--b + c + "" "}.

(a; b @ b ) ( a ; c:[~--c) "" :~-- (a; b c ' " :~-- b c ' " ) . 2 o )

3 o )

Definit ion of ao;b , the "a-chain of b":

ao ; b = II u = II (u + a; u + b:(=u). u,(a ; u + b ~ u) u

b:~-ao ; b.

a; ao ; b:~-ao ; b. (a; c + b:~--c) :~--(a o ; b:~--c). ( a ; x + b ~ - x ) II {(a; u + b @ u ) =(= (x:(=u)} = (x = a 0 ; b).

u

a; b=(=a; a0; b. a ; b:~--ao ; b. ( a ; b :(=b) = (a 0 ; b = b).

".~52. ",~53.

~ ,~55 .

x~56 .

( a :(~-- c) :~-- ( b :~- a o ;c). (b ~--a o ;c) :~--(ao ; b:I~--a(, ;c). (b:~--c) :(=(a 0 ;b:~-a, , ;c). (b :(=a 0 ; c) :(=(a; b:~--a o ; c). (b ~:a,, ; c) :(= (a ; a 0 ; b @ a ; a o ; c).

4 o )

~ 5 7 . T h e o r e m and Definition: a; a 0 ; b = ao ; a; b (= a00; b def ined) .

304 SCHRI~DER'S LECTURE IX

.~58. ,~59.

",~60. ~ .~61. ~

ao ; b = b + aoo ; b. T h e o r e m of ma themat i ca l induct ion:

[a; (a,, ; b)c + b=~--c} =(c--(a,, ; b=r Explana t ion of the last as s u c h ~ s e e later.

ao; (b+ c + ' " ) = a o ; b + a o ; c + " ' . [I a,, ; bc"" =~-- a,, ; b " a,, ; c ' " .

5 o )

P ropos i t ion ,~51 is only given as p re l iminary subsumpt ion .

These thirty n u m b e r e d propos i t ions form, t oge the r with thei r proofs, the c o n t e n t of Dedek ind ' s theory of chains.

We will have to get to know this theory thoroughly ; thus we will discuss the p ropos i t ions at leisure, so to speak.

A l t h o u g h some of these propos i t ions may be used occasional ly later, the "main purposg' of stating them, and for us here the only p u r p o s e of listing t h e m - - n e v e r to be lost sight of, is to p r e p a r e and make possible the proof of the theorem of mathematical induction ~ 5 9 , which contains no circular argument; that is to say, where the p r o o f p r o c e d u r e for i nduc t ion i t s e l f - -no t the least for a "def ini t ion by i n d u c t i o n " - - i s never used on the way.

It goes to Mr. Dedekind's credit to be the first to have stripped the proof

procedure, widely used and known by the name of "inference from n to n + 1," of its arithmetic additions, to have peeled out its logical core, and to have formulated the "proposition of mathematical induction" as a proposition of general logic,

Page 356 which can be represented and understood independent of any number concepts and even before the series of numbers is introduced.

Mr. Dedekind may also claim credit for proving for the first time this proposition correctly and within the rigorous requirements above ment ionedmand indeed in a wonderfully elegant way!

This proof will not lose its value even if we later succeed in simplifying it considerably.

T h e p ropos i t ion is writ ten, as we see, by means of only nine letters.

With the excep t ion of those signs which we know f rom the genera l

theory of relatives, only one special sign is used a 0, and this always in

the re la t ion a0 ; b---a sign which is pecul iar to this special b r a n c h of ou r discipline, the "chain theory."

To establish p ropos i t ion ",~59, a def in i t ion of a0, or, at the same time, ao;b, will have to precede ; and this de f i n i t i onmgive n by D e d e k i n d with ",~44 (or also in the form of , ~ 4 8 ) m r e p r e s e n t s the punc tum saliens, the s tar t ing point , of the whole theory. The cons t ruc t ion of the chain theory d e p e n d s considerably on this choice.

Before talking abou t the difficulties which have to be o v e r c o m e with this def in i t ion (restr icted only by the main pu rpose we have in view), I would like to weave in some remarks which a im- -ac tua l ly f rom a pure ly

Page 357


externa l po in t of v i ewBa t reduc ing (by m o r e than half) the system of propos i t ions that will be cons ide red later.

With the definition ~36 terminology is introduced which Dedekind himself does not use in the chain theory, although it appears frequently in later parts of his essay.

We may have to question whether this terminology ought to be kept as suf-

ficiently adequate when by b, not necessarily a "system" but a relative is understood, just as when by a, not only a "single-valued mapping" but also a general relative is understood. Would it perhaps not be better to say: "a embeds b to itself or "maps b to itself" and so on (instead of "ab")? l

In any case, we can refrain from using such terminology and ignore definition "~36 in the following.

If we also postpone the explanation for "~60, only 28 propositions remain out of 30.

F rom the r e m a i n i n g proposi t ions, it seems that e ight can be omi t ted ,

which I have m a r k e d with a small circle: ,~22, 23, 24, 38, 39, 54, 61,

62. These can be u n d e r s t o o d f rom the genera l propos i t ions of our a lgebra

1), 4), and 5) of w 6; that is to say, they c o r r e s p o n d exactly to the three propos i t ions of Peirce, or are only special cases, par t icu lar appl icat ions, of t h e m m ' , ~ 5 4 is based on the assumpt ion that a 0 finds or has found its def ini t ion as a binary relative. If this assumpt ion refers only to "a0 ; b," we have to keep the last three of the e ight proposi t ions . But the first five do not deserve to be n u m b e r e d and m e n t i o n e d as special

proposi t ions .

Or only for the purpose of getting acquainted with them in their verbal form--by practicing the expression "a-image o f - , .... chain with respect to a"

(~37), and "a-chain of - . " In this sense, we may note:

"~22. If b is con ta ined in c, then the a-image of b is also c o n t a i n e d in the a-image of c, or

The image of a part is a part of the image of the whole.

'~23.

~ 2 4 .

'.~38.

The a-image of a sum is equal to the sum of the a-images of its summands. The a-image of a product is contained in the product (is a " c o m m o n

part") of the a-images of its factors--sum and p r o d u c t are, of course, always u n d e r s t o o d as identity sum and product .

The universe of discourse 12 is a chain with respect to each relative ( a B t o itself).

We already mentioned the known proposition a; 1 :(= 1 ; according to 1) of w

Translator's note: Schr6der's objection is that "ab" suggests an image for each element of b.

Page 358

3o6 SCHRODER'S LECTURE IX

6 it can also be inferred from a =6= 1 in the form of a" 1 :(:: 1 �9 1 according to the abacus.

",~39. The a-image of a chain with respect to a is a chain with respect to a.

This stands to reason when the conclusion is drawn in the form of a; (a;b) =(~--a;b from the premiss and is compared to schema ~37.

.~54. The a-chain of a part is a part of the a-chain of the whole.

.~61. The a-chain of a sum is equal to the sum of the a-chains of its summands.

.~62. The a-chain of a product is a common part of the a-chains of its factors.

From the same point of view, ,~56 also appears as too obvious an in fe rence f rom ,~53 to be stated in a separa te proposi t ion. A similar

po in t ( m e n t i o n e d previously in a d i f ferent context) is t rue for ~ 5 4 ,

which seems to be given afortiori with ".~52 and 53. W h e n e v e r we talk abou t the complex of formulas of the chain theory, the r e a d e r should

cons ider the three lines of g roup (4~ con ta in ing the five propos i t ions

.~52- ,~56 , as taken away and replaced by the two lines of the following group:

"~52, 53. (b@c) =(c--(b:~-ao ; c) :(=(a o ; b@ao ; c)=(=(a ; a,, ; b @ a ; a o ; c). ~ 5 5 . (b@ao;c) : (=(a ;b@a, , ;c ) .

60 )

- - w h i c h contains only three proposi t ions, since of the th ree proposi- t ional subsumpt ions in the first line only the first two need to be proved.

Finally, also ,~49 as m e r e corollary to ",~45 can be e l iminated .

These changes are only minor, but contribute to increase the beauty of the theory; the excessively large number of small propositions is almost confusing.

Now I will tackle the "mock-duality" of which I spoke in the in t roduct ion .

It is visible in six propositions--first, in the formulas of the second and last line of our list--recognizably so, if only they are expressed in Dedekind's mode (or any other adopted to it). When the formulas are written

'~23. ( b + c + ' " ) ' = b ' + c ' + ' " , []"~24. ( b c ' " ) ' ~ b ' c ' " ' ,

,~61. ( b + c + " ' ) 0 = b 0 + c 0 + ' ' ' , ]] '~62. ( b c " ' ) o @ b o c o ' " , then the juxtaposed formulas indeed appear to have to be dual to each other

(with reference to the unintendedly, yet unqualifiedly maintained "image" and

"chain" concepts, respectively, represented by the prime accent and suffix 0,

respectively, in Dedekind). Inconsistent with such a dual correspondence is the difference of the relational signs, that is to say, signs of equality on the left, and signs of inclusion on the right. The "paradox," if I may say so, of this pseudo- duality stems only from an inadequate notation and can be clarified as soon as the propositions are made sufficiently "expressive"--as we did in (1 ~ and (5o).

Then we see immediatelymwhat could, for example, not be seen with the prime

FROM PEIRCE TO SKOLEM 3o7

s ignmthat the concept "a-image o f - " like the "a-chain o f - " is not at all a dual to itself, and that the juxtaposed (as "pseudo-dual ') propositions originate from

very different formula groups of the general theory; the one on the left from group 4), the one on the fight from group 5) of w 6, the latter necessarily being

Page 359 governed by signs of subsumptions instead of signs of equality. Similarly, propositions ~ 4 2 and 43 may only be called pseudo-dual, although

both have the same sign of subsumption.

If we wan t to a r r a n g e the cha in t h eo ry in its s imples t fo rm, whi le

a d h e r i n g closely to D e d e k i n d , a n d fit it in to o u r g e n e r a l d isc ip l ine , we

m a i n t a i n o r r e m e m b e r only the de f in i t ion o f .~37 f r o m the who le g r o u p

(1~ which , incidental ly , we i n c o r p o r a t e d in w 22 u n d e r 5) in to o u r

disc ipl ine . Now we will have to deal o n c e a n d for all with the f o u r p r o p o s i t i o n s

o f g r o u p (2 ~ which p r e c e d e the " p u n c t u m saliens" ".~44.

First, I would like to take propositions ~ 4 2 and 43, because they are elementary and of general interest, whereas ~ 4 0 and 41 are only justified by their

use as lemmas in later proofs; they are also immediately connected to the ob-

servations which begin with ~ 4 4 and which serve to support their purpose, thus

showing their relatedness.

'~42 , 43 state: The sum, respect ively the product, o f chains with r e spec t

to a relat ive a is a chain with r e spec t to a.

In order to prove their formulas in (2~ one has only to combine the premises

by addition or multiplication; the result is

a ' b + a ' c + " " = ( e b + c + ' " , resp.

and then use the schema 4) or 5) of w 6:

a" b " a" c ' " =g~-- b " c ' " ,

a ; b + a ; c + "" = a ; ( b + c + " ' ) , resp. a;bc"" ~- a;b" a ; c ' " .

The conclusion of the left proposition is then more ofa pariter(i.e., an equivalent

transformation); the conclusion of the right proposition is gained a fortiori.

The two propositions could be combined and, at the same time, be gener-

alized to: The result of any combination (by means of identity operations) of any chain

with respect to one and same relative a, is again a chain with respect to a.

If, for example, b, c, d, e, f a r e chains with respect to a, so is bc + def, as well

as (b + ccl)e + f, etc., again are such a chain. The proposition would even be valid

without the words in parentheses.

'~40. (a; c:~-c)(b:~--c) :(=(a; b:~--c)

means : The image of a chain par t is a par t o f this chain, m o r e precisely: the

a- image a ; b o f the pa r t b o f a cha in c with r e spec t to a is c o n t a i n e d in

this cha in c.

The proof is given by relatively premultiplying on both sides by a the second

3o8 SCHR(~DER'S LECTURE IX

Page 360 of the two partial premises into which the given premiss is split, according to 1) of w 6: a,b=~-a; c, and putting it together with the first partial premiss in

order to apply the subsumption inference.

'~41. (a ; c~-c) (a ; b:~--c) ~-- ]2 (a; u~--u)(b:~--u)(a ; u~--c) u

m a s the p r o p o s i t i o n reads af ter spl i t t ing the s t a t e m e n t s on b o t h sides,

a n d means :

I f the a-image of b is a part of a chain c with respect to a, then there is a chain u ( a chain with respect to a), such that it contains b in itself and its a- image is contained in c; in o t h e r words, b is a pa r t o f u a n d its a- image is

pa r t o f c.

The proof (which may begin with the words: "Since it is ...") can also be

expressed as a continuation of the proposition: "Namely" u = b + c is such a chain.

It fulfills indeed the three conditions of the previously divided assertion, that is to say, the third by virtue of the still undivided premiss of 2~ the second

identically because b =(= b + c, and the first afortiori considering the third plus

c =(v-- b + c.

T h e fou r p ro p o s i t i o n s e x p l a i n e d he rewi th can still be c o u n t e d a m o n g

the "introduction" of the cha in theory, which now really begins---with the

punctum saliens " ,~44--and, if o n e wants, c o n c l u d e s a l ready with ".~59.

T h e r e r e m a i n to be s tud ied 1 4 - - I say four teen!- -de f in i t ions or p r o p o - si t ions (or 15, if o n e coun t s the coro l la ry de f in i t i on to p r o p o s i t i o n ~ 5 7

separa te ly ) , which is no t m o r e than ha l f a page. We will now look at this series of p r o p o s i t i o n s from two very different

standpoints---one cou ld say with a lmos t p e r f e c t prec is ion: we c o n s i d e r

t h e m in two d i f f e ren t d i rec t ions , to a n d from, or "forward" a n d "backward." T h e "backward" pa th is far s h o r t e r and , f u r t h e r m o r e , easier, especial ly

s ince it does n o t have to be ru n t h r o u g h comple te ly ; the p r o p o s i t i o n s

o f g r o u p 2~ m e n t i o n e d in the " in t roduc t ion , " can a l ready be o m i t t e d .

For d idact ic reasons , it s eems best to us to take this pa th first. O n the o t h e r h a n d , only the "forward" pa th is cons i s t en t with the m a i n

p u r p o s e of the cha in theory!

In addition, Mr. Dedekind also recommended to his readers to take the

backward pa thmpage 40 under ~ 131.

Thereby one would have to start with the result ,~58, which can be written

as follows when considering the preceding definition "~57:

a o" b = b+ a" a o' b.

Page 361 It then leads to the following "infinite" (unlimited) expansion, when one writes

on the right side continually the value of left side of the equation"

ao'b = b + a ' b + a ' a ' b + a ' a ' a ' b + " "

- (1 ' + a+ a ' a + a ' a ' a + "")"b. o)


It is suggestive to define the relative in parentheses as a 0.

We shall considerably simplify this procedure when we start, not with the

examination of ao;b, but with the immediate introduction of a 0 (and a00).

T h e backward pa th . T h e fo l lowing modus procendi will lead us to the goal , if we m e r e l y

wan t to prove the f o r m u l a .~59 as a g e n e r a l p r o p o s i t i o n in the t h e o r y

o f b ina ry relatives, ignoring its significance as the "theorem o f m a t h e m a t i c a l i n d u c t i o n , " a n d t h e r e b y n o t hes i ta te to state its f o r m a t i o n law by m e a n s

o f the "proof by m a t h e m a t i c a l i nduc t i on , " w h e n inf in i te e x p a n s i o n s oc- ( : u r - - t o p r o c e e d as we always d id in p rev ious l ec tu res w h e n a so lu t ion

c o u l d n o t be given in c losed f o r m (and as we will do aga in in the fu tu re ,

a l t h o u g h t h e n with strict jus t i f i ca t ion) .

We o m i t the weighty p r o p o s i t i o n s ,~44 a n d 48 f r o m D e d e k i n d ' s cha in

t h e o r y (to ob t a in t h e m f r o m the p r e s e n t p o i n t o f d e p a r t u r e , r e fe r to

the n e x t sec t ions) a n d de f ine the "a-image chain" aoo a n d the "a-chain"

a 0 o r "cha in o f an a rb i t r a ry relat ive a" as follows:

a o o = a + a ; a + a ; a ; a + ' " , al l = a(aJ- a)(aJ- aJ- a ) ' " , 1)

= 1' O' 2) a o + aoo, al = al l .

Now the fo l lowing is natural"

1' :(= a o, a I ~ 0', 3)

a :(= a(,(, :~- a,,, a 1 ~ al , :(= a , 4)

and , f u r t h e r m o r e , the fo l lowing are valid:

a ; a o o = a o o ; a o o = a o o ; a : ( = a o o , al l : (=a J-a l l = a l l c t a l l = a l l , a , 5)

a ; a o = aoo = a o ; a :(= a o, a 1 :(= a ~ a I = al l = a I ~ a , 6)

Page 362

a o ; a o = ao, al d" al = a l , 7)

as can be easily seen, by subs t i tu t ing the series set d o w n for aoo a n d

ao a n d t h e n app ly ing relat ive mul t ip l i ca t ion .

We have, for example,

a'aoo = a ' ( a + a ' a + a ' a ' a + ' " ) - a ' a + a ' a ' a + ' " ,

aoo" aoo = (a + a" a + a" a" a + "") " (a + a" a + a" a" a + "") = a ' a + a ' a ' a + a ' a ' a ' a + ' " ,

its terms are incrementally obtained by tautological repetitions because every

3 1 0 SCHR~)DER'S LECTURE IX

term of one series has to be j o ined with every term of the other. In both cases,

the sum results from the first term of the terms adding up to at0 (which could

also be called a000, and so on).

With 6), we also have

a ' a o = a ' ( l ' + a + a ' a + ' " ) - a + a ' a + a ' a ' a + " " =aoo,

because a" 1'= a.

To prove 7), it is not necessary to make the analogous considerat ion again,

but one can refer the proposi t ion back to 5), without opera t ing again with the

infinite ser iesmas follows:

�9 ~ �9 , , 1 / I at at ( l ' + a o o ) ( l ' + a o o ) =1' l ' + a o o + 1 'aoo + a o o ' a o o

= 1 / + aoo + aoo" aoo = 1 ~ + aoo = a t ,

because the third term of the last line is absorbed by the previous term due to

the subsumpt ion aoo'aoo ~ aoo proved with 5).

One should not unders tand aoo as the "chain of at," that is to say, (at) o, which

would be false. The following proposi t ions anticipate such misunders tandings:

(a0) = a,,, (al) l = al,

(a~,,,) 0 = a,, = (a0)00, (all) 1 = a I = (al)ll , 8)

(a~,,,) 0,, = a,,,, (all) l~ = a l l"

Thus, the "chain of the chain ofa" is no th ing less than the "chain ofa." It would

be silly to represent it with a double suffix 00 which we want to keep free for

o the r notat ional use . raThe reader can easily under s t and proposi t ions 8) from

7) and 5). It must, for example, be

(at) o = l' + a o + ao" a o + a o" a o" ao + ''"

= l ' + ( l ' + a 0 0 ) + a 0 + a 0 + ' ' ' = a 0.

The operat ions for taking chains or image chains can always be immediate ly

executed with an image chain or chain.

To p r o v e .~59, we n o w n e e d o n l y t h e three p r o p o s i t i o n s .~45 , ,~55 ,

a n d ",~47.

~ 4 5 m e a n s �9 each relative b is par t o f the a-chain o f i tsel f a n d is u n d e r s t o o d

f r o m t h e p r o p o s i t i o n "

b =~--ao ; b, am d" b =~-b,

b=~--b ; a o, b d- al =~--b , 9)

Page 363 f r o m 3) , obviously. By re la t ive m u l t i p l i c a t i o n wi th b o n b o t h s ides, we

o b t a i n

l ' ; b = ( = a 0 ; b t h e r e f o r e b:~--ao; b, q.e .d .

L ikewise , c o n s i d e r i n g 6) a n d 4) , we have

F R O M P E I R C E T O S K O L E M 3 a l

(b~-a, , ;c) = ( = ( a ; b ~ a ; a , , ; c = a,, o ; c ~ a o ;c),

a n d with it have p r o v e d .~55, o r the p r o p o s i t i o n :

{ (b~--a~ ( a l j ' c ~ - - b ) = ~ - - ( a l j c a ~ - a j ' b ) ' 10)

( b =~- c ; oF) =(= (b ; a a~- c ; a o ) , ( c j- a I a~- b ) a~-- ( c j- a I a~- b d- a ) ,

t h a t is, the a-image of a part b of the a-chain of c is also a part o f the a-chain

ofc. Finally, t he p r o p o s i t i o n ,~47 b e l o n g s to the g r o u p :

( a ; c + b : ~ - c ) ~ - ( a o ; b @ c ) , {c:~-(a~c)b}:~-(c:g~--alj-b), 11)

(b + c; a:~-c) :~-(b; a o =(~--c), {c:~--b(cj- a)} =~-(c:~-bd- al) ,

a n d its p r e m i s s spli ts i n to

(a; c=~--c)(b ~-c).

It thus m e a n s : I f b is part of a chain c with respect to a, then the a-chain of b is also part of this chain c. For proof we d e d u c e f r o m the s e c o n d p remiss ,

with a c o n s t a n t view to the first, the inf in i te ser ies o f i n f e r ences :

b=~--c, a; b a~-a ; c, t h e r e f o r e

a; b ~--c, a; a; b=~--a ; c, t h e r e f o r e

a ; a ; b~--c, a; a ; a ; b=~c--a ; c, t h e r e f o r e

a;a;a;b=~--c , a n d so on .

S u b s e q u e n t a d d i n g o f t he l e f t - h a n d i n f e r e n c e s g i v e s - - c o n s i d e r i n g b =

1' ; / u - t h e c o n c l u s i o n a o ; b ~ - c . * We can see t h a t with t he "and so on" t he "inference f rom n to n + 1" has

b e e n m a d e , w h i c h is in fact absolutely necessary.

In o rde r to make it very clear, we only have to assume that a";b=gr be proved,

based on the premisses of our propositions, for a definite nmwhich is already

the case for n = 1 and 2rowe then show by means of the inferences:

a;a";b=(~-a;c=g~--c

that also a"+l;b=g~--c has to be valid. Now the inference a"; b=g~-c is valid for each Page 364 n u m b e r n, because, if it is valid for a definite number, it has to be valid for the

next higher one; and it is valid for n = 1 (consequent ly also for n = 2, and again

n = 3 and so on in infinitum). The above proof rests on these partial in fe rences .m

Incidentally, one can also give a proof of the whole proposi t ion in one stroke

by means of an infinite series of equivalent proposi t ional t ransformations.

For that purpose we write the schema of' .~40 "more completely" in the following

way:

* T h e i n f e r e n c e a;b ~ c is a r epe t i t ion of the o n e m a d e with 3 4 0 , a l t h o u g h it does no t have to be fo rmal ized as a p ropos i t i on here .

312 SCHRC)DER'S LECTURE IX

(b+ a;c=~-c) --{b+ a ; ( b + c) =(=c} = ( b + a ; b + a;c =(~-c). 12)

The hypothesis of the proposi t ion in the thesis is m e n t i o n e d once more, the

premiss of the assertion is repeated, and the two are j o i n e d ~ w h i c h is permi t ted

according to the principium identitatis of the proposi t ional calculus. The prop-

ositional subsumpt ion passes into a proposi t ional equivalence or equat ion be-

cause the inference from the premiss is admissible "backwards" for the assertion,

since the latter includes the former.

This remark, based on which

( a ; c + b~(~--c) =(={a; (b+ c) ~-c} 13)

must also be obvious, makes a bridge from ~ 4 0 to ",~41, whose thesis must also

be a conclusion of the hypothesis of ,~40.

If we observe that the third s ta tement in 12) has the same form as the first,

the only difference being that the term b + a; b acts in place of the term b in

the first, then we see that proposi t ion 12) gives us the right to rewrite its state-

ments equivalently, for as long as we want to, by replacing b by b + a;b.

Now we only have to "observe" that the effect of such a rep lacement , if it is

consistently done with b, is exactly the same as when done only with the last b;

thus we easily gain the following as equivalent to proposi t ion 12):

(b + a; b + aZ;b+ a; c=(~--c) - - (b+ a; b + a2 ; b + a:~;b + a; c=(~-c) . . . . .

which, with the permissible suppression of the term a; c m t h a t is to say, the omis-

sion of the proposi t ional factor (a ;c .~- -c )~presen t s the conclusions const i tut ing

our proposi t ion ~ 4 7 when ex tended to an infinite set of terms.

To justify this "observation" in general , the following proof suffices: if

then

(1' + a + a 2 + a :~ + "'" + a n ) ' b = f " ( b ) ,

f " (b + a; b) =f"+'(b)

must h o l d m w h e r e finally f=(b) = ao;b. This proof is subject to no difficulties.

Signifying by f l ( b ) =fib) = b + a ; b = ( l ' + a ) ;b here and not ing that (1' +

a)"; (1' + a) = (1' + a) "+l must hold- -cf , the "inductive" or "recursive" definit ion

Page 365 of power, given in w 13 by 8 ) m one can fur ther simplify the p roof by showing

by means of the inference from n to n + 1 that

(1' + a ) " = 1' + a + a 2 + a :~ + a 4 + " " + a" 14)

must hold. If this is true for a definite n, then one obtains by relatively mult iplying

both sides with 1' + a on the right:

(1' + a) "+l = 1' + a + "'" + a " + a "+1,

with a tautological repeti t ion of all terms between the first and the last.

Page 366


Then also

is ob ta ined .m

3x3

a,, = (1 '+ a) = 15)

As l o n g as the p r o o f by i n d u c t i o n ob ta ins its j u s t i f i ca t ion on ly f r o m p r o p o s i t i o n ,~59, wh ich o n its pa r t is only d u e to the p r o o f o f ,~47, the

p r o o f o f ~ 5 9 (with wh ich we will dea l later) is a circle; its on ly va lue is

to give us the c o n f i r m a t i o n : if this i nduc t ive i n f e r e n c e is m a t h e m a t i c a l l y

j u s t i f i ed in this one c a s e m a t least in . ~ 4 7 m i t can c la im f o r m a l o r general validity for e ach case o f its app l i ca t ion .

Thus all o ther propositions of our survey are dealt with from this point of

view. ~46--- to be written in the simpler form of aoo;b=~-ao;b---is unders tood

from 4); likewise ~ 4 9 from a;b,~-aoo;b, and ~50 ; ~ 5 7 from 6), ~ 5 8 from 2).

~ 5 2 is unders tood from the consideration: (b=g~-c) ~-(ao;b=(~ao;C) because ~ 4 5 or 9) afortiori, and therefore also ~ 5 3 by means of

(b =(= a,, ;c) =(= (a,, ;b =(= a 0 ;a,, ;c = a 0 ;c) from 7).

It only remains to prove equation ~ 5 as a forward and backward subsumption.

Because of (b =(= b) = 1, we have

(a; b=(~--b) = (a; b =~- b)(b =(= b) = (a; b + b =(= b) =(= (a 0 ; b ~= b)

Dacco rd ing to ~47 , taking c = b--in case one does not want to repeat entirely

similar inferences there. According to ~ 4 5 or 9), the converse subsumption is

in any case valid, the equation stated in ~51 , right side, is proved from the

premiss, left side. Conversely, the latter follows from the former; because of .~50 or 4) a; b=(=

ao; b=g~-b has to hold. So easy was it to take the backward pa thDwhere even the inferences were a

luxury!

T h e proposition of mathematical induction ~ 5 9 b e l o n g s to t he g r o u p :

{a ; (a 0 ;b)c + b ~ c} =(=(a 0 �9 b ~= c), ~ [c =(= {a ~ (a, 0 ~ b + c)}b] =(= (c =(= a, ct b),

{b+c(b;a~,)'a~-c}=g~-(b'ao=g~--c), [ [c=(~--b{(c+ b~a~,)~a}] ~ (c=(~-b~ al).

16)

Its p r o o f is to be e s t ab l i shed as follows, given tha t its p r e m i s s splits

in to

I. a ; (a0 ;b)c ~-- c a n d II. b @ c .

Proof . b~-ao;b, by ",~45 o r 9); this, c o m b i n e d with p r e m i s s II, gives the

i n f e r e n c e

III. b @ (a,, ; b)c.


On the other hand, we have

IV. (a 0;b)c=~-a0;b,

which leads to, by .~55 or 1 0 ) ~ u s i n g b instead of c, and (a0 ; b)c instead of bm

a ; (a 0 ; b)c :~c-a o ; b,

and this, combined with premiss I, gives

V. a ; (a o ; b)c @ (a o ; b)c.

Uniting III and V results in

a; (a,, ; b)c + b @ (a 0 ; b)c.

This now falls under the schema of the premiss of ".~47 or 11 ) - -where only c has to be replaced by the compound expression (ao;b)c; taking this proposit ion as a model, it permits the inference:

ao ; b=~-(a o ; b)c.

Since the converse s u b s u m p t i o n m I V ~ i s admissible in any case, we have gained the equation

VI. ao ; b = (ao ; b)c,

from which, because (ao;b)c=~c--c, the conclusion follows:

ao ; b =~--c,

which had to be proved. If the formal foundations 9), 10), 11) of this proof are later gained

in a noncircular, respectively, unobject ionable way, the previous proof Page 367 is rigorous and need not be repeated.

The proposition ,~59~f i r s t viewed simply as a theorem about binary relat ives--can also be stated as follows:

To prove that the a-chain of a relative b is completely contained in a third relative c, one only needs to show two things, namely:

First, that b itself is contained in c; Second, that the a-image of each of the pairs of elements contained in c,

belonging to the a-chain of b, must be contained in c. In o ther words, a0; b must be a part of c if b is a part of c, and the a-

image of each common pair of elements of a o ; b and c is part of c.

[The a-image of the sum of all pairs of elements contained in the identity product (a0; b)c, the pairs constituting this relative in an additive waymthus a; (ao;b)cmis equal to the sum of the a4mages of all these pairs of elements.]

In the case that b and c later represent "systems," we can replace the words "pair of elements" by "element."

F R O M P E I R C E T O S K O L E M 3 x5

In s t ead o f ,~60, we would like to exp la in to the r e a d e r why proposition .~59 o r 16) in fact fo rms "the scientific foundation for the mode of proof known by the name of mathematical induction (the inference from n to n + 1) "

For tha t p u r p o s e it is only necessary to assign the fo l lowing interpretation to the un iverse o f d i scourse 1 a n d the le t te r relat ives a, b, c in the p r o p o s i t i o n .

T h e un iverse of d i scourse 11 is c o m p o s e d of the ind iv idua ls i , 2, 3,

4 . . . . o f the inf in i te number, series; we d is t inguish its first m e m b e r , o r the

number one, with the dot , 1, f rom the m o d u l e 1 of o u r theory.

In the universe of discourse 1 z or 1, each element of this number series can be represented as a relative, each row being a full row, when corresponding to

the latter, and all other rows being empty; each system of numbers will appear as the relative of which the row corresponding to the matrix of its elements is

a full row, the others being empty rows. We simply mention this fact to bring

it back to memory; it is not very important for the following argument .

T h e n u m b e r series has to be c o n s i d e r e d as well o r d e r e d , by m e a n s

Page 368 of a assignment-or mapping rule, which leads f rom o n e to the next , a n d

thus allows all n u m b e r s o r ig ina te f r o m the first a m o n g t h e m as the "base number" 1. This m a p p i n g rule is the relative"

a = "greater by i than -."

In o t h e r words , the n u m b e r s following, i a re to be t h o u g h t o f as

de f ined : the " n u m b e r " two as "g rea te r by 1 t han 1," which is

2 = i + i, a n d l i k e w i s e 3 = 2 + i , 4 = 3 + i . . . .

- - w h e r e b y the plus signs have to be. u n d e r s t o o d a r i thme t i ca l ly - - -o r in

the n o t a t i o n of o u r discipline" 2 = a ; 1, 3 = a ; 2, 4 = a ; 3 . . . . , tha t is, e ach n u m b e r is d e f i n e d as tha t "a-imagd' of its p r e d e c e s s o r s - - w i t h the ex-

c e p t i o n o f the base n u m b e r i itself, which does no t have any p r e d e c e s s o r

d u e to the previous ly i m p o s e d res t r ic t ion on o u r un ive r se o f d i scourse .

The matrix of this binary relative a is, incidentally, constructed as follows: its

first row is an empty row; in each following row there is one (and only one) filled

circle at the point which is left of (or below) the main diagonal, the closest grid point.

T h e whole n u m b e r series, o r number system, as the totality o r ident i ty

calculus (bu t no t a r i thmet ica l ! ) sum of all individual n u m b e r s , is repre-

s en t ed as the a-chain of the base number 1. In o t h e r words, it is the module

1 = a 0 " l .

In any case, the a-chain o f any given n u m b e r is n o t h i n g m o r e t han

the total i ty o f n u m b e r s b e l o n g i n g to the n u m b e r series, beginning with

the given n u m b e r ( i n c l u d i n g itself)"


a 0 ; i = i + ( i+ i ) + ( i + 2 ) + ( i + 3 ) + ' "

----on the o the r hand, the "a-image chain" of such a n u m b e r i,

a,, o ;i = (i + i) + (i + 2) + ( i + 3) +""

would e m b o d y the later n u m b e r s (excluding itself). This assumed, the relative b means the e l e m e n t 1 itself, there fore ,

b=i.

This interpretation may suffice at present for the most specialized application

of proof by induction which aims at finding a convincing reason for the following: if a proposition is valid for the number i, and if, as often as it is valid

for number n, it must also be valid for the next greater number n + i or a" n,

then it is actually valid for all numbers�9 In a formally more general application

of proof by induction, b can also stand for a greater number than 1, or actually even any (finite or infinite) "system" of humbert, for example, such a system which is constituted in any way (possibly also with omissions) of certain numbersmfrom

Page 369 and including m to, including, r. We will take such a case into consideration

later.

Thus ao;b represents again the ent i re n u m b e r series. Finally, the relative c means the totality, the "system" of numbers ,

which have the defini te proper ty ~, or, in o the r words, for which a def ini te proposi t ion ~ , con ta in ing an indefini te n u m b e r n, is valid.

In o r d e r to prove that the propos i t ion ~ is valid for all numbers , that

is, that ao;b:~--c, we simply show, accord ing to .~59: First, that the proposi t ion is valid for the n u m b e r b = i, that, conse-

quently, b ~=c; Second, that this proposi t ion must also be valid for the image a of

every n u m b e r (n) for which our proposi t ion ~ is a p p l i c a b l e E w h i c h e v e r

n u m b e r s appea r inc luded in the express ion ( a o ; b ) c n t h a t is, a ; (a0; b ) c ~ c must hold. In simple terms, this means that if a p ropos i t ion

is valid for a definite n u m b e r n, it must also be valid for the next higher n u m b e r n + 1.

On the o the r hand, i f n t o be even m o r e g e n e r a l E b represen ts a

def ini te n u m b e r m or any system of n u m b e r s which includes m as its

smallest number , then ao; b represents the totality of numbers beginning with m.

In o r d e r to proye that the propos i t ion ~ applies to all n u m b e r s in

this series m, m + 1, m + 2 . . . . . in infinitum, it suffices "first" to show that it is valid for the number s of system b; actually, it is only necessary to show its validity for the n u m b e r (b =) m, and "second" etc. (same word ing as before) . [The proposi t ion n e e d not apply for the n u m b e r s below m].

We have thus expla ined the defini t ion ,~60 of Mr. Dedek ind r ega rd ing

Page 370

FROM PEIRCE TO SKOLEM 3x7

the m e a n i n g and ex t en t of p ropos i t ion ~ 5 9 . To recap i tu la te in the

words of the au thor : "'.~60. In o r d e r to prove that all e l emen t s of the chain ao;b possess

a cer ta in p rope r ty (~, it suffices to show: First: that all e l emen t s of b have the p rope r ty @. Second: that the same p rope r ty be longs to the image a; n of each such

element n "of ao ; b" which has the property ~. In o r d e r to recognize .~60 as the verbal t rans la t ion of f o rmu la ~ 5 9 ,

we only have to see c as the system of all e l emen t s which have the

p rope r ty (~."

In this connec t ion , the following r e m a r k may no t be super f luous , as

it h ighl ights the compa r i son of ,~59 with ,~47.

If we had previously left ou t the s t ipulat ion "of ao;b," we would have the verbal t ranscr ip t ion and usage of ,~47. Indeed , this fo rmu la can be

used to justify fully an assert ion, thesis ao;b~--c, based on the proofs of b ~--c and a; c=~--c.

This second partial r e q u i r e m e n t of the hypothes is goes fu r the r with

,~47 than with ~ 5 9 - - b e c a u s e of (ao;b)c =r162 By omi t t i ng this s t ipula t ion we requ i re m o r e than absolutely necessary to establish the thesis; the

t h e o r e m ~ 5 9 suffices for ',~47 with formally fewer cond i t ions tha t will

have to be proved. If, in some cases, t h e o r e m ,~47 suffices to reach the

goal, o n e can general ly only posit .~59 as the logical core of the p r o o f

by induc t ion .

The "forward path."

I have to ask the reader first to accept the name "a0;b" as a simple or un-

prejudiced, neutral sign for a relative to be defined independent of a and b mas, if you want, x or, at most, ~0(a,b)--for the time being, we should ignore

the manner in which this name is denoted. The end of our analysis will show that one may represent the assumed x as a relative product in b, as a known relative a 0, taken from b, and to state the function ~o(a,b) especially in the form a0; b---such as we did under 0) on page 361. But to do this beforehand would hamper our procedure and has to be rejected.

".~44 def ines the "a-chain ofb" as a b inary relative d e p e n d e n t on a and

b---as m e n t i o n e d in anticipation--denoted by "a0 ;b"; it is the ident i ty

p r o d u c t II with respect to u of all the relatives u of the universe of

d iscourse 12 which fulfill the condi t ions in 3 o ) for ex t ens ion cond i t i on

a ; u + b@u, set u n d e r the (first) H-sign. This cond i t i on splits into

a; u @u and b=C-u,

the first requi res that u be a "chain" with respect to a, the last that u also inc ludes b as a par t of itself; thus one can express the def in i t ion

with the words:


We understand by the a-chain of b the identity product ( t h e i n t e r s e c t i o n )

of all the chains with respect to a of which b is a part.

Page 371 Such a product really exists; it is not merely a meaningless name, lacking any factor;, this is clear because it will have at least the factor 1, because u = 1 fulfills

the extension condit ion.

"Fo r this very important n o t i o n the f o l l o w i n g p r o p o s i t i o n s a re val id."

~ 4 5 . b:~-ao;b. [Verbal ly e x p r e s s e d in 9).]

Proof. Because, according to the second part of the extension condi t ion, b is contained in every factor u of product I I u ~ i s a "common part" of these f ac to r s~ i t

must also be contained in their products. The scheme of the identity calculus to be used for this inference would be

II (b =6 u) = (b =6~ II u) u u

- - -compare 3 . ) of volume 1, used for an infinite n u m b e r of factors, here in e),

page 39.

~ 4 6 . a ; (a0; b) ~=a0; b. The "a-chain" of any relative is a "chain with respect to a."

Proof. According to the first part of the extension condi t ion of our IIu, each

factor u of this p roduct is a "chain with respect to a," and t hus - - acco rd ing to

"~43--also the product IIu of it.

.~47 . (a ; c:~--c)(b:~--c) :(=(a,, ; b:~--c).

Proof. According to the condit ions of this proposi t ion, c represents a value

of u which fulfills the extension condit ion. It thus falls u n d e r the factors of our

IIu, and since an identity product has to be included in its factors, we have

IIu~-c; the conclusion seems justified.

~ 4 8 is not expressed as "proposit ion" in D e d e k i n d ~ i n addit ion to the ex-

ternal deviations in its n o t a t i o n ~ b u t reads:

"',~48. R e m a r k . O n e is easi ly c o n v i n c e d t h a t the notion of the a-chain of b, defined in ~ 4 4 , is fully characterized by the previous propositions ~ 4 5 , 46, 47."

This "remark" is a luxury because it informs us casually, in a new and not

un in te res t ing way, how to formulate the definit ion ~ 4 4 of a0;b; it can be left

out in our course. We have clad it in our notat ional language and will justify it

in this form as equivalent to "~44.

In o rde r to save space, we would like to abbreviate the f requent ly used

expression

II u by I I u ) , .( a ; u + b , ~ u) N B

Page 372 ~ t o allude to the extension condi t ion a; u + b~--u merely by a NB (nota bene), and unders t and by rI,, as always, a p roduc t which possesses full extension over

all possible relatives u.

FROM PEIRCE TO SKOLEM 3 X 9

Fur thermore , the subsumption NB :(= (x=l~--u) of ~ 4 8 or

{(a; u + b=l~--u) =l~(x=(=u)} -- S

shall be declared an abbreviation.

If we then write, in t roducing x for ao;b, the definition of ~ 4 4 in the form

( X = a o ' b ) =(x=.xIIau ),

the comparison with ,~48 shows that we have to justify the following proposi-

tional equivalence:

of which the left side can be called L, and the right side R, and which immediately

follows, if one introduces the value of ao;b from ,~44 into ~ 4 8 .

We have thus to show L = R, or L =(= R and R =(= L.

According to the definition of equality, we have now

R- (x + , , u =e + u).

and we can now write the right side, according to the known (explained by

,~45) proposi t ional schema:

(x :(= ~ u ) - 8 (x ~r u) =H{NB =(= (x :(= u)} =H,S,

by rightly marking the extension condit ion

NB = (a; u + b:g~-u)

after (instead of under) the H-sign, as a condit ion to be fulfilled by u; this is

permissible since, not a relative, but ra ther a proposi t ion stands as the factor

after the H.

We thus know that

(x=(c-- H ul - H s and R=(c- H S.

As a consequence of R ( ~ 4 4 ) we had already proved the proposi t ions ~45 :

b=(=x and ,~46: a;x=gc--x, which, when combined, give us the conclusion

R =(c-- ( a ; x + b =(c- x) .

If we combine this with the above proposit ional subsumption, we have R=I~--L

(whereby the theorem ,~48 appears as subsumpt ion from right to left derived

from ~ 4 4 ) .

On the o ther hand, the following conforms to ~47 :

x + b :+ x) + x).

which has to be presented not as a consequence of ,~44, but as obvious from

Page 373 it; according to the premiss, the x on the left has to be, as a value fulfilling the

extension condi t ion "NB" of u, an effective factor of this IIu; this is why it is

subordinated. This last subsumption, multiplied with the above

320 SCHRC)DER'S LECTURE IX

u)

gives us the conclusion: L=(=R, whereby L = R is proved.

.~49. a;b=(c-a; (a0;b) is u n d e r s t o o d by relative p remul t ip l i ca t ion

with a on both sides f rom ~ 4 5 ; it o u g h t no t to be reg is te red as an i m m e d i a t e corol lary t h e r e o f B a s m e n t i o n e d be fo re ; j u s t as one does no t m e n t i o n in geometry , for example , the p ropos i t ion c 2" = (a z + bZ)"mnot to m e n t i o n nc2= na 2 + n b Z B n e x t to the Py thagorean t h e o r e m .

.~50. a; b =~--a o ; b. The a-image of b is part of the a-chain of b. Proof. A fortiori from the corollary (mentioned here for the first time)

a;b=(e:a;ao;b (,~49) to ,~45 in connection with ~46.

,~51. (a ; b=~--b) = (ao ; b = b). I f b is a chain with respect to a, then b is the a-chain of itself, and conversely.

Proof . .~47 gives (a; b + b=g~--b) = (a; b=~--b) ~ (a o ; b=~--b), which combines with

,~45 to give the equation ao;b--b. The converse follows from ".~50. We excuse this repetition from the "backward path."

.~52, 53.. . (b=~--c) :~-(b=(~--a o ;c) =~--(ao ; b=~--a o ; c). The part as well as its a-chain is also contained in the a-chain of the whole

Proof. By ~45, c=(e:ao; c; therefore, from b=~-c follows afortiori also b=g~--ao; c. If b=g~--ao;c (although perhaps not b=ge:c), it can be combined with

a; (a0; c) =(=a 0 ; c, which is valid by ~46: a; (a0; c) + b=gc-ao; c. It provides, by ,~47 (taking a0; c for c), the conclusion ao;b=g~-ao; c. q.e.d.

"~55. (b=~--ao ; c) =(= (a ; b=~--ao ; c).

Dedekind gives us two proofs:

Proof 1. By ",~53, we already have the following conclusion from the premiss:

ao; b=ge:ao; c. Together with "~50, it provides the conclusion afortiori. Proof 2. From the premiss, it follows that a; b=g~-:a;ao;c, the latter being =(=

ao;c by ,~46. Or, to express it differently: by the premiss and ~46, we have a; (a0;c) + b =~-ao;c, from which the conclusion follows also by .~40.

If one would like to further increase the number of small propositions, one

could add as an appendix to .~55 the proposition which follows from the latter and ~52 afortiori:

( b =ge: c) =(~ (a ; b =g~-- ao ; c).

At this point we can supply the p r o o f of p ropos i t ion ~ 5 9 , m e n t i o n e d

on page 366-- -a l though its s ignificance as a basis for the p ropos i t ion of

ma thema t i ca l induc t ion will only be u n d e r s t a n d a b l e later. The p ropo-

Page 374 sition is thus rigorous, namely, wi thout the circle crit icized above, p roved and may thus be used f rom now on.

But in the m e a n t i m e we mus t comple t e ou r "forward path."

",~57. Proposition and Definition. It is


a; (a,, ; b) = a 0 ; (a; b),

t h a t is, the a-image of the a-chain of a relative b is also the a-chain of the a- image of this relative b.

O n e can t h e r e f o r e d e n o t e such a re la t ive by

a00; b

(with t he s a m e r e s e r v a t i o n as o n p. 370) , a n d call it e i t h e r t he a-chain image or t he a-image chain of b.

Proof. Applied to a; b instead of b, ,~45 and .~46 are easily written separately;

however, they can immediately be combined as

a;{b+ a 0; (a;b)} :(=a 0; (a;b).

This r equ i r emen t now has the form of the premiss of',~41 where only cseems

to be represented by the right-hand expression. According to this proposi t ion,

there is a u m b y the way, the expression in braces on the left-side would be

one - -wh ich has the propert ies indicated after the sum sign--or , if we proceed

according to proposi t ional calculus, the following conclusion is valid:

E (a; u + b =(=u){a; u :~- a 0 ; (a; b)}, which, by .~47, t t

E(ao;b:g~-u){a;u ~ : a 0 ; (a; b)} u

:(= E {a; (a 0 ; b) =(= a; u}{a; u ~- a 0 ; (a; b)} u

:(= E {a; (a 0 ; b) :(= a 0 ; (a; b)} -- {a; (a 0 ; b) =(= a 0 ; (a; b)] u

- - a s we can see successively from the obvious consequences of the first propositional factor and then from the subsumption inference [according to the law

of tautology for proposit ional addition, the sum sign could have been omitted,

as long as the general summand is constant with respect to the summat ion

variable u- -which creates the impression that the sum :(= in the consequences

had been inverted with its summands!] . Thus the last successive subsumpt ion

is proved.

On the o ther hand, ".~46 gives the consequence , premul t ip l ied on both sides

by a, that reads, when combined with ~50 :

a ; {a ; (ao ; b)} + a ; b =g~-- a ; (ao ; b )

and shows the form of premiss "~47, whereby only the b there is represen ted

by a;b here, and c on the right side. According to the schema of the conclusion

of this proposit ion, we have

a0; (a;b) :(=a; (a0 ; b),

Page 375 which represents the converse of the subsumption proved previously. Thus, the

equat ion, which we set out to prove, is justified.


.~58. ao; b = b + aoo; b, tha t is, the a-chain of a relative b consists of

this relative itself and of its a-image chain.

Proof. The subsumption ,~46, to be written from now on in the simpler form

of a00 ; b=~-a; b, combines with -~45 as

b+ aoo;b ~-ao;b.

Thus, we only have to illustrate the backward subsumption. For that purpose,

we may abbreviate the previous subsumption given in the proof of .~57 as

a ; (aoo ; b + b) .~c-- aoo ; b, formerly : ( : :a00;b+b,

and may combine it with the self-evident b=~-aoo;b + b as

a;(aoo;b+ b) + b=~-aoo;b+ b,

which provides the conclusion, according to the schema of ,~47, where the right

side takes the place of c:

a 0 ; b =(= a00 ; b + b,

which was the last one to be proved, q.e.d.

O n the basis o f p r o p o s i t i o n ,~58, we can now, as we have s h o w n on

p a g e 361 u n d e r 0), give a b e l a t e d jus t i f i ca t ion o f the c o m b i n a t i o n o f

the t e rms a0; b a n d aoo ;b; a 0 a n d a00 cou ld , be e x p l a i n e d as rela-

t i v e s m a l t h o u g h in the fo rm of inf in i te series, whose f o r m a t i o n p r i n c i p l e

can on ly be jus t i f i ed by m e a n s o f the i n f e r e n c e f r o m n to n + 1; this

i n f e r e n c e has now o b t a i n e d its full civic r ights in o u r d isc ip l ine! Thus ,

o u r " fo rward pa th" has essent ia l ly r e a c h e d an end .

It is interesting, however, to see that we can prove the two propositions ",~61

and .~62 with Dedekind's theory before we recognize the relative, provisionally

called "a0;b", as a relative product of a 0 in b.

F o r ~ 6 1 . a 0 ; ( b + c + ' ' ' ) = a 0 ; b + a 0 ; c + ' ' " o n e has

a ; ( a o ; b + a o ; c + ' " ) + ( b + c + ' " ) : ~ - a 0 ; b + a 0 ; c + ' ' '

- -as a summary of the individual propositions contained in .~46 and ~ 4 5 and

used for b, c . . . . (instead of b); for the inclusion of the first term on the left

side, we can also use ".~42. According to the schema of ,~47, the inference is

Page 376

a 0 ; ( b + c + ' " ) = l ~ a 0 ; b + a 0 ; c + ' " .

On the other hand, because b~-b + c + "", c~-b + c + "", according to ".~52,

53), afortiori schema ,~54 is

ao ; b=~-a o; (b + c+ ""), ao ; C ~ a0; (b + c+ '") . . . .

which can be reduced to


a 0 ; b + a 0 ; c + ' - ' , ~ - a 0 ; ( b + c + ' " )

--whereby the equation stated in the theorem is proved forward and backward

as a subsumption.

For ,~62. a0; ( b c " " ) :~--ao ; b �9 a o ; c " " is

a ; ( a 0 ;b" a 0 ; c ' " ) : ( = a ; ( a 0;b) �9 a ; ( a 0;c) " " : ( = a 0 ;b" a 0 ; c ' ' ' ,

where the last subsumption is justified by multiplication of the propositions in

,~46; the middle conclusion can be replaced by ",~43 (the product of chains has to be a chain). On the other hand, it also follows from the multiplication

of the propositions given by ,~45 that

b=(~--a o ; b , c=(~--a o ; C, " " has to be b c " " ,~ - a o ; b " a o ; C " " .

If we combine this with the previous, we have

a ; ( a o ; b " a o ; c ' " ) + b c ' " = ( ~ - a o ; b " a o ; c ' " ,

and the assertion follows immediately from the schema of .~47. q.e.d.

We have r eached o u r g o a l . Let us discuss it and the pa th that led us

here . T h e r e is u n d o u b t e d l y s o m e t h i n g g r a n d in the way that the fatal o b s t a c l e

of the c i rc l e , proving the p ropos i t ion of ma thema t i ca l induc t ion , has

b e e n c i rcumnaviga ted .

The perspicacity, the care, and the genius of the originator of this theory are

all the more striking and admirable because he had no knowledge and not even

a vague notion of the existence of our discipline--which, up to now, has been

scattered in American papers which are hard to come by, or in notes which are difficult to understand, or insufficiently explained--and it could easily miss the recognition it deserves--not to mention De Morgan's attempts, which are rather

useless despite their merit as pioneering work.

Nevertheless , we may wish to reach the same goal on a less artful and sho r t e r path. We have especially to ask ourselves w h e t h e r it is necessary

to base the chain theory with ~ 4 4 on the def in i t ion of such a c o m p l e x relative as a0;b? It seems m u c h s impler to s t ipulate a def in i t ion of a 0

itself; c o m p a r e d to this p r o c e d u r e , the previous one has to be cal led a

r o u n d a b o u t way. If i n d e e d we succeed in simplifying the chain t heo ry in this sense, I

Page 377 have to expla in why I led the r eade r a long this l eng thy pa th and did no t p r e sen t my simple solut ion r ight away. T h e r e are several reasons:

I seem to be protected against a reproach in any case---even if these reasons

are not accepted--because the student who is not interested in the historical

development of this theory and who would like to reach the essence--in its

simplest form--as quickly as possible, is sufficiently informed through the head-

324 SCHR()DER'S LECTURE IX

ings that he may skip the introduction and jump right to the inferences and their promised simplification.

First of all, I wanted to see Dedekind 's theory expressed and fixed as such in a symbolism which I l i kedmand which shows it in its fullest generality, exceeding Dedekind 's own claims. Thereby I wanted to reveal the advantages of our system of notation, which is much more expressive and extensive; I am convinced that it alone will prevail and maintain a ruling position on the global market of science. On the o ther hand, the beauty of the theory should become apparen t in a way not previously possible with the symbolism of its originator.

I also wanted to show Dedekind's achievements to the reader who only intended to read this book without studying Dedekind's own writing: it has to be clear how much they contributed to my simplification of the theory. To the mathematiciansmand there are many--who wriggled more or less extensively through Dedekind's essay, I wanted to give a kind of "interlinear translation" of one of its important chapters, and provide them also with the foreign symbolism of Peirce's discipline. Furthermore, I wanted to voice some criticism and compile material to show the inadequacy of Hoppe's criticism.

Finally, the following reflection motivated me. Even if a simple chain t heo ry - -whe the r one similar to my own or any othermwil l soon de- throne Dedekind's , there remains much in it which is of more than historical value. Especially, the main point: the formulat ion and proof of ~ 5 9 (and 60), as represented by 16) on page 366, will survive unchanged. A fur ther simplification of the "formulation" of the proposit ion of induct ion is not thinkable; increased simplification of this "proof"-- l im-

Page 378 ited as such- -does not seem advisable or possible, since it is already very simple. The "main point" thus remains un touched by simplification, which can only apply to the abbreviation of the long and arduous path which leads from the basic definitions in Dedekind to the confirmation of the three lemmas ",~45, 47, and 55, on which rests the proof of ,~59, and which then has to continue until the end of the theory is reached.

And if indeed we manage to make do with fewer and much simpler propositions, if we get more quickly and easily to our goal with propositions involving~instead of two or t h r ee - - a t most one or two relatives, Dedekind 's theory nevertheless maintains its advantage of usually operat ing with more general propositions, among them proposit ions that may still prove serviceable for o ther goals, even if, for purposes of our

immedia te goal, they can be dispensed with. Of course, they would cont inue to be easily and quickly available to us whenever an occasion calling for them might arise . . . . 2

Equations 17)-20), summarizing the rules of duality and conjugation, have been omitted here.


Simpl i f ied cha in theory.

For de f in i t ion of the "a-chain," a 0, or of the cha in o f a relat ive a, we u s e

H u = a o , 21) u,NB

w h e r e the e x t e n s i o n c o n d i t i o n "NB" of the p r o d u c t II is u

NB = (1' + a; u =(=u). 22)

The "chain of a" is thereby defined as the identity product-----or to put it more

graphically: as the most comprehensive or maximal, "greatest" common region,

the intersection [Gemeinteil]---of all relatives x that fulfill the condition 1 '+

a;x=gc--x;, this is that intersection, therefore, which every intersection of the total relative in question comprehends, and is compounded from them all taken together.

Including the term 1' in the extension condition appears to be a bit arbitrary.

This disconcerting aspect might be alleviated somewhat by the remark that,

aside from 0, 1' distinguishes itself among the modules in that every relative

must be a chain with respect to it. Hence it will not appear so unheard of to

ask for those chains with respect to a which as "Aliorelativenegatd' contain the

module l 'mto then to take their product.

Since the ex t ens ion c o n d i t i o n splits in to

1' =(= u 22),~ a n d a ; u =~u, 22)t 3

m a k i n g it possible to apply the schemata :

I I ( l ' ~ : u) = (1' =(= IIu), a;IIu=~--II(a; u) ~-IIu,

w h e r e b y the f inal s u b s u m p t i o n is no t valid as a gene ra l fo rmula , bu t mus t be valid only in respec t to the s econd par t 22)~ of the e x t e n s i o n

Page 380 c o n d i t i o n , f rom which it results by mul t ip l i ca t ion on b o t h sides. With that, however, we have

1' =(= a 0 23)~

which can be c o n t r a c t e d to

a n d a ; a 0 ~=a 0, 23)~

1' + a ; a 0 =(=a 0. 23)

The two parts of 23) appear as the simplified ".~45 and 46: that is to say, as their subcases for b = 1'.

F r o m 23)~, it follows directly: 1'; b=~--a o ; b, therefore*

b~-ao;b, 9)

* Repeated from earlier.

326 SCHRt~DER'S LECTURE IX

whe reby p ropos i t i on ~ 4 5 is p roved , which, incidental ly, also e n c o m -

passes the m i n o r proposi t ions :

a : ~ - a o ; a a n d a 0 : ~ : a 0 ; a 0.

It f u r t h e r follows in r e f e r ence to 23)~:

(b:(=a0 ; c) :(=(a; b:(c--a ; ao ; c:(c--ao ; c), whereby

(b :(= a,, ; c) :~: (a; b ~ a0 ; c), 10)

t h r o u g h which p ropos i t i on ",~55 is won. O f the th ree i nd i spensab le

p ropos i t i ons that we n e e d to prove ,~59, we a l ready have two at o u r

disposal.

It would be just as easy to gain a few other propositions, some of them actually

parts of Dedekind's chain theory and some of them the corresponding simpli-

fications of propositions belonging to the theory, which, however, we can do

without here.

Thus it also follows from 23)~ that b; l '=l~- -b;a o, or the proposition b=gc-b;a o,

which also involves a f o r t i o r i a~(c-a; ao and then, in reference to 23)t~, a~=a0-- the

latter two simplifications of .~49 and 50.

With the following two variants of the previous idea,

( b =gc- c) =(1' :~- a ,, ) ( b =(c- c) =(c- ( l ' ; b =gc-- a ,, ; c) = ( b =g~- a o ; c) ,

(b =(= c) = (% ; b =(= ao; c) - (b =(= a o ; b)(a o ; b =(c-= ao; c) =(c-- (b =(c- ao ; c),

the propositions ~52 (b=r from 23)~, or ~ 4 5 are easy to prove.

However, if until now in parallel with Dedekind's theory we have been able

to advance only to the easiest goals with ideas which, at bottom, amount only

to a repetition of the author's ideas for a simpler special case (which, to be

sure, was necessary to do only for that purpose), this procedure now reaches

its limits. For us, Dedekind's proofs fail for -~47, 51, 53 because they essentially

rest on his more general (and more complicated) fundamental definition ~44.

And if we try to make do with Dedekind's parallel ideas, for example, to prove

the corresponding simplifications of these propositions, we only end up in a

Page 381 circle among the three named propositions, from each of which in fact the other

two could be easily derived. To move on from here, it is necessary to take up

an entirely different idea.

To get the p r o o f we are still lacking of the th i rd a n d last of the

i nd i spensab le propos i t ions , ~ 4 7 , I have to en r i ch the cha in t heo ry by

o n e l emma . This is


{ (a;b:~-b) ={a; (beb) :~ - -be~} , ( b @ a e b ) = ( b ; b @ a e b ; b ) , (b;a:~--b) = {( /~eb);a~--~eb}, (b~--bea) = ( ~ ; b ~ - - ~ ; b e a ) 24)

- - i n words: if b is a chain with respect to a, then be/~ is also a chain with respect to a, and conversely.

Proof. Let a;b~--b. Because by 9) of w 17, b= ( b e g ) ; b , we have

a ; b = a ; ( b e ~ ) ;b~-b,

and f rom the last subsumpt ion it follows by t ransposi t ion of the final relative f a c t o r - - b - - a c c o r d i n g to the first inversion t h e o r e m 4) of w 17, in fact,

a ; ( b e [) ) :~c-b e [).

Conversely, the latter subsumpt ion can be rewri t ten equivalently as the previous one and itas the first one; that is, the final series of conclusions is comple te ly reversible, q.e.d.

On this basis it is poss ib le - - in analogy to . ~ 4 1 - - t o establish yet an- o the r lemma. But it is not exactly imperat ive to have fo rmula t ed it as such or even to r e m e m b e r it. Much bet ter would be to b lend its p roo f with that of the following main proposi t ion ('.~51). It states:

(a; b:~--b) =(= E (1' + a; u =(=u)(u ; b:~--b), 25) u

in words: I f b is a chain with respect to a, then there exists a chain u with respect to a that includes within itself all of the individual self-relatives and with respect to which b is a chain.

And indeed: (Proof) u = be/~ is such a chain. For, first of all, because of 3) of w 8, namely, 1' :(= b e/~, it fulfills the

condi t ion 1':~-u. Second, accord ing to the previous l e m m a 24), it also fulfills the condi t ion a; u :(=u, as well as, third, accord ing to 9) of w 17, the condi t ion u ; b:~--b, since (be/~) ; b and b are even equal. Cor respond- ingly, in 25) the last inclusion sign can be writ ten as an equal sign.

It is possible to put proposition 25) in nicer form. The condition l':(=:u, namely, is satisfied in the broadest possible way using the statement u = 1' + v,

Page 382 thus giving rise to

(a" b=~-b) ~ E {a" (1' + v) :(:: 1' + v}(v " b=~-b),

in that with the last partial postulate, the term 1" b or b fell away, as self-evidently _

contained in predicate b. The condition v" b=~-b or v :(= bj- b can once again be satisfied in the most general way by using v = (be/ ) ) , leaving

(a;b:~--b) =~--~ [a;{l ' + ( b e ~ ) w } ~ l ' + (be/~)w]. 26) w

328 SCHR()DER'S LECTURE IX

This m e a n s i f b is a chain, then there is also a relative w such that 1' + (b j- ~) is also a c h a i n - - w i t h respect to a. One such relative is w = b j- ~ in fact ,

f o r which 1' + (bj- ~)w = 1' + (bj- b) also holds, a n d we return to 24) .

Now o u r pa th takes us via ,~51 to ,~47. To prove the p r o p o s i t i o n .~51, or

(a; b:~-b) = (a o ; b = b),

(b; a =(=b) = (b; % ---b),

(b =~-a j- b) = ( a I 0" b = b),

(b=gc--b J- a) = (b ~ a 1 = b), 27)

it is essent ial , f rom the premiss a ; b:~--b, to der ive ao;b=(c--b, w h e r e a 0 = II u, e x t e n d e d over the values of u, which satisfy the c o n d i t i o n "NB."

u

Now, certainly,

a , , ' b = ( I I u)" b=(c- II (u" b) =dc--u" b, ' �9 u , N B ' u , N B ' '

a n d it is so regard less of which relative satisfying the ex t ens ion c o n d i t i o n u n d e r the last u m i g h t be u n d e r s t o o d . This is because of p r o p o s i t i o n 5) of w 6 and because the p r o d u c t mus t be c o n t a i n e d in each o n e o f its factors.

Now, the last l e m m a - - u n d e r 2 5 ) - - s h o w e d tha t a m o n g these factors t h e r e will be at least o n e tha t is par t o f b, namely, tha t t h e r e is a u tha t satisfies N B - - i n the fo rm b j-~---for which u;b=dc--b, and in fact t h e r e f o r e

it also follows ao; b :~--b afort iori .

This subsumption can immediately be combined with ~ 4 5 or 9) in the equation a0; b = b, whereby the propositional subsumption

(a ; b=gc-:b) =6--(a o ; b = b)

is justified. To justify the reverse propositional subsumption--which is incidental to our primary purpose and which even has to be valid already with the weakened premises as

(a(, ; b=6--b) =~--(a ; b - b)

- - i t is necessary simply to call on a=6-a o, ergo a;b=gc--ao;b, that was already

Page 383 justified in the context of page 380.

T h e p r o o f of .~47 or t

(a; c + b=~--c) =(c--(a o ; b:~--c) 11)

can finally be a c c o m p l i s h e d qui te easily as follows:

(a; c + b=6-c) = (b=(=c)(a ; c=(c--c) =(=(a,, ; b::~--a o ;c) =(c-.(ao ; b=~-c )

- - -on the basis of 27) or ",~51.

* Repeated fi'om earlier.


With this, all of the p recondi t ions of the nonc i rcu la r p r o o f of .~59, 60 have been gained.---cf, page 366.

It remains to simplify ~ 5 8 , or prove the proposit ion"

a o = 1' + a ; a o 28)

- - a recurs ion f rom which the power series for a0 can easily be derived. This is d o n e without difficulty, as follows. By 23), we have

1' + a ; a o =~-a o = I I u =~--u, thus

1 ' + a; a o ~ u

for every u that satisfies NB. There fo re , u is, in any case, of the form

u = l ' + a ; a 0 + v ,

and by pu t t ing in the extens ion condi t ion for v in 23),

1' + a + a ; a ; a o + a ; v = ~ - - l ' + a ; a o + v,

which, however, immedia te ly reduces to NB0 =)

a ; v = ~ - l ' + a ; a o + v,

because the first three e l ement s of the subject can be cance led out as

obviously already con ta ined in the predicate; then a = ~ - - a ; a o follows di-

rectly (as a ; 1' =(= a; a0) f rom 23)~ and a; a; a0 = a ; (a; a0) =(=a; a 0 f rom 21)t~. This allows us to conc lude

a 0= 11 ( l ' + a ' a 0 + v ) = l ' + a ' a 0 + 11 v = l ' + a ' a 0, v,NB0 ' ' ~:,NB0 '

since v = 0 satisfies NB0, t h u s / I v = 0 is, q.e.d. As an extra, we want to prove these two proposi t ions as well, w i t h o u t

using the power series for a0:*

7) ao ; a o = a and 6) a ; a o = a o (which def ines = a00).

The first one, 7), results as a conclus ion f rom 23)t~ accord ing to the Page 384 schema of 27) or ,~51, the lat ter by assuming b - a 0. It is, so to speak,

the core of propos i t ion ,~53: (b=~--ao;c) = ~ - ( a o ; b = ~ - - a o ; c ) , f rom which it

resul ted for b - a 0, c = 1', where, then, because the p remise is valid, the

conclus ion must be valid too. Conversely, ".~53 results easily f rom it, with

( b @ a o ;c) @ ( a o ; b=~--a o ;a,, ; c=(~--a o ;c).

To prove the latter 6) we already have a" a 0 = a 0 and a o ' a = a o.

[The former was 23)~, and the latter follows, for example, for b= a 0 based

on 7) from the second proposition on the left in 27) backwards.] Thus we also have

* Repeated from above.

33 ~ SCIIRODER'S LECTURE IX

a" (a" a o) =~-a" a o, (a,, " a) " a=g~--a o " a.

By the left schemata in 27), one can write this equivalently as

a, ," (a" a o) = a" a , , , (a o" a) " a o = a o" a,

according to which, because the left-hand expressions agree (with themselves

and with a o ' a " a o ) , the right sides must also be equal to each other, q.e.d.

With this we can now prove the following proposition"

a oo" a o0 =(= a 00 5)

which I would like to call the pivotal point of the entire chain theory, w i t h o u t

the use of series expansion as follows:

a00" a00 = a" a0" a0" a = a" a0" a=(=a0" a = a00.

Those, finally, for whom it is impor tan t to go from our s impler point of

depar tu re to derive 21) as a theorem from the general definition ~ 4 4 as well

will do best to turn to the considerat ions following unde r 11) of w 24; that is,

they need only verify the general solution we gave for the proposi t ion a ; x +

b=g~--x as such (which again, without using the infinite series, is feasible on the

basis of already established propos i t ionsmsee there) and from that take the

p roduc t II.

From now on, however, the series expansions that follow from 28)

a0 = 1' + a + a 2 + a :~ + "", a00 = a + a 2 + a :~ + "", 29)

may be put to immediate use.

Finally, t he n o t i o n s o f the a -cha in a n d the a - image c h a i n o f b m a y be

d e m o n s t r a t e d with a pa i r o f f igures .

Just in reference to comments such as that by H o p p e on p. 30: "On the

contrary, there would scarcely be any way to fill in the empty frames,* in o rder

to have someth ing to think about, o ther than by the known numbers" such a

step might not seem entirely superfluous.

I had Figures 22 and 23 prepared, however, while I was still using Dedekind 's

notat ion, and so must ask the reader to imagine the letters as having been

changed somewhat, namely, to consider the small a, a t, b, b t, c, c t as having been

Page 385 replaced by the same capitals: A, A', B, B', C, C', but taking for A, A', A", A", and

B in the figures b, b t, b", b ' , and c, so that the letter a remains free for the

relative to be chosen as the mapping rule.

T h e a c c e n t s h e r e a re to be u n d e r s t o o d as a b b r e v i a t i o n s a c c o r d i n g to

t h e s c h e m a

b' = a " b, b" = a ; b ' = a ; a " b = a 2 . b, b at = a ; b " = a ~ " b,

* In Dedekind 1888 (footnote 1).


FIG. 22

331

c h o s e n ad hoc for t he p u r p o s e o f s impl i fy ing the e n t r i e s to be m a d e in

t he f igure .

It should not be disturbing to anyone, but simply more instructive, if from

now on we make use of ano ther me thod of geometr ic demons t ra t ion than the

one illustrated in w 4.

For the first universe of discourse to be considered 1 ~, this t ime let us keep

in mind the entire collection of points in a sector of a circle, or the correspond-

ing angular area if one wants to include as well the points of the whole plane.

Any of the figures in the latter, e.g., a shaded piece of the surface, we will

therefore unders tand as a "system," and every point will be an "element" of such

a system (and not, as in w 4, a pair of elements!)

As a "relative" a, with respect to which the formation of the chain will be

illustrated, I choose a "single-valued assignment," an actual "mapping" from our

theory as opposed to the more restricted circle of ideas in Dedekind ' s text--with

the precise intent ion of showing that it already has a substratum project ing

broadly over the n u m b e r system.

In the context of the three sectors to be seen in Figure 92, this mapp ing is

also an invertible one, thus in ~ ' s terminology "similar" (or "d is t inc t" )mbut a

different one in the sector going to the left than in the two going to the right.

In the latter, the a-image of (an e lement- represent ing) point A is the point

A'= a;A, which lies half as far from the center of the circle on the ray of A; for

the first, however, it is the point lying double the distance along the same ray.

Taken now as system b (in the figure A) is the port ion of a concentr ic circle

that falls in the sector, and in the sector going up and to the right, in the infinite

series of continually shrinking, shaded "quadrangles," we see the a-chain ao;b Page 386 of a system b such that at the same time we also see, going from the ou te rmos t

quadrangle b inward, the a-image chain aoo;b of the same. The dimensions of

33 2

C J

~ t

2?

A~r

FIG. 23

SCHRODER'S LECTURE IX

R

]a..

b have been chosen here so that each of the objects appears to be disjoint from

its image. For the sector runn ing down and to the right, b is chosen so that the image

is adjacent to its object. The a-chain of b here is the entire shaded sector, a n d

the inner half measured radially must manifest the a-image chain of b (regardless

of its image chain). The success would be similar in the case of an object ex-

ceeding its image. For the sector or angular surface going down and to the left, ao;b is the limit

of the ou te r infinite series of successively larger shaded quadrangles , and

aoo;b such, however, that the innermos t one is l ack ingmwhereby it does not

mat ter that divergence would exist here in the series of measures (in the sense

of area) of these shaded surfaces. In Figure 23 the ass ignment selected as mapping rule a is single-valued, but

it is not invertible. As universe of discourse I i (or Dedekind 's system), we see here the point

system of a segment of a circle composed of two sixths of the circle or sextants

I and H and a half of such, Ill. Let the following be the mapp ing rule a.

The image of a point A (a in the figure) in I would always be the point

A ' - a ;A (a' in the figure) on which A comes to rest when we simply move the

sextant I over the "sliding edge" [Rutschkante] MR (without fl ipping it over),

until it coincides with II. The image of a point C (c in the figure) in III would be the point C ' - - a ; C

(c' in the figure) on which C comes to rest when we flip the half of the sixth

of the circle III over the folding edge MF until it coincides with the adjacent

half of II. The image B '= a;B of a point B in H (respectively b' of the b in the figure),

in contrast, is de t e rmined similarly to [that described] above as that point which

lies hallway from the midpoin t on the same ray as B.


T h e r e u p o n it is obvious that what we are working with is a m a p p i n g of the

e n t i r e d o m a i n of discourse or system 1 ~ to itself.

In the shaded por t ion of Figure 23 then is the a-chain a 0 ; (b + c) of the system

b + c of points mak ing up the area of both circles b a n d c (A and Bin the f igure) ,

and we r e c o m m e n d that the r eade r look at the four segments into which the

Page 387 "edges" MF, MR divide these circles and their images, etc. It will then be easy

to check the validity of o the r of the proposi t ions such as ~ 4 5 . . . . ~ 5 6 f rom how

they look by imag in ing a region of points that is s o m e h o w l imited in the surfaces

b or c (A, B in the figure) or in their images or, as the case may be, chains or

image-chains.

w 24. Collateral Studies o f Chain Theory

T h e i n t r o d u c t i o n o f t h e c o n c e p t o f t h e "chain with respect to a relativg'

t h r o u g h , ~ 3 7 s u g g e s t s t h e f o l l o w i n g two p r o b l e m s .

P rob lem 1. To d e t e r m i n e the most genera l relative with respec t to which a

given b is a chain, that is, to solve the subsumpt ion :

x ; b=g~-b.

This p r o b l e m is only a special case of the one we solved in w 17 with the first

inversion t heo rem. Accordingly, the solut ion takes only a m o m e n t : x=~--b3-~ or

x = u(ba-~) for an u n d e t e r m i n e d or arbi t rary u; that is, the p ropos i t ion can be

written:

(x; b=g~--b) -- ~ {x = u(b j- /~)}. 1) u

P a r t i c u l a r r o o t s a r e x = 0 as wel l as x = 1'. T h e r e f o r e : with respect to the modules 0 a n d 1' every relative is a chain.

Prob lem 2. To d e t e r m i n e the most genera l relative with respect to which a

given a is a chain; that is, to solve the subsumpt ion for x:.

a; x=~-x.

We have already given the solut ion to this p r o b l e m in 5) of w 22 (page 325),

f inding two express ions as the genera l root: x = a0 ; u and x = a ~ j- u, which can

easily be recogn ized as essentially different; namely, a l t hough these two terms

becomes the same if u is a root x (and are equal to x in that case), they usually

r e p r e s e n t d i f ferent values for arbi t rary u. For otherwise, the fol lowing would

have to be valid of u-- 1' for any given a:

a0; 1'= a0= 1' + a + a 2 + a :~ + "'" = a l j - 1'= 1' (a3- l ' ) (a3- aj- 1 ' ) " " ,

which is obviously false, since the u n d e t e r m i n e d a does not have to be =~= 1'.

If we may be al lowed once again to poin t out the most i m p o r t a n t aspect for

us of 5) of w 22, we may write the proposi t ion:

3 3 4 SCHR6DVR'S LECTURE IX

(a; x=gex) = ~ (x = a,, ; u) = E ( x = a , j- u). 2) u u

This p roblem has probably provided us with the most natural oppor tuni ty to

in t roduce tile concept a 0 of the a-chain, as well as that of the a-dual chain [a-

Gekett]: the unknown must be the a-chain of some relative umsimilar ly the a-bar-

converse dual chain to one such undef ined relative.

As part icular roots for any given a, only x--0 and x = 1 can be given from the

g roup of modules; that is, it can be said (in conformity with ~ 3 8 ) that the

Page 388 ent i re empty, as well as the full ( e n t i r e ) u n i v e r s e of discourse is a chain with

r e s p e c t t o any rela t ive--any one relative can map the universe of discourse only

"to itself."

In the choice of u = a 0, respectively, a, we also get x = a 0, as well as x = a00, as

par t icular roots from the first form of the solution; similarly, an infinite n u m b e r

of solutions can be given generically, x = a000, e tc . - - i f what we designate by this

symbol is the sum of row 6), page 325, without the first two elements , etc. Likewise

for the dual.

T h e a-chain or chain o f a, like the a- image chain, accordingly, is also always

a "chain wi th respect to a. "Etc .

We already did the two proofs with both solution forms 2) on page 331; for

the first one, Proof 1 resulted in the proposi t ion a; a 0 =(=a, the consequences of

which have also been registered already in 6), page 361. For the second solution

torm, Proof 1 has already been done at the bot tom of page 331; but we have

not yet registered as a proposi t ion the fact that it is a solution. If we say b for

the u that appears there, then we get the p roof of the proposit ion:

a ' ( a , ctb)=(= a, ct b, ao " b =~-- act ao " b ,

(act ~l) " b =(e act ~,, a" b,, =(e a" b oct ~, 3)

and especially for b =0', etc.:

a ' a, =(e=al, a 0 =(= act a0, 4)

a,. a+a, , a~, + - a , , s a.

To derive the cor responding dual formulas from each other, it is necessary

to take the contrapositive, as always, replacing the letters by their negations, but

at the same time, a by a (etc.) ...:~

H a v i n g m a d e it poss ib le , wi th t he suff ixes 0, 00, 1, 11, to p r o d u c e

e x t r e m e l y c o n c i s e n a m e s fo r e x p r e s s i o n s tha t , wh i l e h a v i n g t he f o r m o f

i n f in i t e e x p a n s i o n s n e v e r t h e l e s s m a n i f e s t t he s i m p l e s t a n d m o s t t rans-

p a r e n t f o r m a t i o n laws, we will a g r e e fo r t he sake o f b rev i ty to say o f a

s o l u t i o n o f a p r o b l e m t h a t it is p r e s e n t e d in "ha l f - c losed f o r m " ( o r

p e r h a p s m o r e exactly, i n s t e a d o f semi-, q u a s i - c l o s e d ) , as s o o n as its ex-

p r e s s i o n is c o m p o s e d by m e a n s o f a f in i te c o l l e c t i o n o f o p e r a t i o n s o f

* The conversion rules for e4~, a,, 0, a I, and all (pp. 388-389) have been omitted here.


t h e six spec ie s a n d the d i s t r i b u t i o n o f such suffixes. T h e s e f o r m s t h a t

we have d e s i g n a t e d "ha l f -c losed" for brevi ty ' s sake a r e t h e r e f o r e in es-

s e n c e the o n e s t h a t o c c u r in s ide the c i rc le o f o p e r a t i o n s o f t h e six spec i e s

as in f in i t e e x p a n s i o n s , whi le o n the o t h e r h a n d r e p r e s e n t i n g c l o s e d

f o r m s a f t e r a d j o i n i n g the n o t i o n o f t he a-chain .

We m o v e n o w to t he s t ep o f g e n e r a l i z i n g t he last p r o b l e m .

P r o b l e m 3. Solve for x t he s u b s u m p t i o n

a; x + b:~-x. 6)

The p rominen t role this plays in various of the Dedekind p r o p o s i t i o n s ~ 4 0 ,

41, 44, 47, 48---seems by itself enough to bring up the problem. It can be solved

in several ways, and we want to follow to the end all of them, insofar as they

lead to interest ing results and compar ing them appears to be methodological ly

instructive. That objective leads us to the following four forms of the solution,

Page 390 written out separately:

a ; x + b @ x ) = EIx = a(, ; (b + u)} 7) u

= E [x = a0 ;{u + g~); (d~ ~f a)b}] 8) u

= E l x = a l j - ( a 0 ; b + u ) } 9) u

= E [x = a o ; b + u{a 1 ~ (a0 ;b + u)], 10) u

o f w h i c h the first is to be ca l l ed " the best ."

On the derivation. Since x = 1 satisfies the r equ i remen t of the problem, then

we have no resultant.

Given that x already appears isolated in 3) fight as a predicate, we can quickly

write down a first form of the solution tollowing proposi t ion 1) of w 13:

(a" x + b=~-x) = E {x =f~(u)}, where f (u) -- b + u + a" u. u

With this f (u) = b + (1 '+ a ) ' u , however, it is easy to prove, by the inference

from r to r + 1, the formation law of the i terated function"

f ' ( u ) = (1' + a)~'u + (1' + a) "-1" b = (1 '+ a) ~-1 ;f(u)

given that (1' + a) ~-l =(=(1' + a)'= (1' + a) �9 (1' + a) ~-l must be valid. With this we

find

f~ = (1' + a)='u + (1' + a) =" b = a 0 �9 b + a 0 "u = ao ' (b + u),

that is, it is solution form 7), gained heuristically. Later on, a n o t h e r course will

lead us to this same result.

Al though unnecessary, we shall also verify it directly. That Proof 1 is correct

is based solely on the proposit ions 1' :(=a 0 and a" ao=~-a o.

Proof 2 coincides with the proof of the proposit ion:

Page 391

Page 402

33 6 SCHRODER'S LECTURE IX

(a" x + b =(= x) = {x = a + 0" (b + x)} = {a 0 �9 (b + x) =(= x}, 11)

the last part of which is true because the last subsumption is invertible anyway

(because of ~45 ) , while the first parts of the forward propositional subsumption

can be justified accordingly from 13) of w 23 (or from ~40) , in that

( a ' x + b=g~--x) ~-{a" (b+ x)=(= x} =(=la" (b + x)=g~-b+ x}

= { a 0 " ( b + x ) - b + x - - x }

according to ,~51, and because b =(= x must be valid, the reverse direction is

a l readyproved by Proof 1 (using it for u = x), q.e.d.

S o l u t i o n 7), which is now jus t i f ied , puts us in a pos i t ion to c o m p l e t e ,

in the m o s t e l e g a n t way, the final o u t s t a n d i n g s tep in the backward

c o u r s e o f the inves t iga t ion in w 23, namely , to arr ive heur i s t i ca l ly at

D e d e k i n d ' s de f in i t i on ,~44 of the a-chain of b. It follows"

II = I I a 0 ; ( b + u ) = % ; b + I I a 0 ; u = a 0 ; b , x , ( a ; x + l , ~ x ) u u

in t ha t II a0 ; u = 0 is obviously valid, as showing the fac to r a0 ;0 (=0) . u

With that , we have now c o m e all the way to the e n d by the "backward

pa th , " as it is ca l led in w 23 .... 4

O n e m o r e word in conc lus ion .

On ~63 . This "proposition," offered without proof by Mr. Dedekind, contains

several claims, some of which concern "proper" parts. We should therefore use

the subset sign C in representing them. It has no effect on correctness, however,

to replace the latter by a subsumption sign :(= and rest content with proving

the corresponding claims merely for "parts" per se, which is what we will do

next. Then the first of these would be valid not only for single-valued mappings

and systems, but once again for any given relative.

The first claim of the proposition (not yet in need of modification) is

( a " c =(~-- b =~-- c ) =(~-- ( a " b =g~-- b )

or (a" c=~--b)(b=(c--c) ~--(a" b=g~-b), 37)

which says that each part b of a chain c (with respect to a) containing the a-

image of the same chain, must itself be a chain (with respect to a ) ~ a n d therefore

a" c=(c--c is also valid, and therefore c will be a chain.

This follows a f o r t i o r i with the utmost ease by means of a" b . ( v - a ' c (from the

second premise: b:~c--c) because of the first a" c=(c--b.

Less obvious is the proof of the following claims by Dedekind.

A second claim says that, under the conditions given above, it must be that

a o �9 [~c=(c--c. 38)

4 Schr6der's three other solution forms for Problem :3 have been omitted here.

Page 403


[More precisely, if even b C c, then a0;/~c C c; the /~c is D e d e k i n d ' s U.]

We recall that we have these propos i t ions at o u r disposal:

(a; c =(=c)(a ; c =~-b)(b=g~-=c)(a ; b=~-b)(a o ; b = b)(a o ; c = c),

the last two of which follow by ~ 5 1 f rom the previous mater ia l . Now

ao ; c = ao ; (b + [~)c = ao ; bc + ao ; [,c, c o n s e q u e n t l y

a~;/~c=(ea0; c, which = c,

and thus is the second claim proved.

A thi rd claim is mere ly a repe t i t ion of the second. If, namely,

a0;/~c = d

is given as an abbrevia t ion, the claim is c - d + c[t (in that cd coincides with

D e d e k i n d ' s U). This can be put m o r e simply, however, c - -d + c and comes ou t

as d=(c--c, which makes up the second claim.

[A four th claim asserts that

b - - a ; d + ccl,

and a fifth (and final) claim says that, if, in addi t ion to every th ing else, b=

a;c , t hen it mus t also be that

cd ~ : a ; cd.

These two claims, at all events, are not as genera l ly valid as the o thers above

for any given binary relative. They shou ld have been omi t t ed f rom the "second

part" of D e d e k i n d ' s t e x t m a c c o r d i n g to my de l inea t ion of the same. And this is

no t the place to check the i r validity for s ingle-valued m a p p i n g s a and systems

b , c . ] m

Last of a l lmsc i ence is i n d e e d u n e n d i n g ! ~ s o m e t h i n g new:

Also va l id fo r t h e c h a i n s a r e t h e p r o p o s i t i o n s

( a : (= l ' ) = (a,, = 1'), (0' : (=a) = (a~ = 0 ' ) , 39)

o r (l~z) 0 = 1', (0' + a) l = 0' ,

w h i c h c a n b e p r o v e d w i th u t m o s t e a s e f r o m t h e i r f o r m a t i o n laws; s ee

6) o f w 22, p a g e 325. I f a:~-- l ' , t h e n a ; a : ~ : l ' ; 1 ' = 1', e tc . A n d

a 0 = (0~)0 , a I = (1' + a) 1. 40)

T h e c h a i n o f a is t h e r e f o r e the s a m e as the c h a i n o f O'a, t h a t is, t h e a l io-

p a r t [ A l i o t e i l ] o f a.

Various types of proofs are possible, the s implest based on 15), page 365, with

a,, = (1' + a) ~~ = (1' + 0~z) = = (0~),,.

W i t h t h e s o l u t i o n o f t h e s o l u t i o n p r o b l e m s , o f c o u r s e D i n h a r m o n y

33 8 SCHRODER'S LECTURE IX

with p a g e 174f f . - -a few e l i m i n a t i o n p r o b l e m s have also b e e n solved.

Namely, in the fo l lowing p r o p o s i t i o n a l s u b s u m p t i o n s

(xoo = a) ~ (a ; a ~ a) = (aoo = a), 41)

P a g e 404

(x,, = a) :(=(a 0 = a), (a - 1' + y)(y;y:~--y) :(=(a 0 - a) 42)

the r i gh t s ide r e p r e s e n t s the full r e su l t an t s of the e l i m i n a t i o n o f x o r y,

respect ively, f r o m the left side.

A direct proof is also easy: Since, according to 5), page 361" x00" Xoo:(e:Xoo, by

inserting this, a" a:g~-:a follows from the premises of 41) as "one" resultant. This,

according to ,~51 is equivalent with ao'a = a, that is, with the final equation in

41), and the latter allows us to see that, when it is fulfilled, x- -a will also be a

root of the equation x00 = a, that our resultant was therefore the complete one.

A c c o r d i n g to this, the concept of an image chain coincides [altogether] with

that of a transitive relative.

The resultant also follows from the premiss of the first subsumption 42)

directly as a conclusion from (x0)0 = x0 by virtue of 8), page 362, and is recog-

nizable from first glance as the complete one, because then x = a suffices. Re-

garding the second subsumption 42), we may rewrite its second premise equiv-

alently, according to schema 41), as Y00 =Y, which can be used to reduce the

first one to a = 1' + Y00 =Y0. Etc. q.e.d.

Also at o u r d isposal for r e p r e s e n t i n g all t ransi t ive relatives, in the

f o r m o f 34), page 339, a re c losed express ions , a n d the q u e s t i o n arises

as to the e x t e n t to which the la t te r are of value for c a l cu l a t i ng i m a g e

chains . As r ega rds d e t e r m i n i n g the i m a g e cha in a00 for a given re la t ive

a, this does no t a p p e a r poss ib le so far.

If the issue, however , has on ly to do with speci fy ing those a m o n g the

b ina ry relat ives tha t are, respectively, i m a g e cha ins o r cha ins at all, t he

s a m e resul ts wou ld be c u m b e r s o m e to p r o d u c e for o t h e r u in the f o r m

of an inf in i te e x p a n s i o n :

= , 1 ' + U00 12 q- U 2 -t- U 3 -q- "" U 0 = U00.

This b e c o m e s an easy mat te r , however , if we take for u00 the e x p r e s s i o n

v ( v # v ) f o r m e d a c c o r d i n g to the s c h e m a c i t ed above, a n d eva lua te

i t - - w h i c h has a c losed f o r m - - f o r o t h e r v.

Appendix 6" Schr6der's Lecture XI

Introduction

In this lecture Schr6der primarily works out rules for sum (existential quantifier) and product (universal quantifier) over domains of the first order [Stufe] (quantifiers over individuals) and of the second order (quantifiers over all binary relatives on the domain, his version of second-order logic). This is wholly interpreted; there is no fixed formal system with rules of inference in which he operates. The previous lectures indicate that Schr6der is perfectly happy to regard these operations as operations on propositional functions, rather than operat ions on formulas. This lecture thus has a highly algebraic flavor, even though it expresses more quantifier rules than one has seen anywhere else in Schr6der (or anywhere else, for that matter). The quantifier rules naturally emphasize interactions with relational operations, a subject probably not taken up since.

The most interesting feature of this lecture is that Schr6der 's algebraic point of view meant that he regarded a universal existential prefix as a product of sums or a greatest lower bound of least upper bounds. He thus leapt from the finite case to the arbitrary case and simply wrote out the most general distributive law. If x and y range over the integers,

he writes (qx)(3y)rh(x, y) as (u where f r a n g e s over all functions on the integers to the integers. From his point of view, this is

II,Zj4(i,j) = ZjIIxrh(x,f(x)). Note that the least upper bound Ef on the right is over a cont inuum of functions. This reflects the complete distributive law (for propositional functions). This device and the formulas it entails is precisely the device used by L6wenheim in his difficult-to- follow proof of the L6wenheim-Skolem theorem. We have used a simpler notat ion than these authors, since they wrote f as a sequence instead of as a function, giving complex-looking subscripts.

This lecture is significant for revealing exactly how far Schr6der got

339

34 ~ SCHRC)DER'S LECTURE XI

with q u a n t i f i e r s - - f u r t h e r than the early Peirce; in addi t ion , the algebraic po in t of view of the comple t e distributive law appears he re for the first t ime in mathemat ica l history, which was the device that L 6 w e n h e i m picked up for his proof.

E l e v e n t h L e c t u r e

S t u d i e s o f E l i m i n a t i o n , P r o d u c t a n d S u m Problems

w 29. On Peirce's So-called "Development F o r m u l a s " S u m m a t i o n a n d Product

Eva lua t ion . On the Inversion Problem.

O n page 190 of "Note B" (Peirce 1883; see also Peirce 1880, p. 55), Page 491 Peirce remarks that in the relative a lgebra there are a n u m b e r of "cu-

rious d e v e l o p m e n t formulas," such as

ab ; c = II (a " uc + b" f~c), (a + b) 3- c = F., {a d (u + c) }{b o'- (u + c) }, ~ 1)

a" bc = II (au" b + ati" c), a d- (b + c) - E {(a + u) a- b]{(a + zi) a'- c}, u u

where the II and E as identical p r o d u c t and identical sum, respectively, e x t e n d over all relatives of the universe of discourse 12 . . . . 1

Page 497 Peirce 's p ropos i t ions 1), and ou r ex tens ion of them, fo rm the "start- up stock" of propos i t ions and m e t h o d s which allow us to evaluate sums

E as well as products II in our discipline. It is of ten useful to be able to give the identi ty p r o d u c t ( the inter-

sect ion [Gemeinheit]) of all relatives x which fulfill a cer ta in condi t ion , for example , roots of a given e q u a t i o n ~ a s we have al ready shown in the n in th lecture. Likewise, the ques t ion r ega rd ing the identi ty sum of all roots can be of impor tance . For this reason alone, the art o f deter- m i n i n g the sum and p r o d u c t ~ w h i c h is no t so easy- -dese rves to be cult ivated and deve loped systematically. Tha t this is f u n d a m e n t a l for e l imina t ion problems, and for in fe rence in genera l , was shown at the e n d of w 28.

I shall now presen t a series of my own findings which aim at increas ing

this capital (science); they no longe r relate to Peirce 's publ icat ions , bu t are never theless relevant.

We are dea l ing with sums E; and produc ts II which have "absolute" ex tens ion , i.e., over the ent i re d o m a i n of discourse. D e p e n d i n g on w h e t h e r the index is an e l e m e n t symbol i or j, etc., and its ex tens ion is the first d o m a i n of discourse 1', or w h e t h e r it appears as the sum or p r o d u c t variable of a binary relative u of any given type with the s econd

' Schr6der's proofs of Peirce's "curious development formulae" (pp. 521-536) have been omitted here.


d o m a i n of discourse 12 as its ex tens ion , we can dis t inguish two o rders [Stufe] in sum and p r o d u c t de t e rmina t i on .

Even t h o u g h Peirce 's p ropos i t ions 1) a l ready b e l o n g to the s e c ond o r d e r [zweite Stufe], we want, first, to cons ide r the p r o b l e m s which b e l o n g to the first o r d e r [erste Stufe].

To begin with, the r eade r can easily prove these g roups of little proposit ions by m e a n s of the coeff ic ient evidence:

Ei =E~= E i ' = E ~ = 1, n i = n ; = n ~ = n ~ = 0, 7)

E i i = 1' = l-I(/+ ~) = H(~ + ;), Ei~ = Ei-/" = 0 ' = II(~+ ~), 8)

~ [ = 1, II(i + i) = 0. 9 , )

T h e index is a s sumed always to be i.

For the proof, one only has to add the common coefficient of the suffix hk Page 498 for E or l-I, respectively, and to discuss it. For example, formula 9), on the left,

becomes

7 7 / I

If the domain 1 ~ has more than two elements, there is also an i for h :~ k. For / !

this i, 0;h0;k = 1 because it is different from both elements h and k, and the last Ei equals 1, q.e.d.

For the domain 1,~ of only two elements, the right sides in equations 9) would have to be replaced by 1' or 0'.

If we take, for the t ime being, J as the represen ta t ive of o n e of the four e l e m e n t relatives i, i, i, ;, we can shor t en fo rmu la 7) as follows:

E J = l , n j = o .

Because II~(i)J~--IIJ, etc., it is clear that m o r e genera l ly we mus t have

{4~(i) + J} = 1, II 4~(i)J = o.

Thus, we do no t have to write ou t the p r o d u c t formulas for express ions such as IIi a ; i " i, IIii" i;b, because we immedia te ly recogn ize t h e m to be equal to 0. Etc.

Propos i t ions such as

E 0 ij '= 1, II 0 ( ~ + ] ) =0,

which refer to doub le or mul t ip le sums or products , we do no t want to take into cons ide ra t ion for the t ime being.

R e g a r d i n g fo rmula 7), the following p ropos i t ion comes f rom 29) o f w 25:

342

" - " / Z i a ; i " i = Z~(aj . i ) i

Z i i " i , a - Z~i( [j- a)

SCHRODER'S LECTURE XI

{ H , ( a ; i + ~ ) = H i ( a o ~ + ~ )

I I i ( ~ + i ; a ) = I I , ( ~ + ~ j - a). 10)

Of course, it is also easy to produce the coefficient evidence for any of the

formulas, for example,

{if, (a" i + ~)}hk = II;{(a" i)hk + '=kh} = H;(ah; + 0~k) = ahk-

Finally, these can be in fe r redmin the form of r.,~a'i, i ; l '= a" 1 ~, for exam- p l e m f r o m the more general proposit ion 14) which we will give later.

F u r t h e r m o r e , we a re ab le to eva lua t e Z i a n d IIi fo r the 16 o p e r a t i o n s

c o m b i n i n g a g e n e r a l re la t ive a a n d a re la t ive o f i, wh ich w e re d i s cus sed

Page 499 in f o r m u l a s 21) to 23) o f w 25. T h e resul t s in q u e s t i o n a re e x p r e s s e d by 32 f o r m u l a s wh ich , i n c i d e n -

tally, also p r o v i d e r e p r e s e n t a t i o n s for the o p e r a t i o n s c o m b i n i n g the

m o d u l e s a n d a. If we a s s u m e a as c o n s t a n t with r e s p e c t to the i n d e x

w h i c h is to be a s s u m e d as i, we have

1 = Z(ij- a) = E(~j- a) = Z(a j- i) = Z(a d- ~), 1 1)

O= I I i ; a = H i ; a = H a ; i = H a ; ~ ,

a" 1 = Za" i = Za" ~-= Za" i'= Z a ' ~ = Z:(aj- [),

1 ; a = E i ; a = Z ~ ' a = Z i ; a = Z ~ ; a = Z ( ~ j - a) ,

a j - 0 = H(aj - i) = I I (a j - ~) = 1-I(a 0, i) = I I ( a ~ ~) = H a ; i,

0 0' a = H(i 0 ~ a) = H(~o ~ a) = H ( i o ~ a) = II( ~j- a) = I I i ; a,

12)

{ E i ( a ~ i ) = ( a j - l ' ) ; 1 ,

I]i (i& a) = 1; (1'0" a),

Hi a ; [ = a ; 0 ' j - 0 ,

Hi i ; a = 0 0 , 0 " , a. 13)

For the proof of 11), we should r e m e m b e r m f o r example, on the right

s idemtha t i ; a - i " 1 ; a b y 21) o fw 25; therefore, IIi; a = 1 ;a" Hi, which vanishes

according to 7). Etc.

For 12), we also only have to take 7), according to which we have

F.,a; i = a;F.,i=a;1, II(a3- i) = a3- Hi = a3-0,

and, finally, a; i = a3- z from 22) of w 25. One part of these formulas we can infer

in like manne r (because of i';1 - 1) as

Z , a ; i = Z ; a ; i " i ; l = a ; 1 , I I~a; i=II~(a; i+ i';0) = a3-0,

from the more general proposit ion 14), which will follow.


Thus, of these formulas, only the last, 13), needs to be explained.Justification is given for the top right formula by the observation that Lhk- lI,(a; ~)hk is different from Rhk only because of the different designation of the product index (i instead of m)--with respect to the proof given for 28) of w 25.

Remarkably simply and impor tant , it seems to me, are the following groups of proposi t ions:

a ; b = F,,a ; i" i; b = F,, (a j- ~')( ~j- b)

, . , v

= E~ia; 1 ;bi = Z,{( ~ +a) j- 0}{0 j- (b + ~)},

a j- b = I-I, (a; i + i; b) = II, (a c~ ~ + ~J b)

=II~{(~+a) j -0 j - (b + ~ ) = I I , ( i a ; 1 + 1 ;bi). 14)

It is a good idea to memor ize the first formulas on the r ight of the Page 500 equal sign for each of these expressions. They teach us how to break

down a relative product into an identity sum, and a relative sum into an identity product.

The proof is achieved fastest with 32) of w 25 where we have Y~ia;i. i; b =

E ~ a ; i b = a ; b E i = a ; b l - a;b according to 7). Etc. The other forms of the proposition are modifications of the one proved from 22) of w 25--and we could state several others as well.

The dual to 14) is

v , . .

I I ~ a ; i . i ; b = ( a j - O ) ( O ~ b ) , E ~ ( a ; i + i ; b ) = a ; l + l ; b , 15)

because of 12), where, on the left, we must have I I ~ a ; i . i ; b = H~a; i �9 II~i'; b, etc.

While we now can also easily evaluate

II~ai ; b = II, a ; ib = (a ~ 0)(0 ~ b),

E , { ( a + z) j . b } = E , l a ~ ( i + b ) } = a ; 1 + 1 ;b 15~)

---cf. 32) of w 25- - i t is not at all t rue for II~a; ~b, and, in general , we still look at the large majority of sum and p roduc t expressions with confusion.

There fo re it seems advisable, first, to have at our disposal as completely as possible the simplest sum and p roduc t formulas f rom the ':start-up stock" of our discipline, and, second, to learn me thods to refer a given summat ion problem, etc., back to the p rob lems solved with the simple formulas f rom the "start-up stock."

Conce rn ing the first task, we believe we should state, discuss, and

344 SCHRODER'S LECTURE XI

prove at the very least the fol lowing set of p ropos i t ions , as duals and s u p p l e m e n t s of 10)"

{ E ; a ; ~ . i'= a;O' , H ; ( a j - i + ~ ) = a j - l ' , - 16)

I];i" ; ; a = 0 ' ; a , II; ( i + i'o ~ a) = l ' j - a ,

, . , , . ,

Eia; i ; i = ~;(aj- ~) i= a ; 0 ' ,

E,~" i; a = E,~(~0 ~ a) = 0 ' ; a,

II,(a ; i + ~') = II,(a j- ~ + i) = a j- 1',

1-I;(i + i; a) = l-I;(/+ ~ a) = 1 '~ a, 17)

{ H,(a ; ~+ ~) =a;O ' ,

H ; ( ~ + ; ; a) = 0", a,

E; (aj- i ) i = aj- 1',

E; i ( i ; a) = 1'~ a, 18)

Page 501

v

Z ; a ; [ ' i = a ; 1 ,

~;~" i ; a = 1 ;a,

H, (a j - i + i) = aj-O,

l-I; (i + i'd" a) = 0 d" a , 19.)

where , w h e n wi thou t the asterisk, r ead a ; 0 ' ; 0 ' for a ; 1 . Etc.,

{ E , ( a ~ i ) ~ = ( a ~ l ' ) ; O ' , II, (a ; ~ + / ) = a i 0 ' ~ 1', _ , . , ~ ,

~ ; i ( i j - a ) = 0 ' ; ( l ' ~ a ) , I I ; ( i + i ; a ) = l ' ~ 0 ' ; a , 20)

In the 32 fo rmulas 10) and 16) to 20), we will f ind the ~ and II o f all (b inary identi ty) p roduc t s and sums which can be f o r m e d f rom a ; i or a;~, as well as f rom a j - ~ o r a j - i and ; o r ~, e t c . - - a s long as at least those E, II that can be r e d u c e d to 0 or 1 at o n e g lance are no t t aken in to cons ide ra t ion .

Formulas of the type which, in their general terms a" i or a" ~ appears instead _ , . ,

of a" i or a" i, etc., can easily be reduced to something we know because a" i or a'z splits into a" l ' i , respectively, a" 1 �9 ~; they do not belong have the same

status as the formulas discussed so far and do not deserve to be stated together

with them.

O f the fo rmulas s tated, f o r m u l a 18) appea r s as especial ly r e m a r k a b l e because it t eaches us to r e p r e s e n t cer ta in relative p roduc t s , such as

a ; 0 ' , as ident i ty products, while, in genera l , this is only ca r r i ed ou t in the fo rm of an ident i ty sum.

For the proof of the propositions, we refer, for 16), to 30) of w 25 and to 7). Formulas 17) are particular cases which result from our theorem 14), in which.

for example, the fight side has to be II,(a" i + z') -- 1-Ii(a" i + f; 1') = aj- 1'.

For formula 18), we transform via the identity calculus" a" i+ ~ --a" f" i+ ~=

a ' 0 " i + ~ = a ' 0 ' +~ according to 30) of w 25, where we must have ri;(a.0q- ~) = a" 0' + II;~= a" 0' + 0, according to 7).


For formulas 19) and 20), we appeal to the coefficient evidence, whereupon the left side is, respectively,

Lhk=Ei (a ' [ )hk ~ - - , , t^k = EiE t amtu, tk^ = E t a h l ~ , i O l i O i k = (a 'O' "O')hk, 7 ! =

Lhk=~;IIt(aht+ ilk)Zkh=F.,iIIt(aht+ lt;)01k R^k, q.e.d.

As dua l a n d c o m p l e m e n t to f o r m u l a s 14), 15), we l ikewise have to cite the fo l lowing p r o p o s i t i o n s wh ich we can see as a g e n e r a l i z a t i o n o f p r o p o s i t i o n s 16) to 20) above:

E , ( a j - i ) ( i j - b ) = ( a j - l ' ) ; ( l ' j -b ) , I I , ( a ; ~ + ~ ; b ) = a ; O ' j - O ' ; b , 21)

. . . . . ,

B, (a j - i)( i ~ b) = r . , (a j , i) �9 i; b

= (a o" 1') ;b ,

~, (a j- ;)( i ~ b) = E,a ; i " ( i j - b)

= a ; (l'c~ b),

II,(a ; ~ + i; b) = II,(a ; i + ~ b)

= a ; O ' ~ b, , 1 . . . .

II,(a ; i+ ; ; b) = R ( a ~ ~+ ~; b)

= a j - O ' ", b,

22)

{ E , a ; i" i'; b = E,a; ~. (~ j . b) ] -- _ _- J = a ; O " b , ~ , a ; i" ~;b = ~,(a ~t i)" i ;b

{ I I i ( a o ~ i + ~ j - b) = I I , ( a j - i + i';b) } 1'

I-I, (a j- i + i ~ b) = l I , (a ; i + i j- b) = a j- j- b, 23)

Page 502 I2,a ; ~. ~; b = a ; 1 ; b, II,(a j- i + i'0 ~ b) = a ~ 0 ~ b. 24*)

T h e II, o f the g e n e r a l t e rms o n the left a n d the ~i o f t hose o n the r igh t a re easy to state a c c o r d i n g to 11) a n d 13) s ince the g e n e r a l t e rms a re c o m p o s i t e a n d can be split up.

Proof. According to 3) and 4) of w 25, we can write

i 1' ', '. = = at ~, i=/at 1 i= 0 i, /' i ; 0 ' ,

whereupon we can then rewrite

{ a at i = a at l t at i= (aat 1')"i, a " ~ = a " O " i , . - . 25)

i a tb=~a t l ' a t b = i ; ( l ' a t b ) , ; ; b = i ; O " b

_

(as well as a a t i = a ' i , ; a t b = i ; b ) , in addition to 23) o f w 25, page 418.

T h e r e u p o n , all o f the f o r m u l a s 21) t h r o u g h 24) fall u n d e r the s c h e m e (o f the first e q u a t i o n o n the left a n d the r ight ) o f o u r p r o p o s i t i o n 14).

We can see f r o m 25) in c o n n e c t i o n with 21) a n d 22) o f w 25 that , in

Page 503

Page 504

34 6 SCHR6DVR'S LECTURE XI

c o m b i n i n g relat ives with e l e m e n t re la t ions , re la t ive a d d i t i o n is always

d i s p e n s a b l e , namely , it can be p l ayed o u t as a re la t ive m u l t i p l i c a t i o n

( even w i t h o u t c o n t r a p o s i t i o n ) , w h e n it d o e s n o t na tu ra l l y t e r m i n a t e in

i den t i t y a d d i t i o n . Basically, o n e on ly n e e d s to l e a rn to ca l cu la t e well

with the e x p r e s s i o n s of the two f o r m s a ; i a n d i ; b . . . . 2

I n s t e a d o f e s t ab l i sh ing still m o r e fo rmu l a s , we n o w w a n t to d e m o n -

s t ra te , wi th a ser ies of sma l l e r p r o b l e m s , h o w we can d e t e r m i n e , with t he p r o p o s i t i o n s m e n t i o n e d so far, n u m e r o u s a n d var ious ly s t a t ed p r o d -

ucts a n d sums. For tha t p u r p o s e , we shall m a i n l y use p r o d u c t s , a n d n o t try to c o m p l e t e the g r o u p s o f a s soc ia t ed fo rmu la s . F r o m the way we

t r ea t t hese e x a m p l e s , the r e a d e r will at least be ab le to ge t an abs t r ac t

i dea o f o u r m e t h o d s , a n d m o r e so in the fo l lowing sec t ions .

Problem 1. We are looking for x = IIi (a ; i'+ ~; b).

We have x = I I ; ( a ; l " i '+~;b) =II ;(a;1 +~;b)( i '+~;b) =(a ;1 + I I i ; ' , b ) I I , ( ; +

f ;0 ' ;b) =(a ;1 + 0 , : t 0 ' ; b ) I I ; i ' ; ( l ' + 0 ' ; b ) by 13), and thus finally x = ( a ; 1 + 0 , r

0' ; b)10 0'- (1' + 0'; b)].

Problem 2. We are looking for x = H i (a; i'+ f; b).

Solution: x=r I i (a ;1 �9 f+ [. 1;b) =(a;1 + 1 ;b)(a;1 + II ih(II i i '+ 1;b)rI,(i+ f) = a ; l " 1 ;b" I I i ( 0 ' ; i + i'; 1')= a; 1 ;b" (0'0~ 1'); therefore x = l " a;1 ;b.

Problem 3. We are looking for x = H i (a; i + i; b) = II i (a ~ ~+ i; b).

x = I I i ( a ; i + i" 1 ;b) = ( I I i a ; i + 1 ;b) I I i (a; i+ i) = (a~0 + 1 ;b)II i (a+ 1 ' ) ; i = ( a ~ 0 + 1;b){(a+ l ' ) ~ 0 } = a ( a + l ' ) ~ 0 + { ( a + l ' ) z t 0 } " 1;b; therefore x = a ~ 0 + {(a+ 1') ~0} ;b.

In particular, if we let a---0' and thereafter take a for b, we get the second formula on the left of the following group:

a" 1 = l-I, (a" i + i; 0') = IIi(a" i+ ~), a j- 0 = Ei (a j- ~) i',

1 " a = I I i ( 0 " i + i 'a ) =II ; (~+ i 'a) , O j -a=Ei i (~o-a) , 32)

which is an interesting dual to 18) in as much as it shows how the relative

product a; 1, etc., can also be represented as a l-I; (instead of, as usual, Ei). The

formulas, incidentally, can also be easily unders tood directly.

Problem 4. We are looking for x = II; (a ~t i + i; b).

Considering 25), it comes as x = IIi{(a j- 1') ; i + i; b} under the previously solved

problem, and, therefore, has to be

x = a j - O + {(a~ 1' + l ') , : t0l" 1 ;b.

We can also de termine x by means of a double product as follows. Because of 14), we have

Schr6der's first-order identities involving negation and converse (pp. 501-503) have been omitted here.

Page 505


x = II,{IIi(a ;j + j-; i) + i; b} = II u (a ;j + j';i + i; b) = IIj{a ;j + l-I, (j-;i + i; b)}.

Now from the scheme of the previous problem, we have

n;(]; i+ i;b) =j'j-0 + {0"+ 1') ~0}" 1 ; b = 0 + ( l ' j - j ) �9 1 ;b=0*

by 3) of w 25, and therefore *x -- Ilja ;j - a3- 0.

Problem 5. We are looking for x = I-I; (a j- [+ i; b) = H i (a; i + [; b).

x=IIa(a;i+ i" 1 ;b) = ( I I~a ; i+ 1 ;b)IIa(a;i+ ~) = (aj-0 + 1 ;b)II;(a + 0 ' ) ; i .

Thus x = ( a j - O + l ; b ) { ( a + O ' ) ~ O } = a & O + l ~ a ; l ' l ; b = a j - O + l ' a ; l ' l ; b = a j - 0 + l ~ t ' l " l ' b .

Problem 6. We are looking for x=l-I~(aj-i+ i;b) =l-I;{(a~ 1 ' ) ; i + i;b)}.

By the scheme of the previous problem, we can immediately state: x = a~

0 + l'(aj- 1') ;1 ;b.

But we can also go through the double product: x = H;Hj(a;j + j'; i + i; b) =

I l f la; j +j-,r + 1)"; 1 "b) = I l f l a ; j + l ' ; j ;b ) =IIj(a;j+j'b) =a~t0 + {(a+ 1')o'-0}" b,

according to problems 5 and 3. The two results coincide because of the second

formula of proposi t ion 30. The II in problems 6 and 3 are the same!

Problem 7. We are looking for x = II; (i; a + ~-; b).

x=IIi(i;a+ ~" 1;b) = ( I I ; i ; a + 1;b)IIi(i;a+ ~) = ( 0 + 1 ; b ) I I ; ( 0 ' ; i + i ;a) ;

therefore, from problem 3: x = I ;a" 1 ;b.

As n u m e r o u s as the p r o b l e m s are wh ich can be so lved in this way, we

still c a n n o t d iscover , for e x a m p l e , Ili a;~b.

Le t us t u r n now to the s u m a n d p r o d u c t p r o b l e m s o f t he s e c o n d

o r d e r [zweiten Stufe]. An i m p o r t a n t p r o b l e m of a g e n e r a l c h a r a c t e r is: to determine the "in-

tersection"[ Gemeinheit] H, as well as the ( c o m m o n o r to ta l ) "domain" [Ber- eich] ~ of all binary relatives x which fulfill a given condition--for e x a m p l e ,

t he e q u a t i o n F(x) = 0 - - a s t h e i r " roots . "

It makes sense to deno te these two unknowns (as product and as sum) by P

and S. But the product is the Subject and the sum is the Predicate to each of

the roots, so that this denota t ion could be misleading. I will therefore call them

P and Q.

By marking the extension condi t ion under the signs H, ~, the mathemat ic ian

would be inclined to write

P=IIx Q=2x x x

{F(x) =0} {F(x) =0}.

But our discipline has the advantage that the extension condi t ion can be in-

corpora ted into the H, E-expressions themselves. How this can be done shall be

explained shortly with an obvious extension of the problem.

T h e p r o b l e m can be c o n s i d e r a b l y g e n e r a l i z e d w h e n we try. to m u l t i p l y


or a d d a given func t i on ~(x) o f the roots , i n s t ead o f the r o o t x itself.

We c o u l d t h e r e f o r e look for

P = II r Q = g r x x

{F(x) = 0} {F(x) = 0}.

Page 506

We give II a n d ~ abso lu t e e x t e n s i o n m o v e r all poss ib le relat ives x o f

the s e c o n d d o m a i n o f d iscourse . Now we only have to m a k e the g e n e r a l

t e r m n e u t r a l w h e n x does not fulfill the e x t e n s i o n c o n d i t i o n F(x) = O, t ha t is, we on ly have to worry that , in e a c h such case, the g e n e r a l t e r m

o f 1-I a n d E is neu t ra l , n a m e l y equa l to 1 if it is a p r o d u c t factor, a n d

e q u a l to 0, if it is a s u m m a n d . O n the o t h e r h a n d , the t e r m ~(x) m u s t

ac tual ly a p p e a r o r be effective for every x which fulfills the e x t e n s i o n

c o n d i t i o n .

We can do this by wr i t ing

P = 11 { ~ ( x ) + F(x) = 0 l , Q = E ~b(x){F(x) = 0}. 33) x x

D e p e n d i n g on w h e t h e r x is r o o t o r not , the fac to r p r o p o s i t i o n

F(x) = 0 in Qwil l in fact have the t r u th value 1 o r 0; on the o t h e r h a n d ,

t he n e g a t i o n which a p p e a r s as a s u m m a n d in P will be e q u a l 0 o r 1,

etc.

In case the polynomial F(x) of our equation would only be able to express

the values 0 and 1 as proposition symbols, for example a coefficient function

or even as a "distinguished" relative, we can simplify the above as follows:

P = 11 {cI,(x) + F(x)], Q = ~ ~(x) F(x). x x

m

In this case, we would have (F #: 0) - (F= 1) - F a n d (F=0) -- (F = 1) -- F. In any

other case, this would be a grave mistake.

In gene ra l , we can now rep lace , in 33), the p r o p o s i t i o n a l t e rm, ac-

c o r d i n g to the s c h e m e of w 11, by a b ina ry relat ive, t ha t is to say, a

d i s t i n g u i s h e d relat ive which t h e n takes on the value 0 o r 1. T h e n we

have

F(x) = 0 = {F(x) 4: 0} = 1 ;F(x) ; 1, m

{F(x) = 0} - 0 ~ F(x) ~ 0.

T h e n it follows

P = II {O(x) + 1 ; F(x) ; 1 l, Q = 1~ O(x){0 e F(x) ~ 0l. 34) x x

H e r e u can be wr i t ten for x.

In the lower p o r t i o n o f the p r o b l e m , which i n t e r e s t e d us initially, we

have, in par t icu lar ,

Page 507

Page 508


II{x + F(x) =0} = H{U + 1 ;F(x) �9 1}, x u

{F(x) = 0} = g u{0 e F(u) e 0}. 35) x u

If we could evaluate H, respectively g, taken over u with the absolute ex tens ion (over all binary relatives), for any given funct ion of u, we would be able, with schemes 34), 35), to obtain the P and Q in ques- t i o n - - e v e n wi thout knowing or d e t e r m i n i n g the root x [of the equa t ion

F(x) = 0 ] .

If, however, we know the genera l root or the solut ion of this conditional equa t ion in the form of

x =f(u),

we are in an even be t te r posit ion to find the solut ion of our p rob lem,

as we have m o r e immedia te ly and simply

P = H ~{f(u)}, Q = ~ ~{f(u)}, 36) u

as well as for the lower por t ion of 35) of our p rob lem:

II,, {x + F(x) = 0} = H f(u), r,x x{F(x) = 0} = ~f(u). 37)

According to the idea of the genera l solution, we have then in fact for e v e r y u:

Fir(u)} = O, Fl f(u)} 1.

W h e t h e r we chose this way or that, the art consists in determining the II and ~ taken over u with absolute extension from any given relative function " ~ ( u ) .

A method of solving this p rob l em in its full and un l imi t ed general i ty is no t known.* But to discover such a m e t h o d is the ideal of this theory which will pe rhaps never be possible. Probably we will only be able to solve it in stages and thus app roach this faraway goal slowly.

For the t ime being, we can only set a defini te goal for ourselves and, in o r d e r to reach it, create a par t of the me thod .

Such a practical g o a l - - a n d , in fact, the closest one f rom a systematic

po in t of view--is the d e t e r m i n a t i o n of H and E of all roots of one of

ou r three e l emen ta ry inversion problems.

Since x= u(aj-~) is the general root of the subsumption x" b 4= a, and, of

course, we must have

IIu =0, Yu=l, u u

* Compare the end of this section.

35 ~ SCHRODER'S LECTURE XI

and since u =0 and u = 1 themselves figure unde r the values over which u ex-

tends, we have to have I I u c = c I I u - - O , I ; u c - - c r . u = c , as well as u u u u

II (x + x" b=6--a) = O, E (x" b=6--a) = a j- ~. x x

quod erat inveniendum.

Whenever 0 belongs to the roots x of the given condit ion, then Hx is equal

to 0; and when to 1 belongs to them, then Z;x is equal to 1 and will not interest

us any longer here.

For example,

I I u ' a = 0 , E u ' a = l ' a u u

is immediate .

T h u s , t h e r e r e m a i n s to be so lved t he i nve r s i on p r o b l e m for t he (ex-

t e n d e d ) s e c o n d case a n d for t he t h i r d case, a n d h e r e it will be im-

p o r t a n t u i f we on ly dea l with o n e r e p r e s e n t a t i v e for e a c h c a s e m t h a t we

l e a r n to e v a l u a t e a p r o d u c t in t he f o r m o f

x = I I [u + a{(b + (t) j- c} ; d]. 38) u

In o r d e r to rea l ize this goal , w h i c h can on ly be a p p r o a c h e d in s tages ,

we n e e d to tackle a ser ies o f p r e l i m i n a r y p r o b l e m s .

Problem 8. We are looking for: x = II (u + 7i" a). u

According to Peirce's proposi t ion 1) or 4), we can immediately state

x = 1 �9 al ' , and so x = II (u" 1' + ,i" a). u

Because of 1 �9 i1~= i; 1~= i, we have, in particular,

II (u + ti ' i) = i, and similarly II (u + fi" i) = [. u u

As a corollary, we now also have found:

I I ( u + a " ~ ' b ) = a ' l ' b = a " l ' b l ' , u

because the general factor can be split into (u + a)(u + (t" b), and therefore the

II can be separated into the II of the first factor,

I I ( u + a ) = a + I Iu = a + O = a,

and the II of the second, which falls unde r the above schema.

Problem 9. We are looking for: x = II ( u ' a + (t r b). u

According to 14), we can represent tij-b as a product (over z), by which we

have won the game! Because we can now conclude

FROM PEIRCE TO SKOLEM 35 x

x = I I { u ' a + II;(~i" i + i;b)] u

Page 509

•II;{II (u" a + ~i'i) + i;b}-II;(1 "ai+ i;b) u

accord ing to 4), and fur ther : x=II~(i;a+ i;b) =I'I;/; ( a + b).

By the last fo rmula 12), we have thus found:

x --0 ~ (a + b)

m w h i c h is, curiously enough , symmetr ic with respect to a and b.

If we assume that a -- 1' and then take a for b, then we have found , in particular,

I I ( u + 7i,y a) = 0 ~ ( a + 1'). u

[However, if we were to assume b =0' , exchang ing u for ti, we would get the

result of p r o b l e m 8 again, a l though in the somewha t d i f fe ren t form of 0j-

(a+ 0').] As corol lary of this result, we now also know

I I { u + a(~i~ b)} =a{Oj- (b+ 1')}, u

which can be in fe r red in a similar way as above.

With exactly the same me thod , we can also solve an expans ion of the previous

p rob lem.

P rob lem 10. We are looking for x = II {u" a + (~i + b) j- c}. u

x = I I I I i { u ' a + (~i+ b)"i+ i;c} =II~{b" i + i ; c + I I ( u ' a + fi'i)} u u

=YI~(b" i + i ;c+ 1 "ai) =II;{b" i + i ; ( c + a)} = b,y (a + c)

by 14), 4), and 14). In particular, we have now found

YI{u+ (7i+ a) o~b} = a,y (1' + b), u

1-[ [u + a{(Ti + b) ,t- c}] = a{b3- (c + 1')}. u

As in 1), by means of a I~ over u, so we can now also r e p r e s e n t a0,- (b + c)mal-

t h o u g h its form is u n s y m m e t r i c a l m b y a H over u.

P rob l em 11.* We are looking for x -- H{u + mi" b}. u

We will not succeed in solving this p r o b l e m in the same way as the two previous

ones because we canno t r ep re sen t the relative p roduc t of the s econd te rm of

the genera l factor as H; but only as Ei, accord ing to 14)" however, a I] c a n n o t

be t rans fe r red f rom beh ind a II to the f ront of it. In the case of b =0 ' , we can

refer to 18). The re fo re , we must, in fac tmsolve only the s implest case:

n(u+ a .0 ' ) :n , l :+n( , ,+ a. ;) l-n;(~+h =n;~=0 u u

accord ing to 7) and p r o b l e m 8 - -wh ich we have al ready u n d e r s t o o d on page

* Because of u = 1 u" 1' its quickest solution comes from 6) p. 496, and is a special case ~ �9

35 2 SCHRODER'S LECTURE XI

494 from Peirce's formula 1). Now we also have aft" b0':(= ft '0 ' , and therefore also

Page 510

II (u + aft" b0') :(= II (u + ft" 0') = 0. u u

If we solve (for x) b =0'b + l'b, and if we bear in mind that, according to 24) of

w 22, we have a~i" b l ' - a f t " 1" bl', then we can split the general term,

u + a f t ' b = u + a f t 'bO' + a" 1 "bl',

suppress ing the factor ft in the last term in the presence of the first. We then

obtain

x = a " l'b,

while the H on the r ight-hand side vanishes by the previous result. In particular,

we have

II (u + aft" 1) = a. u

Please observe that, in spite of the p lacement of the parentheses which are

different from those in the corollary to problem 8, the end result of the two is

the same.

We can also immediately recognize the new value as the lower limit for x

because

y = II (au + aft" b) = II (au" 1' + aft" b) = a" l'b u u

according to Peirce's theorem 1, but, because of au =6- u, we must have y =6= x.

The same lower limit can also be de t e rmined by means of

Xhk = II [Uhk + E t ahtfthtbtk ] = II E l (Uhk + ahtbtk)(Uhk + fthl) u u

because we must have EtH :(= HE t according to the proposi t ional scheme o) on

page 41. Because we now have I luhk=0 , we get, as we will show in the next p rob lem,

II(uh, + fth,)= l't,, SO follows E, ah, b,,l'~,= (a" l'b)hk=C--Xhk.

[It is conceivable to set an upper limit for the lower limit for x. The reasoning

goes thus: aft;b=6--a;b" ft;b, therefore x=6--a;b" 1 ;b l '= a ; b ; l ' b , and the propo-

sition a; l'b=6--a;b; l'b has to be valid, which also can easily be proved directly.

However, we have already recognized the lower limit as the exact value.]

P r o b l e m 12. We are l o o k i n g for: x = II {u + (7i 0 ~ a) ; b}.

T h i s is a d i f f icu l t one ! Its s o l u t i o n will o n l y be poss ib l e fo r c e r t a i n

spec ia l cases, s u c h as

I I{u + (7i0~0) ;b} =0, t t

I I { u + (720~ a ) ; l ' b } =00~ (a0 ' + bl ' ) , u

for w h i c h we ge t t he r e su l t easi ly f r o m w h a t we have l e a r n e d so far:

b e c a u s e t h e s e c o n d t e r m o f t he g e n e r a l f a c to r spli ts i n to (7i0~0) �9 1 ;b


or (~20 ~ a) �9 1 ;bl ' , r e s p e c t i v e l y ~ w h e r e u p o n the g e n e r a l f ac to r i tself a n d

its II split up. This last resul t is o b t a i n e d by m u l t i p ly ing the c o m b i n a t i o n

00 ~ (a + 1') with 1 ;bl ' , which is = 0 ~ (b + 0 ') . An important special case in which the solution is easy is the case b = i.

If we let

y =II{u + (tij-a)"i} u

and deal first with this subproblem, we have

yhk--~{Uh, + Et(tij-a)mit,} =~{Uhk + (~i3- a)hi}

=IIluhk + IIt (liht + at;)} = IItlati + II (u^k + ~iht)} =l-It(at; + l~t) u u

: ( l ' j- a)ki = {(l'J- a) "i}kh.

Page 511 That in fact we obtain

II (uhk + ~ih~) = l't, u

can be explained as follows: for l = k this II equals 1, but for l ~ k it has to be

equal to 0; the latter vanishes as a factor of II because then we also have among

the u (that is, among all possible ones) some for which u^, =0 and, at the same

time, uht= 1; therefore also 7iht =0. We have now found

y = i; (d0 ~ 1').

In the g e n e r a l case we can (again) f ind two limits b e t w e e n which the u n k n o w n (bu t fully d e t e r m i n e d ) relat ive x m u s t be. To d e t e r m i n e these

limits b e f o r e h a n d is wor th the t r o u b l e for two reasons . First, they give u s m a s in the p rev ious special c a s e s m a va luable way o f c h e c k i n g the

exac t value o f x which we shall f ind la ter with c o m p l e t e l y new m e t h o d s .

Also, we can t h e r e b y d iscover r e m a r k a b l e t h e o r e m s by s h e e r luck. By 14), we have x = IIF,~{u + (~ i j -a ) ; i " i;b}, a n d s ince, by the p rop -

u

os i t iona l s c h e m e 0), page 41, we have EII :(= l iE , we m u s t ge t

E~ I I{u+ (Tiz~ a ) ; i } I I ( u + i;b) = ~ i ; (~izj 1') �9 i;b=F,~i;(gtztl ')b:~--x u u

- - - c o m p a r e 26) o f w 25; t h e r e f o r e , by 12), 1 ;(~i# l')b:~--x, which gives

us the lower limit. In o r d e r to f ind the u p p e r limit, we write, also a c c o r d i n g to 14)"

x = II [u + {II~(~i; i + i ;a)};b]. But h e r e we are n o t p e r m i t t e d to i g n o r e I t

the p a r e n t h e s e s . S u p p r e s s i n g t h e m would , on the cont ra ry , a m o u n t to

d i sp l ac ing t h e m a n d wou ld set t h e m over, a c c o r d i n g to s c h e m e

{IIa} ; blc--IIa ; b, which = II{a ; b}. Thus ,

v ,,,

x:~--II~[H(u+ ~i ; i ;b ) + i ;a;b] =II~{1 ; ( i ; b ) l ' + i ;a ;b}

354 SCHRODER'S LECTURE XI , . . , , ,

---cf. p r o b l e m 8. But because of i ; b = i . 1 ; b and 1 ; i 1 ' - i ; 1 ' - i , the first t e rm can be c h a n g e d into i" 1" b and we obtain:

x ~: II~i; (1' + a;b) �9 (1 ; b + f l~ i ;a;b)

= { 0 f f ( l ' + a ; b ) } ( 1 ; b + 0 j - a ; b )

= 0 e a ; b + { 0 ~ ( a ; b + ]~)}. 1;b.

H e r e the first term, which is c o n t a i n e d in the second one , can be omi t t e d because we have 0 0 ~ a ; b:(=0 0 ~ (a ; b + 1'), as well as 0 o ~ a ; b=(c--a ; b=~- 1 ; b. T h e r e f o r e , the second te rm remains as the upper limit that we sough t and is c o n f i r m e d by the whole express ion, which mus t be

1 ; ( d j - l ' ) b ~ x ~ { O ~ . ( a ; b + l ' ) } . 1 ; b = 0 j - ( a ; b + l ; b . 1').

In the last express ion for x [Translator 's note: x = II [u + {IIi(~; i + " u

i; a)} ;b]], we c a n n o t pull the /I~ ou t o f the braces by an equ iva len t t r ans fo rmat ion , and, by the same token, we c a n n o t push the E; in f ron t

Page 512 of the II, in the first express ion for x [Translator 's note: x = HE~{u + u " u

(~ 0 ~ a) ; i" i; b}], as was asserted above, in o r d e r to f ind the lower limit. This cou ld only succeed with a cou ra ge ous p roc e du re : T h e m e t h o d would be to ope ra t e with infinite (or un l imi ted) multiple

produc t s H; even with one whose H-sign could possibly fo rm a cont in- u u m (in case we would write it down in detail); for example , if we assign to each point of the linea lI c o r r e s p o n d i n g to a p r o d u c t variable specifically chosen. For such produc ts and sums we may also wi thout hes i ta t ion transfer and apply the in fe rence rules which are g u a r a n t e e d by the p ropo- sit ional s cheme based on the dictum de omni.

This is probably the first t ime in ma themat i c s that this has b e e n done . I will t he re fo re guide the s tuden t heuristically a long the pa th on which the m e t h o d first occu r r ed to me.

I first t r ied to ex t end the par t icu lar case y of ou r p r o b l e m - - w h e r e the solut ion was f o u n d - - b y trying to obta in (as a s u b p r o b l e m ) :

z - H{u + (~j- a) ; i + (~o" a) ;j}. u

We have

Zhk = II{Uhk + ({tJ" a)h , + (~J- a)hj} u

"- I-I {Uhk + t im( Uhm "~ a,,,,) + 1-I,,(~h,, + a,q)} u

= II,,,,{ami + a,q + II (Uhk + ftnm + ~h,,)}" u

A n d n o w

II (Uh, + Uhm + Uh,,) = l'mk + l',,k, u

Page 513


namely, equal to 0 for (m :/: k)(n :/: k) because, a m o n g o t h e r things, we

will have a factor with Uhk = O, Uhm = 1, Uh, , = 1, and equal to 1 for (m = k) + (n = k). It follows that:

Z,,k = H,,H,,(I',~ + am, + 1~,,, + a,,j) = Hm( l/kin + a,,,) + H,(ak, + a,j)

= ( l ' j - a)k, + (l 'J- a)ki = {(1' J- a) ; i + (1' ~ a) ;j}kh,

whereby we have found

;

As we now found a solut ion to P rob lem 12 easily in the case when b = i is an e l emen t , as well as when b = i + j represen ts a system of two

e lements , we c a n n o t foresee why it should not also work in the case where b = b ; 1 is a system and the re fore a sum of any n u m b e r of e l ement s that could possibly as points cont inuously fill a line. We immedia te ly observe that the investigation only has to be genera l ize quantitatively, and, indeed , we will find

H{u + ( f t j -a) ;b;1} = l ;f~; (Sj- l '), H{u + (( t j -a) ;1} = l ; (Sj- l '). u u

Looking back at Zhk, we observe that our inferences would not have

been possible if we - -wh ich s e e m e d feasible at f i r s t - -had used the same let ter m for the index n of the last H as for the previous H.

If, however, all terms are products II in a sum with mutual ly inde-

p e n d e n t indices, n a m e d independent ly , then it is possible to advance all of the H, each affixed with its index as suffix, to the left; it is possible to use this insight also for a sum rep re sen t ed symbolically by g.

We now in tend to deduce the fo rmula on the left, on page 512 (bottom) (of which the one on the r ight is mere ly a special c a s e ) - - b y calling s the p r o d u c t H over u we are looking f o r - - b e c a u s e this will give us g rounds to formalize our p rocedure .

For that pu rpose it is conven ien t to use for the system b;1 = b the fo rmula ob ta ined f rom w 27 as b = E~b~i or, shorter, ~ii; where we only have to keep in mind that the sum over i, the ~ which does no t have

the index a d d e d as d e p e n d e n t suffix, but where it appears (ad hoc)

wri t ten u n d e r n e a t h , that this sum does not have the full, bu t a s o m e h o w given ( l imited or un l imi ted) ex tens ion f rom the d o m a i n of discourse

11 of the e lements . For

s = I I{u + (~2o~ a);E;}, t he re fo re Shk=II{u + E;(~20~ a);i}hk u

=HlUhk + E,({tJ- a)h,} = IIluhk + Z,flm(5h,, + am,)}. u u

Withou t a new idea, we c a n n o t con t inue because we c a n n o t br ing

the II m before the ~i and thereby before the II. The idea (already a l luded

to) which will get us fu r the r is the one which~we now general ly fo rmula te

Page 5 1 4

35 6 S C H R C ) D E R ' S L E C T U R E X I

in the main text and conf ron t it with its dual c o u n t e r p a r t - - w i t h o u t m e n t i o n i n g the lat ter very much.

If we have a E i of a lI m of a genera l te rm f(i,m), and we wish for some reason to push the E beh ind the II in an equivalent transformation, this

is no t immedia te ly possible. Because of ~II :(= II~, we could only do so by drawing weakened con-

c lus ions - - i f we want to be satisfied with such a p rocedure . Otherwise , n o t h i n g h inders us f rom r e n a m i n g the index of the IIm in all the o the r

terms of the ~i, that is, "to differentiate" all these indices as m, (m with

the suffix i o t a ) m w h e r e we only have to r e m e m b e r that L changes in "parallel" with i.

It appears suggestive to take for ~ the le t ter i itself as suffix for m. Dis regard ing the fact that m~ already has a fixed m e a n i n g as relative coeff ic ient of the e l e m e n t m in w 27, it still would not be correct . As we

will soon s e e - - i n case we s u c c e e d m w e may not choose for L a symbol which contains the name /-----such as 4~(i).

This will have the advantage that we now can push every single II to be taken over a certain m, to the front, in f ront of our Z. We can now justify the impor t an t formula

{ ~,Ilmf(i,m ) = E,IIm,f(i,m,) = II, (IIm,)]],f(i,m,), I I ,~. f ( i ,m) = II,~m,f(i,m,) = l-I , (~ml)II,f(i,m,),

39)

by which we have a t ta ined our goal of having pushed all II in front of the ~.

To facilitate the pr int ing, we have set the indices i and m as if they, as e lements , would have full ex tens ion over 11. I am sure this is permissible. But it is not at all necessary for the p roo f of our scheme. The indices i and m may have any given extensions in l ~ - - p r e s u p p o s i n g of

course that the extens ion of m is i n d e p e n d e n t of i, the same for each

i and is also t ransfer red to each m,, that is, associated to each of these

(a l imitat ion of which even the last par t of each of the doub le s chema ta

is i n d e p e n d e n t ) . In o the r words, we may also write Ei for Zi or 1-I m for

I-I m .

O u r schema would remain in force even if the two indices, or one

of them, were not e l e m e n t letters but would have their ex tens ion in 12 as a u or v. But we do not want to discuss this possibility here.

The last par t of our s cheme needs a fu r ther explanat ion , but has to be understood first.

If L (in parallel with i) would have to run t h rough a series of values

1,2, . . . , we could explain the m e a n i n g of the mysterious o p e r a t o r in f ron t of the last ~ in 39) by writing it in the ord inary way, expl ic i t ly--by not m e n t i o n i n g the genera l te rm or factor, in the form of


II,(II,,,,) =IIm II,, IIm.~''' o r Hmlm2m3.." = H i l t m t

and def ine it as p roduc t symbol for a (possibly un l imi ted) "mul t ip le product . " And then

Page 515

II,(E,,,) = EmtEm2E,,~... or E,,~m2m3... = En, m,

would be n o t h i n g but the summat ion symbol to indicate a " mul t iple s u m . "

Since the lat ter cor responds , dually, to the mul t ip le p roduc t , we can see that in the new symbol (which is shown to be indispensable for abbrevia t ion here) the II, may not be t ranscr ibed, dually, to E,, but will r emain as II, in the dual c o u n t e r p a r t to the scheme.

It r emains open whe the r the theory will ever make use of symbols

such as E,(IIm, ), E,(Em,), by which only some, any, but at least one of the II or E over m, could be set.

Of the given represen ta t ions or m e t h o d s of expressions, the ones on the r ight are less good, even misleading, for the reason that the com-

posite suffix m 1 m.2m3"" of a II or E should not be a real p r o d u c t (ne i the r an identi ty nor a relative one) , but stands convent ional ly for the "series"

m l , m 2 , m 3, "" (cf. p. 24). It is t rue that our II, does not po in t to a real p r o d u c t ei ther, but to

a succession of signs (of the type indicated after it in pa ren theses ) which may also b e c o m e a con t inuum.

If the L in parallel with i has to run th rough a c o n t i n u u m of values, such as all the points of a line, we can no longer write the m e a n i n g of

II,(IIm,) explicitly. Ar i thmet ic allows us, however, to n a m e t h e m all and differently by assigning each of those points to a real n u m b e r f rom an interval. For example , we could let m, be the n u m b e r c o r r e s p o n d i n g to po in t L.

Thus, we can only say with respect to the exp lana t ion of our symbols: we have to assume f o r each point L of the line a II

m t �9

The o r d e r in which such II over d i f ferent indices are taken (if we want to assume them in a defini te sequence , which is no t always necessary) is of no consequence , as is well known. B e c a u s e ~ a c c o r d i n g to the broadly used dic tum de omni: what applies at each m for each n necessarily has to apply at each n for each m, etc.

To prove our scheme 39) on the left-hand side, we want to think, for didactic reasons, of a discrete series of i and L, where we choose the

names m l, m2, m3, . . . for the index m of IIm refer r ing to the values A,

B, C, ..., or also il, i2, i3 . . . . . of i. T h e n the left side of our s cheme is

L = I l m l f ( i l , m x ) + I lmzf( iz ,m,2) + I I , , 3 f ( i ~ , m 3 ) + ...

- rlm,n rlm, { f ( i~ ,m, ) + f ( i z , m 2 ) + f ( i~ ,m3) + ...} = R.

For its proof, we only have to say that the sum of the te rms of L,

Page 516

Page 517


which, for l-I, p r ecede or succeed a def ini te m• do no t con ta in this index m• and, as a cons tan t with respect to it, can be des igna ted as a or b, ad hoc. T h e n it is only necessary for the t r ans fo rma t ion of L to R for each of the d i f fe ren t ia ted ( that is, different ly n a m e d ) rn to use the

scheme"

a + IImf(m) + b = rl m {a + f ( m ) + b},

where the II m can be e x t e n d e d over the p r e c e d i n g or the succeed ing cons t an t addend . This very scheme---cf. 26), page 1 0 0 ) - - c o u l d easily be p roved by the initial p ropos i t iona l s cheme 3,), page 40.

In a similar way, we would have for the s cheme 39), the r igh t -hand

side"

L = r ,m, f ( i , ,ml )E , , .~ f ( i z ,mz)~ ,m~f ( i3 ,m 3) ...

= E, ,Em, E,,:~ . . . f ( i ~ , m ~ ) f ( i z , m z ) f ( i ~ , m 3 ) . . . . R .

W h a t we have said shall no t jus t be jus t i f ied and stated for an arbi t rary discrete value series of i and L (for which we i l lustrated it above, so to s p e a k ) - - f o r example , by p r o o f by induc t ion , bu t we want to use it for

all i, L accord ing to the i. If the i, L would have to run t h r o u g h a c o n t i n u u m of values, we can

assume for each i x, X of their value, as we have p roved f r o m / 3 ) , page 37, that accord ing to 18), page 98, each te rm of a "E" can also be r e p r e s e n t e d as a real te rm of a (binary) "sum" (in the nar rowes t sense) , the o t h e r te rm of which, des igna ted as a ( i n d e p e n d e n t of the m appea r i ng in this term) must to be subject to the s c he me I lmf (m) + a =

IIm{f(m ) + a}. Etc. q.e.d. F u r t h e r m o r e , we can see that ou r s cheme would be false and illu-

s ionary if we would replace the index n a m e m, by m i or by 4)(i). Because, in its last part, F, i f ( i ,m ) would appea r as the genera l factor o f II, and this would have to have a value totally i n d e p e n d e n t of i since the le t ter i only funct ions as p l a c e h o l d e r for the values which are assigned to it f rom the ex tens ion of i. The re fo re , we can also no l onge r use m i in its eva lua ted expression. (In analogy to a def ini te integral which is inde- p e n d e n t o f its in tegra t ion variables!) The re fo re , we can totally omi t the

o p e r a t o r Ili(IIm;) p r e c e d i n g the term, accord ing to the law of tautology Ha = a, and ou r scheme would then be very m u c h simplified. T h a t such simplif icat ions are general ly no t admissible cou ld be i l lustrated with

examples . T h e appl ica t ion of ou r s chema to ou r p r o b l e m is now:

Shk = I ' I luhk + II,(IIm,)~-,i(Uhm ' + am, i)} t t

= nn,(nm,)lu,,, + r,(a,,~, + am, i) } = II,(IIm,) H (Uhk + ~,~2,,~, + Z;,am,,)

= am,, + n +

Page 518


Again, we can easily see that we have to have

II (Uhk + I]iuhm,) = I]il' u kmt

359

40)

which, in case even only one of the m, is equal to k, makes sense in the form of the equa t ion 1 = 1; in the case, however, that all m, of the sum over i are unequa l to k, to be recognized in the form of the equa t ion 0 = 0, then a m o n g the admissible values of u there will be one such for which uh, = 0, as well as i (and the L chang ing in parallel with it) and each Uhm ' = 1, that is, each Uhm, = 0, and the factor of 1-I thus vanishes.

T h e n we have

Shk = II,(1-Im,)E,(a'k,,, + am,,) = E,II,,(lkm + am,) = E,( I '~ a)k,,

where we used our scheme 39) again in reverse. Tha t is,

S,,k = E,{(I' ~ a) ; ilk,, = 1(1' ~ a) ;F,,iik, , = 1(1' J- a) ;bi,h = 1/~; ( ~ 1')1,,,,

whereby it is f o u n d consis tent with the above p r o b l e m s = 1 ;/~; (5~f 1'). Now that we have the method for the solution, we want to tackle and

solve the genera l p rob lem, ra ther than the special p r o b l e m 12, which we charac te r ized in p rob lem 8.

Before we start, we have to say that by specializing the result of the genera l investigation, the solution to p rob lem 12 can be easily found a s

x = l : ( d 0 ~l')b.

Tha t is, the lower limit for x found on page 511 represen ts in this case the exact value of this unknown. There fore , the check given by the previous de t e rmina t i on of this limit cor responds to our result.

This result also gives us an answer for the par t icu lar cases of the p rob l em sett led previously. Thus, it is immedia t e for a - 0. If, on the o the r hand, we have bl' for b, we have to u n d e r s t a n d

1 ; ( ~ - l ' ) b l ' = 0 0 ~ (a0 ' + b l ' ) .

To that end, we can (and I ant icipate a little), accord ing to the already stated proposi t ions 47), 46), separate the lef t-hand side into 1 ; ( ~ 0 ~ 1 ' )1 ' - 1 ; (1'0 ~ a ) l ' = 0 cr (a + 1') and 1 ; b l ' - 0 el- (b + 0 ') , then the product of these two expressions 0 el- (a + l ')(b + 0') can be expressed as the r ight -hand side, q.e.d.

If, finally, b ; 1 stands for b, we immedia te ly have 1 ; (~0 ~ l ')(b ; 1) - 1 ;b; ( ~ 1), q.e.d.

At times, one obtains in teres t ing proposi t ions even if one makes a mistake! I have been led to wrongly change (i0 ~ a) ;b into i'cl- a ; b in the deduc t ion of x by inaccurately r e m e m b e r i n g propos i t ions 27) of w 29,

36o SCHRODER'S LECTURE XI

and thus obtained the value 0 ~t (a;b + 1') for x with which all of the four checkings, except the last, are correct. Strangely enough, this in- correct result lies correctly between the previous determined limits, and by checking it, one obtains remarkable propositions!

Indeed, we must have

1 ; ( d d - l ' ) b ~ O ~ ( a ; b + l ' ) = ( = O ~ ( a ; b + 1" 1;b).

Because the major has already been split into {0~ ( a ; b + 1')}{0 ( a ; b + l ; b ) } = x " (0c t l ; b ) =x" 1;b, we only have to represent x=(= x . 1 ; b o f x ~ = l ; b b y 0 ~ ( a ; b + l ' ) : ( = 0 j - ( 1 ; b + l ' ) = 0 j - l ' + l ; b = l ; b .

More valuable is what the minor, the first partial subsumption of our double summation, teaches. According to the first inversion theorem, it can be transcribed equivalently into 1 ;1 ; ( ~ t l')b:~--a;b + 1', or into the first proposit ion of the following pair:

0' �9 1 ; (g~t l ' ) b ~ : a ; b ,

0' a(l'j-/~)" 1 :,~--a;b, a j -b~-- l '+ Oj. (d ;0 ' + b),

a~b:~--l' + (a + 0' ;/~) j-0, 41)

of which the conjugated propositions, combined with each other and with what is already known, enclose the relative product and the relative sum between the following limits:

{ 0'{a(l'ct/~) ;1 + l ' (d j - l ' )b}:~--a;b:~--a; l" 1;b,

actO + Oj-b~--a~b::~--l' + {(a+ 0' ;/~) cr 0}{0 ~t (~ ;0 ' + b)}. 42)

To prove the first proposit ion 41) from the coefficient evidence, we have to show, bringing the right side to 0, that 0 ' ' 1 ; (ac t l ')b" (dot /~) = 0; t h e r e f o r e 0~j~hI-lk(akh + l'kj)bhjII l (di, + btj ) = 0, or that we have ~hI-IktOtij(akh + l'k~)(d,t + bo)bhj = O.

Since j ~e i, k g: j in the effective terms and factors, the value k = i is represented, and for each h a factor of IIkt with k = i, l = h as O0(a~h +

I 10)(d~h + bhj)bhi equals 0, and therefore each term of I; h vanishes, q.e.d. Afor t ior i , we also have

0'{a(l'ct/~) + ( ~ t l')bl~--a;b; therefore, e.g., 0'(a~t l')b(d~/~) =0,

whereby certain identity products are proved to be those which are at Page 519 least contained in the relative product. Etc.

Problem 13. We are looking for x = II [u + a{(~2 + b) ~ c} ;d] as in 38), u

page 508. It comprises the previous Problems 8 to 12 as special cases--but Problems 9 and 10 not fully, that is, only the minor cases. We have--because of 39)


x,, k = II {Uhk + I]~II,,ah, (~ih. , + b,,m + C,,~)d~k] t t

= I],(I'Im,)[r, iahi(bhm ' + Cm, i)dik -k- ~ (Uhk -t- F.,iahidik~thm,)]

36x

a n d w o n d e r wh ich va lue the last II has. At this p o i n t we have to obse rve

tha t m, is n o t c o n s t a n t with r e spec t to i, b u t c h a n g e s in para l le l with i

in E;, t e r m by te rm. If over L all m, are u n e q u a l to k, Uhk = 0 will o c c u r n e x t to all Uhm ' =

1 over L, a n d o u r II vanishes .

If, however , over L s o m e m, is equa l to k, the last I2~ o f the fac to r

7ibm ' (= ~ihk) in all a c c o m p a n y i n g t e rms will be s u p p r e s s e d in the p r e s e n c e

o f the s u m m a n d Uhk, a n d r, iahidi, l',m ' occurs as a n o n e x p r e s s i v e com- p o n e n t o f the g e n e r a l fac tor in o u r lI, to wh ich the w h o l e II is also

r e d u c e d , b e c a u s e n e x t to Uhk = 0 the ot"her ~ihk ' (in w h i c h m, is ~ i f f e r e n t

f r o m k) = 0 will also o c c u r - - s i n c e all poss ib le values f r o m 12 have to be

wr i t t en for u. We m u s t t h e r e f o r e have

}] iahidik 1kin,, II (Uhk + )2iahidik~ih.,, ) = ' 43) u

which also gives the c o r r e c t value, 0, for the p rev ious case. T h e s u m o n

the r i g h t - h a n d side, o f course , c a n n o t be r e d u c e d to a s ingle t e r m ac-

c o r d i n g to s c h e m e 12), page 121, b e c a u s e in it m, is n o t c o n s t a n t with

r e spec t to i b u t its m e a n i n g c h a n g e s in para l le l with i; this s u m can have

any n u m b e r o f effect ive te rms. We thus o b t a i n

x,,~ = II , ( II . , , )~,a, , , (b , , . . , + Cm,, + l',,,k)d,k = F,,ah,IIm(bhm + Cm, + l ' k ) d , k

if we use o u r s c h e m e 39) in reverse (above it was u s e d in the fo rwa rd

d i r e c t i o n ) . Now we can wri te c,,,~ = i~,,, = (i; c)hm = (c;i)mk in any way we w a n t a n d

also a s s u m e the t e r m as tau to log ica l ly d o u b l e d , a n d c h o o s e for t he o n e

the fo rmer , for the o t h e r the la t te r fo rm. Af ter this, we o b t a i n

/ /

II ,, ( b,, ,,, + c,,, + l mk ) = {(b + i; c') e 1 }hk

: {b j- ( c ; i + l')}hk = {(b + i; c') ~ ( c ; i + l')}hk,

as well as ahi = (a ; i)hk, dik = ( i ; d)hk, a n d we have

, . , , . ,

xj, k = F,~[a;i" {(b + i; ~) o ~ ( c ; i + 1')} �9 i; d]h k

or t h e r e f o r e

v ..,

x = F,~a;i" {(b + i; c") j- ( c ; i + 1')} �9 i ; d ,

w h e r e o f the two t e rms i; ~ a n d c; i , the o n e or the o t h e r ( bu t n o t b o t h )

can be s u p p r e s s e d . We may write o u r resu l t m o r e s imply as

Page 520

Page 521

362 SCHRI3DER'S LECTURE XI

x = ~ a ; i " {brt (c ; i+ 1')} �9 i;d. 44)

T h e x, however, is not complete ly r ep resen ted in closed form, but the II over u has been reduced f rom the second o rde r to a I2 over i of the first level or order.

T h e latter can also be given in a s impler form:

x = E j . a{b~ (c + i )};d 45)

- - w h e r e again the s u m m a n d i can be separa ted f rom c and added as i" to b as a s u m m a n d .

This can be easily verified by establishing the genera l coeff icient Xhk for the last ~, as a result, we have in fact immedia te ly the f o r m e r expression of Xhk----only l is taken for k.

On the o the r hand, we can also derive systematically the last expression of x, 45), f rom the previous, 44)ruby passing th rough a doub le sum. To that end, we write the middle factor in 44) as (b + i;c) ~t 1' in the form of e~t 1' and choose f rom the four represen ta t ions for e~ 1' which we have according to 14) or 17), 16) or 22), and 18):

e0 ~ 1'= IIj(e ; j + ]) = II~(ectj+ ]) = IIj(ertj+]) = ~ j (e , j ) ] ,

the last because the two summat ion signs can be exchanged . T h e n we get (b + i; i) ~ j = be (c; i + j) = b e ( c + j ) ; i = { b ~ t ( c + j ) } ; i

because j =j; i , etc., and we have

x = ~ j ] " ~ , a ; i ' { b r t ( c + j ) } ; i " i ;d = F~i]" E,a{brt (c + j ) } ; i " i; d = Zj]" a{6e (c + j ) } ;d ,

which is the represen ta t ion 45) of x, except for the des ignat ion of the s u m m a t i o n variable.

[We had to cons ider the proposi t ions (because i = i ;1 ) 10) of w 27, then 27) and 26) of w 25, and, finally 14).]

T h e r e are many ways to check our result. In particular, we want to derive the solution of Prob lem 12 already

checked. For that purpose , we let a = 1, b = 0 in 45), and there fore write a and b for c and d. Thus, we have now

x = I],i" {0ct (a + i)};b = E, i . (i0~ a) ;b =l~,i" i; (1 '~ a) ;b;

cf. 32) of w 25, and 25). Accord ing to 26),

x = 1 ;{(1'0 ~ a) ; b}l' = 1 ; 1'{/~; (gj- 1')},

because 1~ = l'c'~ Now this r emarkab le proposi t ion is valid:


l ' (a; b) ; 1 = a/~; 1,

1 ; (a; b)l ' = 1 ; ~b,

l '(a j- b) ; 1 = (a =/~) 0 ~ 0,

1 ; (aj- b ) l ' - 0 j- ( ~ / + b),

363

46)

of which the second formula on the left (necessary here) can be proved with the coefficient evidence by means of

L o = l~t 1;t~ h au, bhjl ~ = E h ajhbhj = I~ h 1 ~h(~/b)h~ = R o �9

This propos i t ion belongs to a g roup of proposi t ions which refer to relatives of the form of l~z; 1, etc., some of which we have s tudied u n d e r 24), 25) of w 22 (p. 335), a special case in the fo rm of 30). Tha t requires a lso- -as obvious f rom aiibii = (ab)ii,

l h ; 1 �9 l 'b ;1 = lhb; 1, l h ; 1 + l 'b ;1 = l '(a + b) ; 1, 47)

1 ; a l ' ' 1 ; b l ' = l ; a b l ' , etc.

which can be immedia te ly e x t e n d e d to more than two terms. By 46), we have found the solution of Prob lem 12: x = 1 ; (~/~t l')b, as

given on page 517. And there fore we have also checked our invest igation's main result

r I - . . = E~... consist ing of sett ing the values of x f rom 38), 44), and 45) u

equal to each other:

, . ,

II [u + a{(b + ~i) ;t c} ; d] = E ~ a ; i . { b j - ( c ; i + 1')}" i; d u , . ,

= E~i" a{be (c + i)} ; d. 48)

As a fu r the r check, it is left to the s tuden t to derive the r e m a i n i n g p r o d u c t values, which fall u n d e r the scheme of our P rob lem 13, and f rom this scheme the results that solve the p rob lem . . . . ~

P rob lem 19. We may now also decide the quest ion which a p p e a r e d on page 268, namely whe the r the general solut ion of equa t ion x ; 0 ' = a ; 0 ' given u n d e r 22) cor responds in essence with that in the first line of 26). The answer to this quest ion is positive.

This also brings us to proposi t ions which are of some interest. First, we notice that the solution 25), page 269, to x ; 0 ' = a is by using

a :t 0 = (a 0 ~ l ' ) a and is fu r the r simplified to

x = (a# l'){d + u + (z/# 1') ;1}, 94)

where it now consists of seven instead of n ine terms wi thout having lost anyth ing in clarity. Instead of the (last) relative factor 1, we could also write 0 ' - -cf . 15), page 229.

We can also br ing the expression of x in 26), page 269, closer to the one in 22) by writ ing

"~ Problems 14-18 (pp. 521-538) have been omitted here.

Page 539

364 SCHR()DER'S LECTURE XI

a ; 0 ' o ` 0 = (a;0 'o` 1') �9 a ; 0 ' (do` l ' )a = (a '0 'o` l ')(do` 1')

There fo re we can also set apart the factor a ;0 'o` 1' in 26) and obtain

x= (a;0 'o` l'){do` 1' + u + (6o` 1') ;0'1 95)

as a s impler expression for the general root of equat ion x" 0 ' = a ;0 ' , which now consists of nine instead of 10 terms.

It has the advantage that only a appears in the combina t ion a ; 0 ' = c, do` 1'= t? in the two expressions of x which have to be proved together.

If we again take a for this c and b for u, we can indeed prove as a universally valid formula that

(ao` l')[b + dl (b+ d ;0 ' ) O 1'} ;0'] = (ao` l '){d + b + (go` 1 ') ;0 '1. 96)

Proof. The two partial subsumpt ions of this equat ion L = R divide because their predicate is a product , and they are therefore obviously valid as partial condi t ions of L ~ a o ` 1' and R ~ a o ` 1'. We thus only have to show that

R=6-b+ a{(/~+ d ;O ' )o ` l ' } ;O ' and L:~--d+ b + ( /~o`l ' ) ;0 ' ,

that is, /~R and aDL relative to =(= of the last term on the right. The latter, b rough t completely to 0, results in

a/~(b;0'o` l ' )(ao ~ 1') �9 a{(/~+ d ;0 ' ) 0 ~ 1'} ; 0 ' = 0,

and the factor/~ proves to be irrelevant. If we replace (ao` l ' ) a by a o 0 we can use the proposi t ion

(ao`O) �9 ab; c = (ao`O) �9 b; c, 97)

which is easily proved based on 24), page 255, and (a o 0)a = (a o 0). We now only have to show

(a o 0)(b ; 0' o 1') �9 {(d;0' +/~) O 1'} ; 0 ' = 0.

Also wi thout the above proposi t ion, the p roof would follow a fortiori f rom the latter.

Fu r the rmore , we can suppress the term d;0 ' , and the last factor now appears as the negat ion of the second but last and makes the p roo f clear and unders tandable .

This is because we must categorically have

(ao`0) �9 {(d" b + c) o d } ; e - (ao`0) �9 (co" d) ;e. 98)

If we multiply with (a o 0) o, namely Hha~h in

F, klIt (F,,,,di,,bml + c u + dlk)ekj,


we e l iminate the whole Em because each dim coincides with a factor

aim of II h. T h e subsumpt ion for a/~L is therefore done, and we only have to prove,

for/~R,

/~R= (a0 ~ I')M~ + (aft 1')/~ �9 (/~0 ~ 1');0'=(= a{(/~ + 4 ; 0 ' ) ~t 1'} ;0',

which divides into two parts because we added the subjects. This is clear even wi thout factor /~--af ter we transcribe the last two terms and mult iply the a j-1' with the negative on the right side as

a �9 b(a ~ 1') ; 0 '0 ~ 1' =(== b ; 0 '0 ~ 1'

because the relative p r e s u m m a t i o n on the left appears as =~= to b ;0 ' Page 540 on the right. If we suppress identical factors, we mus t necessarily get

c o n t a i n m e n t [ Obergeordnete]. T h e o the r part, b r o u g h t to 0 on the right, likewise requires

dbla" b (a~ 1 ' ) ; 0 ' # 1'} = 0,

where the factor a can also be omit ted, as we can see, and this brings b, a for d, /~ to a more accessible proposi t ion:

ab :~-{(a + b ;0 ' ) j- 1'}0', a(b j- 1') ; 0 ' 3 1' :~: a + b, 99)

etc. In o rde r to prove it, we have to use the coeff icient evidence since ne i the r a~-- (a j - 1') ;0 ' nor b:~-(b ;0 ' j - 1') ; 0 ' - b ;0 ' is i n d e p e n d e n t . If we state S:(c--P for the first subsumpt ion , we have to show S 0 -aob!i:~--P O, where

' ' + 1 ' . P0 = ~J, 0hjIIk (aik + F'lbilOtk kh)

To p roceed in a complete ly analytical way, we adjoin to the universal factor o f l I k the te rm 0'k~l'kj which is = 0 , and we analyze f rom the s cheme a + bc = (a + b)(a + c), evaluat ing also IIk of the first factor accord ing to 12+), page 121, and we have"

PO = F.,,, O,'o(a 0 + F., t bitO' ' ' ' O)IIk (aik + F., t bitOtk + lk, , + lkj)}.

If we fu r the r mult iply the general term of the last I~ t with 1' 0 + 0~i, which is =1, and we evaluate the I] t which comes f rom the first term, accord ing to 12x), page 121, we get the s u m m a n d boO~k which can be simplif ied to b o because of the following l~)k, a n d we can write this term, independen t of k, in f ront of II k. We now have

Po = F"J, Ojo(ao' + F'tb,Oo){bo' + IIk(aik + Et bitO'tk...O~3' + lk ht + lkj) } , .

' = ( a b ) �9 1 = S O , and eve- By mult iplying we easily get the terms aobuF, hO~o it rything is thus proved.

In 99) we could, of course, also add b ' 0' left of the subject.


Better than the formula for x given on page 272, this formula,

x=(a;b&{~)[(dj-[O ;/~;b + u + (u,,.'l"/~) ;/~1,

comprises empirically the results which we obtained for b = for one of the four module values on page 268, etc., for the general root of x ;b = a;b. However, I have not succeeded in simplifying the solution on page 266 of the third inversion problem with an arbitrary b in a similar way.

Exercise 20. Also in regard to partial solutions of the general (third) inversion problem, I can add a few observations to the result obtained i n w 19.

In order to prove that substituting u = a" a ; b ; b in the general solution 64) of our problems must also give x = u, and that therefore this u represents a particular solution, root, we had to prove the following, almost monstrous equation as a universally valid formula:

(a;bj-{))[a " a ; b ; / ~ + (a;b)({d + do~/~+/~;/)

+(dj-/)) ;/~} ~/~) ;/~] = a " a;b

In it, the under l ined term must vanish, according to the formula d + (d0~/~) ; /~=d ment ioned on page 267 and given from a;b=~c--a;b. Since we then have a ( a ; b ; b ) ' b = a ; b b y 9 ) o f w 19, we now have

and therefore the second term in parentheses equals

(a;b)(d&f)) ;/~= 0 ;/~=0,

Page 541

and we only have to show that

(a;bj-{))a" a ; b ; b = a " a ; b ; / ~ o r a" a ;b ;D~-a;b j - { ) .

This can be done with a(a; b; D)b=~--a; b from 5) of w 6, q.e.d. However, we also get involved with the curious circumstance, trou-

blesome for our discipline, that the theorem 9) of w 19 which we had to discover as a particular case of the general third inversion theorem, had already to be used in the process, so that we cannot vouch for its i ndependen t discovery and justification (as stated earlier)!

In this connection, we would like to draw attention to the fact as well that x = a" a;b; 1 represents an independen t solution, so that in addition to the formula already ment ioned, we also have the pair of formulas,

Page 5 4 2

367 F R O M P E I R C E T O S K O L E M

a(a;b;1);b=a;b, (a+ a~b~O) j-b=aj-b, 100)

a ; ( 1 ; a ; b ) b = a ; b , aj-(Oj-aztb+ b) =a~b,

because we have i m m e d i a t e l y

a(a;b;1) ;b=a;b;1 �9 a;b=a;b, q.e.d.

Bo th g r o u p s o f f o rmu la s can be co l l ec ted in the g e n e r a l p r o p o s i t i o n

tha t x = a �9 a ; b ; (/~ + w) r e p r e s e n t s a class of pa r t i cu l a r so lu t ions o f the

p r o b l e m x; b = a ; b for any a rb i t r a ry w, etc., a class o f c o n s i d e r a b l e gen-

erality, yet s imple in its e x p re s s i o n of the root .

P r o b l e m 21. Finally, we also wou ld like to m a k e a c o n t r i b u t i o n to the

"determination" of o u r g e n e r a l p r o b l e m 64). T h e m o s t i m p o r t a n t ques t i ons are: W h e n ( tha t is, u n d e r which con-

d i t ions for a a n d b) does the x, d e t e r m i n e d by the r e q u i r e m e n t x; b =

a;b, r e m a i n Comple te ly arbi t rary? An d w h e n is x pe r fec t ly d e t e r m i n e d

by this r e q u i r e m e n t ? T h e first q u e s t i o n can be easily a n s w e r e d by saying b m u s t be equa l

to 0, if x = u m u s t r e m a i n u n d e t e r m i n e d . Because if we have to have u ; b -- a ; b for each u, we mu s t have a ; b = 0 - - a s the a s s u m p t i o n u = 0

i n d i c a t e s - - t h e r e f o r e also u ; b = 0 for each u, thus 1 ; b = 0 o r b = 0, q .e .d.

To aff i rm the s e c o n d ques t ion , we a c c e p t e d b = 1, w h e r e we h a d to

have x = a, as suff ic ient c o n d i t i o n , bu t it is n o t at all necessary. To f ind

the necessa ry a n d suff ic ient c o n d i t i o n for the fact tha t t h e r e is on ly o n e roo t o f the e q u a t i o n x; b = a;b, or tha t the so lu t ion 64) is c o n s t a n t in

re la t ion to u, we have to go deepe r . T h e suff ic ient o r a d e q u a t e a n d necessa ry o r r e q u i r e d c o n d i t i o n for

the fact tha t a f u n c t i o n f(u) of a ( res t r ic ted o r u n r e s t r i c t e d ) var iable

relat ive u is c o n s t a n t with r e spec t to it is

~f(u) = Hf(u) 101)

(where in the first case in p a r e n t h e s e s , the E a n d II a re e x t e n d e d only

over the d o m a i n of variabil i ty o f u, in the s e c o n d - - p r e s e n t - - - c a s e , they

have abso lu t e e x t e n s i o n ) . If we have f(u) - e for all u of the s ame value, the above e q u a t i o n is

ce r ta in ly valid b e cau s e of ~ e a n d IIe. By the s ame token , if this is valid, t t u

we can call the c o r r e s p o n d i n g value o f the i r two-sided e x p r e s s i o n e ( and

it b e c o m e s i n d e p e n d e n t o f u because u only a p p e a r s as i n d e x in t h e m ) .

Because o f f ( u ) ~ E f ( u ) a n d 1-If(u)::(c-f (u) we also ge t for e ach u, f(u) ::~--e a n d e::~--f(u), a n d thus f(u) = e, as we have shown.

By app ly ing this s c h e m e ' s 101), we now f ind by co l l a t ing 65) with 66):

c = cE~..., o r c :~: E~ ... . tha t is,

c :~- Z , i . (c; b)lizt (/~ + i)}; 1 102)

368 S C H R I ~ D E R ' S L E C T U R E XI

as the necessary and sufficient cond i t ion for the fact that roo t x o f the equa t ion x; b = a ; b is comple te ly d e t e r m i n e d by a n d (= a).

With this r e q u i r e m e n t , the c de f ined as a;b : t /~ has to be equal to a, which resul ts - - inf in i te ly m o r e easily than f rom i t se l f - - f rom the observation that also these have to coinc ide because c and a are always roots.

In any case, the general , known par t icular solut ions of the e qua t i on have to co r r e spond , which gives us--cf , page 260 - - c = a = a" a ; b ; b = c" a ; b ; b and leads to the doub le subsumpt ion :

a; b j- ~:/~--a:~--a; b; b, 103)

o f which the first par t has the s t reng th of an equa t ion . If we ignore the midd le term, we get

(a ; b j- ~) ( d j- [~ j- ~) = (a ; b ) ( d j- [~) 0~ /~ = Oj-/~=(=O,

which converts into /~ct 0 = 0 and thus supplies a result for b:

b; 1 = 1. 104)

Page 543 It also tells us that b may not have an empty row. O b t a i n i n g this result directly with the e l imina t ion of a f rom 102) may have its difficulties.

In 102), we can write a for c. If we do this, after cons ider ing all four ways of writing from 66) and reduc ing the r ight to O, we get four forms:

{ aII,{(~/~) ; i + a ; O ' ( b ; i ) + i';/~} =0, an,{i+(~t~+ a;~b)o~/~} = 0 105)

- - w h e r e the u n d e r l i n e d te rm can be e l i m i n a t e d - - a s express ion of the r e q u i r e d condi t ion . However, it can no longe r be called "sufficient," only in c o n n e c t i o n with the first s u b s u m p t i o n 103) which has p roved c = a. The sufficient or comple t e cond i t ion was still expressed in 105), if we r e i n t r o d u c e d c for a and then wrote for c its value a; b0 ~/~). As coefficient , this last r e q u i r e m e n t 105) is mos t simply wri t ten as

ahkIIm{( t~ ~" b)ltm de_ ~,anlOtk~lm} = 0 . 105 ~

We can draw many inferences f rom 105). T he r e q u i r e m e n t has to exist a fort iori if we suppress any terms after the Hi or, possibly, add some factors.

We have, for example" aII~{(50~/~);i + i;/~} =0, i.e., a cco rd ing to 14)

a(53-/~0~/~)=0 or a : ~ - - a ; b ' b

as a conf i rma t ion of the second subsumpt ion 103). It also immedia te ly _

follows ou t of the last form of 105) as a(d0 ~ b~0 ) = 0 or a:~--a; b; 1 which can be c o n f i r m e d with 104). Etc.

Page 544


Further, we must have aII~( ~+a ; ~b # 0) - 0. According to 7) of w 6, we have, however, a ; (~)b 0 ~ 0) :(= a ; ~b~ 0, and here the subject equals a; (~:t0)(bo~0) = a; ~(bc~0) = a(Oj-b) ;~, thus a fo r t i o r i aIIi{a(Oj-b) ; i + [) =0, which results in, according to 18), a" {a(0 j-/~) ;0' or

a �9 a ; 0'(b o ~ 0) - 0 106)

as a further, necessary condition. It is possible to satisfy the first subsumption 103) independent ly in

the most general way with a = oL;b~/~, as already, shown in 4 ~ of w 19, and the second subsumption with a = ol 'o~;b;b. As I have found in a more painful procedure, both requirements 103), involving also 104), can be satisfied in the most general way with i ndependen t parameters c~,/3 with the following formula which is easily verifiable,

a=ol ; ( /~j -O) +c~; /3 j -O+ (c~;f3j-/~) �9 1;/3, b=/3o~0+/3, 107)

and it may be possible to continue to gradually satisfy fur ther partial requirements or subrequirements of the p rob lem--as 106)--by determining the parameters.

But as long as we cannot evaluate the products II~ in 105) in closed form by transforming them equivalently as functions of a and b, which are only construed by means of the six species of these arguments and perhaps also by the modules, there is little hope that we can completely solve our difficult "determinat ion problem" (for the third inversion problem) in this way.

We therefore have to leave the problem here as it is. To evaluate II~ hardly promises success as long as it did not succeed

with much simpler products, such as

x = II~a; ~b = Ilia;~b. 108)

Concerning this last problem of the first order [erster Stufe], with which we return to the main theme of this section and which--a l ready for b = 0'---cannot yet be solved, much could be said that is of interest. But we have to abstain and r ecommend it as a problem, next in difficulty among the still unsolved problems, for further investigation.

In view of the importance, the fundamental significance for the com- pletion of elimination, for inference in general, which the problem of sum and product on the second order [zweiter Stufe] gained at the end of w 28, we will devote some time to this problem. It will be shown by mod- ifying (in a practically insignificant way) what was said on page 468 that our algebra has also methods for the solution of this problem which

Page 545

37 ~ SCHRODER'S LECTURE XI

are theoret ical ly usable in a genera l way, but which seem to be e n t a n g l e d in u n s u r m o u n t a b l e technical difficulties or ma themat i ca l compl ica t ions when we try to use t hem practically.

W h e n the ex tens ion of H, E is absolute (over the ent i re universe of d iscourse 12), w e can assume as self-evident that

H f ( u ) = H f ( 5 ) = H f ( ( z ) - H f ( u ) 109)

and, similarly for E - - w h e r e the p r o d u c t variable is r ep laceab le by each of its relatives. If u can take on any value, then also z~, fi, etc.

If, f u r t h e r m o r e , the express ion o f f ( u ) only appears as u and fi, bu t no t z~, u (or reverse), after its p r o p e r r educ t ion by execu t ing all nega- t ions on c o m p o u n d subexpress ions (next to any p a r a m e t e r relatives or cons tants re la t ing to u), then we know

nf(u,(,) =tip,o) = n f (~ ,u ) ,

which we can write in a s impler way as

H f ( u ) - f(O). 110)

T h e n we have f (0) in our rl as the min imal factor, c o n t a i n e d in all o thers , the p r o o f of which could be given in detail by using the t h e o r e m s u0 :(= uv, u + 0 :(= u + v, u; 0 :(= u ;v, u 0 ~ 0 :(= u 0 ~ v and its con juga t ions in c o m b i n a t i o n , or mainly accord ing to t h e o r e m 1) of w 6 by cons ide r ing 0 :~:v. We thus have, for example , immedia te ly

rl [a{(u + b) j. c} ; d + e; (tf] = a(b j- c) ; d.

We leave it to the r eade r to state the dually c o r r e s p o n d i n g p ropos i t ion for I2.

As a p rob lem, the discovery of only such H, E is o f in teres t in which in the genera l te rm f ( u ) not only u or ~ but also z~ or u a p p e a r essentially.

We have solved p rob lems of this kind in 1) and 6). These can be m a d e into the m o r e genera l t h e o r e m by m e t h o d s which we will cons ide r later:

u

111)

where the sums over K and X e x t e n d over any series or systems of suffix values, such as 1,2, 3 . . . . .


Before discussing the me thods , we want to state some c o n c r e t e examples which are in teres t ing for their der ivat ions and results.

P r o b l e m 22. W h e n we look at 5), we are incl ined to inqu i re af ter the value of the following p r o d u c t l-I, after which we immedia t e ly state

I I ( a u ; b + c;gtd) = a d " 1%;1;b1', I I ( a ; u b + o i ' d ) =bc" 1 ~ ; 1 ; d 1 ' , u u

112)

which gives the a n s w e r m w h e r e the second result can be o b t a i n e d by e x c h a n g i n g letters in view of 109).

For proof , we call the x the first, sough t for II and Uits genera l factor,

and we have

x - n U . . , ; - n v,j. u

U 0 = r,h a~hbhiu ~h + Ek qkdkj~ikj"

We now "develop" this express ion for u 0 by mak ing the lat ter (and its nega t ion ) "prominent" or by "bringing it into evidence." For that p u r p o s e it is necessary and sufficient to let the terms where u or ~i are i n d e x e d with this par t icu lar suffix ij to e m e r g e or to let c o m e ou t whe reve r they

can be found . This can be d o n e by pu re calculat ion, in our case: by mul t ip ly ing the

genera l e l e m e n t of ~j, with (1=) (l'hj + 0,'0), the ~k with (1=) (l'h, + 0~,,), t hen resolve them, then apply to the first terms the s c h e m e 12x) f rom page 121. Now we have indeed:

U 0 = aobiiuii + c,d~gt 0 + ~,,O,',~a~hbjou, h + ~kO~kc~kdk~(%,

where we only have in the last two sums u-coefficients which are d i f fe ren t f rom each o t h e r and f rom u o. These may also a p p e a r in H with 0-values

Tt

u~h = 0 (at h r j) , ~ikj = 0 (at k r i), so that only the first two terms can con t r ibu t e s o m e t h i n g to the value of x. T h e actually o c c u r r i n g min imal value, which is c o n t a i n e d in all values, o f a h o m o g e n e o u s l inear function"

Page 546 ~ u + ~ i - - c ~ + c~ju + o ~ i

must be the p r o d u c t of its coefficients, t he re fo re be ol/3, since the last two terms vanish i n d e e d when we assume u = o~/3.

T h e r e f o r e , the U# f o r m e d for all possible u conta ins at least the te rm

aobjjc~d 0 and will, for cer ta in values of u, no l onge r compr i se its terms, so that we have f o u n d

U!i = a ijd ~jc.bjj.

We now have

372 SCHRt~DER'S LECTURE XI

x o= (ad)o(l~c;1)~i(1 ;bl ' ) 0 and x = ad" 1~; 1 ;bl' , q.e.d.

The p rocedure is certainly impeccable; however, the condi t ions for developing it so smoothly and simply are almost never as favorable.

We can gain a deepe r insight into the p roduc t m e t h o d which promises general success, with the derivation (of the first) of the following results:

P rob lem 23. We have to discover that

II {au ; b + c( ~t j- d) ; e} = (ar id)c; ( [t j- b)e,

I~I {a ; ub + c; (de 7i)e} = (a j- it)c; (d j- b)e, ~{(a+ u) rtb}{(c+ ~i; d)0~ e} = ( a ; d + c) e ( d j - b + e), 113)

~ { a ~ ( u + b ) I l c J - ( d ; f z + e ) } = ( a ; d + c ) ~ ( d ; b + e), t t

- - w h e r e the ones s tanding below each o the r are immediate ly omi t ted by exchanging d with d in one proposi t ion.

Derivation. By formulat ing the first p rob lem again as x = IIU we get

x 0 = II U 0 and U 0 = Eta,boU,l + ~kc, kek~IIt(~i, + dj, l).

The p rob lem is simple insofar as the u consistently appears with the first index i.

We now make uih p r o m i n e n t for any one definite h. We have shown previously how to use Et to that end (we multiply the general term by l'u, + O~h ). In dual cor respondence , we only have to add (0=) Ou, l u ; ' to

Page 547 the general factor of II~, break it into O'u, and l'u, by the dual equivalent of the distributive law in (72. + dtk + O~h)(~i . + dtk + l'th), and take IIt individually of these two factors, where for the first factor the scheme 12 • ) of page 121 can be used.

We thus obtain

U 0 = a,,,b,ou,, , + ElO~,,a,,bljU . + EkC, keki(fZ,,, + d,,k)IIt(l'u, + 7i,t + dtk).

This has the (linear) form which we have already developed:

U, i = c~ u ~, + 3 czar, + 7 ,

which we are bet ter to leave in the n o n h o m o g e n o u s state. Prior to writ ing the specific values of or, 3, "t' which are here not visible, we want to in t roduce a somewhat more accessible symbolism that we r e c o m m e n d for all similar problems.

A sum of the form Et01h4)(/) represents no th ing more than the sum of all 4~(1) without do(h) and can also be expressed complete ly by I2~ -h4~(l). By analogy,

I2~-"-k4)(/) = I; t O~,,Oikch(l)l

becomes the sum o v e r / o f all 4)(1) wi thout 4)(h) and 4)(k), and so forth.

Page 548


In dual c o r r e s p o n d e n c e , we can also write

H,{I',,, + r H-f" r H,{I' + l',k + r /it-n-* r = ' lh = '

and so forth, where the express ions on the left r e p r e s e n t n o t h i n g m o r e than the p r o d u c t of all 4)(1) wi thout ~(h) and 4)(k), respectively, etc.

This small modi f ica t ion of the symbolism which is legi t imate in ou r discipl ine has the advantage that the genera l t e rm of E a nd of 1-I always has the same (a constant ly simple) express ion (whereas in the n o r m a l way of r ep resen ta t ion , it would grow and increase in bulk) , as m o r e and m o r e terms are omi t t ed f rom the sum, factors o f lI; t he re fo re , also the gene ra l term, which of course remains the old one , does no t n e e d to be m e n t i o n e d and repea ted .

W h e n finally all terms of Et, all factors o f / I t are exc luded , that is, e l imina ted , the f o r m e r will be equal to 0 and the lat ter equa l to 1.

If we use these results, we now have

-h a = aihb #, 13- ~ks , 3" = ~-~h q_ ~kCikekjdhkI- i1 h '

where the genera l t e rm of E t is now a,~Ou ., and the gene ra l t e rm of lI t is u~h + dhk, jUSt as in the first express ions of ou r U 0.

In whatever fo rm the o t h e r u , (without uih) are given, we now can d e t e r m i n e , choose , u~h in such a way that the above l inear func t ions of it take U 0 as its min imal value which must be

which can be r e d u c e d slightly and results in

3/' + OL~ = ~27 h + ~,, Cikekj(aihbhj + d,,k)/ii -h .

We now make a n o t h e r u~h, p r o m i n e n t in this 3' + c~/3, whe re we have to assume h' 4: h because h does no t occur in it. We now have

3" + ol/3 = E~-h-h' + a~h, bh,jU~h, + Ekc~kekj(a~hbhj + dhk)(6~h, + dh,k)II~ -h-h' I i ~ l - I

= O~ u ih, + uih, + 3" ,

and, again, the min imal value of this express ion in view of the variables (or func t ions of) u~ z is

3" + ol '{3 '= F.,[h-# + F.,kcikekj(aihbhj + dhk)(aih, b~l j + dn,,)/i-/h-n'.

This cou ld have been wri t ten immedia te ly wi thout the i n t e r m e d i a r y

calculat ion on the basis of the observa t ion that 3' + o~j3 with respec t to u , again gives the same form as u o - - e x c e p t for the absence of o n e t e rm ( that is, up to the ex tens ion) in E and /i over l, and excep t for the c i r cums tance that the p a r a m e t e r p r e c e d i n g the iIt as a factor---or express ion of the c o n s t a n t s % % in U 0 ~ h a s now b e c o m e m o r e compli- ca ted in 3' + c~/3 (as we can see it the re now).

Page 549

3 7 4 SCHRODER'S LECTURE XI

This observat ion is also valid for "y' + a'{3', and the law of fo rmat ion

has b e c o m e clear. If we assume these conclusions to be un l imi ted until all terms of the

Etare e l iminated , thereby, at the same time, all factors of IIt have b e c o m e

ineffective, = 1, we have found x 0 as the last express ion of the min imal

value, 3 ,~~176 + a~~176176

Xij = ~kCikekjI-Ih (aihbh~ + dhk)

= E~c, JL(a,,, + ~)rI,,(~,,~ + bhj)ek) = {(a0 ~ d)c; (dj- b)e} o,

and therefore , x = (a ~ d)c ; (d j- b)e, q.e.d. As we can see, the p r o c e d u r e leads us to a "de t e rmina t ion of the

limiting value" and is charac ter ized as a kind of "method of exhaustion": the E and H over I were gradually "exhausted"--as we would e l imina te

in a p r o b l e m of e l iminat ion of x the un l imi ted double series which could possibly form a c o n t i n u u m of its coeff icient Xhk by progressive e l imina t ion of one after the other. In the same way, we have taken out

of cons idera t ion or "annihi la ted" one U~h after the o t h e r by keep ing only what it canno t avoid con t r ibu t ing U 0 to H; in o t h e r words, we were looking for the minimal value of the min imal value of the min imal

value, etc., of U O with respect to one of its a rgumen t s u~h (of the proposit ional funct ion) after the o t h e r - - b y a "minimal value)' of a funct ion

4~, "with respect to an a r g u m e n t u," we unders t and , now briefly speaking, a value which is actually ob ta ined for some u, while it is con ta ined in

all values which the funct ion can take on for any u - - n a m e l y we have to con t inue this until the last a r g u m e n t uih, if it exists; m o r e generally, then when this has been d o n e with respect to each a r g u m e n t uih. Thus the resul t ing expression for the min imal value of U O has b e c o m e in-

d e p e n d e n t of all a rgumen t s u~h, in shor t of u; and the re fo re we could e l imina te the sign H in f ront of this cons tan t by the law of tautology.

Incidentally, for a = 1, b = 1', by chang ing some letters, our first result 113) becomes 50) - -4ha t fo rmula which previously we could only find

in an artful but convolu ted way th rough an infinite n u m b e r of mul t ip le

II. O u r result for d = 0' also includes our t h e o r e m 6) and appears as

one of the most genera l ones of H which could so far be writ ten in

closed form. What we have l ea rned in the last p rob l em can now easily be gener-

alized "th eo re tically."

If we have to know a x = II U with the "absolute extension," where

U =f(u) is a given relative func"tion, we have to lookforthe general coefficient x,,k = II U,,k.

u

We will first have no difficulties, following the st ipulat ions (10) to (13) of w 3, r ep resen t ing the genera l factor Uhk of the last II explicitly

as a "proposi t ional function," which is built f rom the coefficients of the

Page 550

VROM PEZRCE TO SKOLEM 375

a r g u m e n t u and its nega t ion 6, as well as of all the possible p a r a m e t e r s a,b,c . . . . of f (u ) using the three species of the p ropos i t iona l calculus and, possibly, the ~ and H signs in a cer ta in way.

We can "develop" this express ion in the fo rm of

U~,k = a u o + 3 a u + %

for the u-coefficient with any given suffix ij. This d e v e l o p m e n t is l inear and also h o m o g e n o u s , but to the h o m o g e n o u s fo rm

(~ + v)u 0 + (3 + ~,)a 0,

the o n e which is no t yet h o m o g e n o u s has to be p re f e r r ed , as we will soon see. T h e po lynomia l coefficients a , /3 , -y are no t yet i n d e p e n d e n t o f u ( they carry possibly the o t h e r u-coefficients), bu t they are inde- p e n d e n t of u O.

To "mak e prominen t , " to "br ing into evidencd' u 0 - - w h e r e the cases i = or r j as well as i , j = or r h,k are to be t rea ted sepa ra t e ly - -we only have to observe: w h e n e v e r the first index 1 of u or ~ is d o m i n a t e d by a Zt, we can mult iply the genera l t e rm of this E by l~u + 01t ( = l ) - - f o r a s econd index, however, we mult iply with 1~ + 0~.. If, on the o t h e r hand , they are d o m i n a t e d by a Ht, we can add 0al' ', (=0), o r 00'1'0, respectively, to the gene ra l t e r m - - i n bo th cases, b reak it into 0 ~th and 1 ~th acco rd ing to the distr ibutive law a(b + c) = ab + ac, or a + bc = (a + b)(a + c), respectively. Thus Z or H falls respectively into a sum or a p r o d u c t o f two of them, and, for one part , the s chema 12) o f page 121 is appl icable whereby we ul t imately obta in u 0 or 5 o as explicit fac tor o r sum. In the o t h e r part , a t e rm previously effective is ineffective, e x h a u s t e d or ex- c luded by the factor 0~t or 0~, respectively, by the a d d e n d 1', o r 1' tj, respectively, f rom the r e m a i n i n g Z t with respect to H t.

Now our Uhk actually takes on its " m i n i m a l va lue" for u 0 = d3, which has to be c o n t a i n e d in all values of the Uhk for all u a nd is

whe re ij no longe r occurs as a suffix of u and ~. If m n is any new, arbi t rary suffix, we can also deve lop

I - ! Vh k = ~ 'U m ,, + 3 Urn,, + ~/ ,

of which as a factor of Xhk only the min imal value c o n t a i n e d in all Vhk for all u and real ized for some Urn,, namely

w, ,~ = v , , : ~ " = vh-,i-m.= "v' + ~'~',

in which now n e i t h e r ij nor m n can occur as an index o f u or ~, and in some E, H even two terms can be exc lude d and a p p e a r exhaus ted . And so on.


It is now only a mat ter of studying the law of constructing minimal values repea ted lymand until all terms have been excluded, exhausted from ~ and II referring to indices of u or 4. This is theoretically pos- s iblemin practice, the complications can quickly astonish us. The completely exhausted ~ vanishes, becomes =0, the 11 is equal to 1.

The procedure leaves fur ther possibilities open for ingenuity: we can try to eliminate the u-coefficient in series (by rows or columns) , or also the uii along the main diagonal or also the pairs conversely to each other, etc.

If this succeeds, we have Xhk as a "propositional function" which no longer appears in any u-coefficient, in front of which the II can be

u

el iminated and which consists merely of coefficients of parameters

a, b, c . . . . of f(u). It remains to "compress" (as I would like to say), to "condense" the

Page 551 proposit ional function, that is, to represent it as a coefficient with suffix hk i ndependen t of h and k, a relative built out of a, b, c, ... by the six species including ~, l-I, a "relative function" X, and we will have found x=X.

Practically, it will always succeed, at least by using sum and product forms of the first order [Stufe] which only extend over the universe of discourse 12~as we will show later in a last problem. However, in most cases, the a t tempt misfires in p r a c t i c e ~ a n d remains open for fur ther research and study.

Problem 26. We are looking for

x=II[a(bu;c) ; d + e{(f+ 4) ~g} ;h], 114) u

where h is not valid as a e lement le t ter - -so that our 11 has eight different relatives as parameters.

With the method of exhaustion, we easily find

x O = F"keikhkjHt(buF',,,a~,,,ctmdmj + f t + gtk), 115)

and we only have to "condense" the expression, that is, to express the relative x with the eight parameters , based on the coefficient relation. We can do this when we replace the index k,l with j,i in the form of

x=2je{(b+ f ) j - g } ; j ' j ; h " I I , {a; ([ ; i )d+ f ; i + i;g;j}, 116)

a result which allows us to derive correctly most previous results as special cases, for example, for a = 1, d = 1' by exchanging letters of 113).

By using l l6 )mabs t rac t ing part of its m e t h o d - - w e transform % = [t~ = ([;/),,f---of. page 423---and ~j is t ransformed into {a; ([; l)d},,j. We, fur thermore , divide the 11 t in IIt (b, + f t + gtk) = {(b + f ) J- g}~k, on the one hand, which can be reduced with the factor % to [e{(b + f ) j- g}]~k~Which may be called for the time being r~k, and, on the

Page 552


other hand, 1-It{a; ([; l)d + f ; l + l; g; k} 0. If r~khkg is written equivalently as (r; k" k; h)i j, we can completely relieve the suffix ij on the right of 115), and then omit it on both sides according to (14), page 33, since the equation 115) was to be assumed under the dominance of the sign II 0. We now have result 116) from 115).

By using the propositions 34) to 36) of w 25 effectively, we can always achieve such "condensation," reduction. It is thus only the el imination or exhaustion procedure which contains unsurmountab le difficulties.

The researcher will immediately become aware of them, when he tries, for example, to find

II {a(u j- b) ; c + d(~t j- e) ;f}. u

We let the problem stand as it is at this point and only observe that it shows certain analogies with the mathematical problem of "ration- alizing" algebraic equations, the elimination of roots which appear in it. Even if each individual root can be eliminated by isolating it on one side of the equation and then empowering the equat ion on both sides with its root exponents, we do not succeed in this way to eliminate all roots. More refined methods are necessary. We unders tand this because when eliminating (with the ment ioned method) a certain root, the number of the other, still to be eliminated roots, is increased.

With respect to the relative coefficients, this is not the case when exhausting, eliminating a certain one of them; however, the difficulty is increased because the other coefficients appear.

Appendix 7: Schr6der's Lecture Xll

Introduction

In his twelfth lecture, Schr6der uses the quantifier rules of the previous lecture and relation algebra computat ions to construct a theory of one- to-one maps and cardinal equivalence within the calculus of relatives. There were no o ther papers at the same level of abstractness in this period. It is likely that the developments of this and the previous lecture, read by Tarski in his youth, were not only the inspiration for Tarski's relation algebras, but also for his set theory without variables. Schr6der 's lectures IX, XI, and XII were certainly the inspiration for L6wenheim's 1915 paper as well.

In the first part of the twelfth lecture (w 30), Schr6der distinguishes types of relations or "mappings" [Abbildung]. Types AI through A4 are def ined on page 561: AI are the mappings that are never undef ined [hie un~utig]; A2 are the mappings that are never multivalued [nie mehrdeutig]; A:~ are the mappings for which the converse is never undefined; and A4 are the mappings for which the converse is never multivalued. Schr6der then says that A~ characterizes the assignments [Zuordnung] that are "mindestens eindeutige," that is, at least single-valued, in the sense that there is at least one value, and A 2 the assignments that are at most single-valued (page 568). Then he defines eindeutige Zuordnung, or function, which is both AI and A2. Thus, the notion of single-valued seems to include the notion of being total. Finally, a "substitution" is a relative of type A~AzA:~A 4 (page 569), in other words, a bijective mapp ing where the domain and the codomain are the same.

379

380

Twelfth Lecture

SCHRtDDER'S L E C T U R E Xl I

w 31. Dedekind's Similar Mapping of One System to Another. Similar or EquipoUent Systems.

Page 596 In the sect ions above, we have looked , so to speak, at the var ious types

o f m a p p i n g s in an absolute sense, namely, as be ing cha rac t e r i zed by cer ta in

p r o p e r t i e s for the entire universe of discourse I. In this sense, for e x a m p l e , an inver t ible m a p p i n g [gegenseitig eindeutige Abbildung] is to be cal led a

"subst i tu t ion."

For o u r i m m e d i a t e p u r p o s e s ~ t h e f o r m u l a t i o n of the concep t s o f

equipollence [Gleichmiichtigkeit], finiteness [Endlichkeit], a n d number [An- zahl]--this way of looking at th ings is no t a d e q u a t e because it imposes

res t r ic t ions f rom the outse t on the m a p p i n g rule or relative x that it by

no m e a n s needs to fulfill, i n d e e d which it o f ten cannot at all fulfill in

prac t ice in that they will confl ic t with essential p r e c o n d i t i o n s o f o u r

inves t iga t ion or be i ncompa t ib l e with t h e m f rom the outset .

Thus, for example, within the domain [Gebiet] of natural numbers, through

the assignment of elements standing under each other as

(as x object : ) 2, 3, 4, 5, 6 (: image for x')

(as x image : ) 5, 6, 7, 8, 9 ( :object for ~)

the two systems a = 2 to 6, b = 5 to 9 will be said to be mapped one-to-one, without our mapping rule x having to be a substitution, whether in the whole universe of discourse of natural numbers or only in the universe restricted to the elements

2 to 9 coming into consideration here. In fact, already in this assignment, the

elements 2, 3, and 4 do not have to be x-images, and 7, 8, and 9 do not need

to have x-images.

That, however, in the universe of positive whole numbers a substitution is not

at all capable of the invertible assignment [gegenseitig eindeutige Zuordnung],

(object) i ,2 ,3 ,4 ,5 . . . .

(x-image) 2, 3, 4, 5, 6 . . . .

which is essential to the proof that this number system is simply infinite, is a

priori obvious: it is necessary, since the number 1 here is the x-image of no

object, for the relative x to have the first row of its matrix as an empty row and

therefore it cannot be a substitution. Mr. Hoppe (1888, p. 31) felt correctly that

this was the case, but used it to charge incorrectly that there was an internal

inconsistency in Dedekind's work~in particular since the "similar mapping" is

not introduced as a substitution at all!

For reasons thus sugges ted we mus t also s tudy mapp ings , especial ly

those tha t are s ingle-valued [eindeutige] (in o n e d i r ec t ion [einseitig], as well as) in bo th d i rec t ions [gegenseitig], in a relative sense, namely, as be ing


Page 597 one such mere ly with respect to a certain system a = a j-0 = a; 1 as object and (possibly also) a certain system b = b~ 0 = b; 1 as (its x-)image.

For the special assumpt ion a = 1 and b = 1, this relative perspect ive turns into the previous absolute o n e - - a n d thus what we will be dea l ing with is a generalization, an extens ion of the ear l ier results (of w 30). P resen t ing those results ear l ier seems to be didactically jus t i f ied because

of their simplicity and impor tance , as well as for offer ing considerably g rea te r ease in their i n d e p e n d e n t derivation, etc.

If we simply take b = 1 or a = 1, then the relative perspect ive, at least

with regard to the image (respectively, object) , turns into an "absolute" one, r ema in ing relative only with respect to the object (respectively,

image) .

Later I want to designate the relative requirements with respect to a as well

as b by numbered letters y (or 6 . . . . ), in order to be able likewise to use the

letter ~ for conditions or requirements referring solely to a,/3 for those referring

solely to b---which comes from the equivalent designations y (respectively, 6 .... )

as illustrated above.

Now in the n in th lecture, we set out for the highly worthy goal of

b r ing ing the most essential aspects of Dedek ind ' s f u n d a m e n t a l studies (Dedek ind 1888) into conformi ty with our theory (which did not prove

to be a l toge the r easy), and in the first par t of this sect ion it is no t the least of our aims to pursue the fu r ther in tegra t ion up to exclusively ".~64, namely, up to ",~ w 5, with the head ing "The Finite and the Infinite."

This i n c o r p o r a t i o n - - j u s t to that p o i n t - - h a s already been very largely

accompl i shed , and not wi thout yielding some gains on both sides (in

particular, a genera l iza t ion and ult imately also a simplif icat ion of chain theory) . In o r d e r to carry it out completely, there remains only to take care of a dozen defini t ions or proposi t ions. Of the first proposi t ions

.~1 to 63 of the oft-cited text, in fact only the dozen inc lud ing ".~21

and ~ 2 5 to 35 have not already been taken care of in the chain theory.

What remains of our task, however, presents a series of u n e x p e c t e d difficulties that should not be unde res t ima ted , and o v e r c o m i n g them will be m o r e than a little instructive and beneficial .

In that I now move on to explain this, I do not want to suggest to

s tudents that I (also) t h o u g h t I was thereby offer ing any simplif icat ion

of Dedek ind ' s conclusions and perspectives (insofar as they are relevant

Page 598 here)! On the contrary: we will have to go to grea t lengths in o r d e r to

prove, by way of strict analytical deduc t ion f rom the fundamental stipulations [fundamentalen Festsetzungen] of our discipline, things which seem

to be obvious to those who are familiar with the na tu re of invert ible

ass ignments and their intuit ion.

We may end up in a position with respect to Mr. Dedekind that is perhaps very similar to the one he had toward mathematicians who contented themselves

382 SCHRODER'S LECTURE xII

with an understanding of the concept of"number" (finiteness, etc.) as something

that was simply given, something (like the air) that was just there, and who

never felt the need to justify inferences drawn by mathematical induction based

on the principles of some logic!

In fact, o u r objective goes b e y o n d this specific advance based on

D e d e k i n d . We also want to reach the po in t of, so to speak, a "pasi-

g raph ic" fo rmula t ion , exp res sed in the no ta t iona l l a n g u a g e of o u r al-

gebra , o f those f u n d a m e n t a l concepts , such as the "similarity" [Ahnli- chkeit] or "equipollence"[Gleichmiichtigkeit] of two systems (as well as the

"finiteness," "simple infinity," etc., o f such a system a n d m i n the case of

the f i r s t m t h e " n u m b e r " [Anzahl] of its e l emen t s ) .

D e d e k i n d ' s c o n c e p t of "similar systems" ,~32 co inc ides with G e o r g

C a n t o r ' s c o n c e p t of "manifolds" [Mannig]'altigkeiten] "having equal cardinality" [Miichtigkeit]. Accord ing to Cantor , such man i fo lds (see Bor- chardt's Journal, vol. 84, p. 242ff) (or systems) s h o u l d also be t e r m e d

" e q u i v a l e n t ' m a n express ion we p robab ly have to refuse to use in o u r

disc ipl ine for obvious reasons. Now this e q u i p o l l e n c e or similarity, for

e x a m p l e , mus t resul t in a re la t ion be tween the systems a a n d b tha t can

be exhaust ively r e p r e s e n t e d by m e a n s of the six species of o u r discipl ine.

T h e first s tep in d e t e r m i n i n g it leads back to an e l im ina t i on p r o b l e m

in o u r a lgebra , and this step will also be solved explicitly for the lowest

universe o f discourse.

Mathematicians are aware of the great importance of the concept of equi-

pollence. I may be allowed here to draw attention to what all it involvesmand,

indeed, to do so in popular language.

Similar or equipollent systems must either both vanish or both be different

than 0. Either they are both finite, or they are both infinite, and in the first

case, both systems must be composed of the same number of elements (either

"equinumerable" sets of units, or, in other words, the units in both sets are "of

the same frequency [Hiiufigkeit]"ma concept that precedes the concept o fnum-

Page 599 ber). In the other cases, either both systems are "simply infinite" or both are

not, thus forming manifolds of the second type in G. Cantor's sense. As examples

of the latter sort, irrational numbers could be men t ionedmand even the tran-

scendental (irrational) numbers- -or also numbers as such, the totality of points

on a straight line, etc. As one of the most striking examples of the first, Mr.

Cantor's works are known for having brought to light the surprising fact that

the system of rational numbers (including even the system of algebraic numbers)

has the same cardinality as the positive whole numbers.

We are therefore concerned with an expansion of the concept of the equal

numerability [Gleichzahligkeit] of finite sets, which will also make it possible to

apply it to infinite systems and at the same time is also propaedeutic in regard

to the c o n c e p t ~ a n d in the process will be led to observations that cannot fail


but to secure the tie connecting the "doctrine of manifolds" and theories of the

"actual infinite" to general logic, and prove themselves fruitful in the latter. May the student, therefore, follow our presentation free of utilitarian concerns,

keeping in mind that one of its main objectives has to do with first taking

systematic control of algebra as an instrument in order to learn to apply it to more subtle tasks.

A grea t deal wou ld be ga ined a n d the Car tes ian a n d Le ibn iz ian idea

of pas ig raphy would seem to have taken a g iant step, p e r h a p s its mos t

s ignif icant a n d difficult step, forward, if (in this v o l u m e ) it tu rns o u t to

be possible to del iver the p r o o f that the no t a t i ona l capital c r e a t e d with

o u r s t ipula t ions (1) to (15) (an overview of which I have p u t t o g e t h e r

in a h a l f m n o t very dense ly p r i n t e d m p a g e in my Annalen no te [ S c h r 6 d e r

1890]) is fully a d e q u a t e to r e p r e s e n t exactly a n d exhaus t ive ly all definitions, propositions, a n d conclusions within the in te l lec tua l compass of

D e d e k i n d ' s t e x t m a n d c o n s e q u e n t l y the fundamental concepts of arithmetical science as a whole--to clad t h e m in the most concise formulas a n d with

absolute consistency.

Because we are going to depart somewhat from the course Dedekind took,

leaving for later our look at only one-directional single-valued assignments [einseitig eindeutigen Zuordnung] of the elements of a system to those of another, we

begin directly with the invertible assignment [gegenseitig eindeutigen Zuordnung] between the elements of two systems and with the concept of "similar" systems

526 , 32, as well as continuing this, establishing the propositions with respect

to the latter.

In doing so, however, we will have to follow out several paths. Various versions

of the definition of similarity will emerge, which are also to be referred back to one another.

T h e necessary a n d suff icient c o n d i t i o n for the "similarity" or "equi- Page 600 pollencd' of two systems

a = a~tO = a ; 1, b = b~O = b; 1 O)

is (always) the existence, the possibility of a "similar (or d is t inc t [deu- tlichen]) mapping' x (respectively, ; ) of the two systems in to each o ther ,

tha t is, an invertible assignment be tween all the e l e m e n t s o f the o n e system

a n d those of the o t h e r ~ w h i c h , however, it r ema ins now to exp la in in

g r ea t e r c o n c e p t u a l detail .

We can first put forward a "rigorous" formulation of this requirement as

minimal, thus requiring no more than what is absolutely necessary; in other

words, that the matter will be strictly regarded as an "internal concern" of systems

a and b, and that nothing will be stipulated concerning the external behavior

of the mapping rule x. It is therefore left completely open which x-images might

still be possessed by the elements of not-a or ~ inside or outside b, as well as

Page 601

3 8 4 SCHR6DER'S LECTUI~E XII

which among the elements outside of b (therefore of/~) might otherwise be x- images whe the r of a or of d. Finally, whe the r there are e l emen t s in a that are

no t x-images of others , as well as whe the r there are e l emen t s in b o f which no

x-image exists, is likewise to be left o p e n and indifferent .

What is needed, then, is the following: For every element h of a, there should be inside b one and only one element k

that is an x-image of the same, and conversely: for every element k of b there should be inside a one and only one element h, of which that k is an x-image.

In our formulation, it will be necessary to avoid the definite number word "one (and only one)," and we will have to put in place of the condit ion with that aim another than the one upon which it is based, which in essence stipulates that other (or various) is to belong to other (various).

The essential task therefore confront ing us is: For every h:~--a there is (one) k:~--b and for every k~--b there is (one)

h:~--a such that

k @ x ; h (thus h ~ : ; ; k ) ,

while fulfilled at the same time is the double requirement , which we will designate by Xkh--SO as not to have to write down its expression repeatedly-- that , namely,

for every element m of a, g: h, k ~ x; m must be valid,

Xk,, = for every element n of b, g: k, n:~=x;h must be valid.

Symbolically, this appears as follows:

II^{(h=(c--a) =~-Ek(k=~-b)(k ~- x; h)Xkh}IIk{(k ~ b) ~- Eh(h =~- a)(k =~- x; h)Xkh}, 1)

where

Xkh = IIm{(m=~c--a)(m ~ h) ~ - ( k ~ x; m)}II,,{(n:~--b)(n ~ k) @(n4~ x; h)}.

Now this simplifies for the coefficients of our relatives to

II,,(a,, =~-EkbkXk,,Xk,,) IIk(b k =~-E,,a,,xk,,Xk,,),

where Xk,, = I-Im(amOtmh :~- :~k,n) II,,(b,,O',,k :(= x,h)" Or, therefore,

n,,{d,, + ~,b,(xX),~ln,{~, + ~ha,,(xX)kh},

where Xkh = IIm( ~km + d m + l'mh)H,,(Y.n + /~,, + l'.k)

= [{;~ (d + 1')}{1'~ (/~ + ~)]k,

in fact appears later to be the coefficient of the suffix kh of a relative X that is independen t of k and h, the meaning of which is obvious.

If, accordingly, we introduce the abbreviation y for xX, where

385 F R O M P E I R C E T O S K O L E M

y = {(a+ ~) o ~ l '}xll ' 0 ~ ( ; +/~)} = 1~ 0 ~ (d + l')}xl(lq-/~) 3- ~}, 2)

would be valid, then remain ing as an expression is ou r condi t ion

I'lh(ah + ~"kbkYkh) IIk(Dk + ~"hahYkh) = I l k h { ( a + 1 ; by)([~ + dy ; 1)}kh

m a s is appa ren t if one thinks of the runn ing index k h of ~k and ~h as having been r e n a m e d I and keeps constantly in m i n d that, for system coefficients, the second index and, for system-converse coefficients, the first index can be arbitrarily a p p l i e d n a c c o r d i n g to convenience .

The latter H is, now, the general coefficient of a dis t inguished relative, which itself is known to be equal to it. Thus is 0 j- ot/3 :t 0 itself, or if one prefers

1 =(= 0 :t ot/3 ~ 0,

where ot = a +1 ; by = a + [~ ; y, j3 = {~ + ~ty ; 1 -- [~ + y ; a, the expression of our condi t ion. This, however, splits into (0 :tot 3-0)(0 :t/3 :tO), which is equal to (ot ~ 0)(0 #/3), because here ot is a system converse and /3 is a system. Moreover, it reduces to ot/3-- 1 or (1 =(=ot)(1 :(=/3) equivalently. Thus we have as an expression of it:

11 ; ( a + by) ~tOllO~t (/~+ 6y);11 = {(a+ 1 ; by) ~tOllO:t (/~+ 6y; 1)1

= ( /~ ;y# d)(/~:t y; a)

= ( a j - 5 ; b ) ( ~ : t y ; a ) = ( a : ~ - - 5 ; b ) ( b : ~ y ; a ) 3)

m w h e r e all that was necessary to attain the factor before last was the convers ion of the condi t ion 6=(=/~;y rewrit ten as 1 =(=ot [and] on the left side was to be observed that a dist inguished relative is equal to its converse.

Formula 1) contains the formulat ion, and 2) and 3) toge the r conta in the solut ion of our problem.

Let us express that a system a is "equipol lent" or "similar" to a system Page 602 b---with G. Cantor, loc. cit., page 249- -by the s ta tement [Ansatz]"

a ~-'' b,

which gives us the following "first version" of the def ini t ion of the similarity or equipollence of a system a with a system b:

(a ,I, b) =

[b=~-{(a+ x-) :t l '}x{l':f (s +/~)} ; a][a:~:{(/~ + x ) 3- 1'}~{1'~ ( x + d)} ; b]. x

(4)

This, however, can be p r o d u c e d in a much more comfor tab le form

386 S C H R O D E R ' S LECTURE XII

through the introduction of y instead of x, and, indeed, in an equivalent t r ans fo rmat ion~we as se r t~ the following "second version":

(a,~ b) = E ( b " y ; ~ + a'~;y~-l ')(a=~--f;b)(b=~--y;a), (5) y

whereby the first b and a on the right can be replaced by /~ and d, respectively, andmjus t as we already saw in (4)Dthe last two subsumptions must have the force of equations.

This can be seen as follows. From equation 2), first of all, x can be eliminated. We conclude

y~-- (a+ x-) ~ 1', y=~--x , y~=l '0 ~ ( ; +/~), or

y ; 0' =(= a + ;, ~/x=(=~#l', Ety~-Ezx, ergo 8y=(=~c~l', ~7;~y:61',

0 ' ;y=(=;+/~, x b =(c- l ' ~ ~, y b =~-- x b, ergo y b =~-- l ' ~ ~, y b ; ~ ~-- l ' .

Because, however, a and b are systems, then

~ ; # = / ; y " ~, y b ; . ~ = b ' y ; y ,

and, accordingly, the two partial resultants combine into

b . y ; ; + d . Y;Y =(= 1'. 61)

It immediately follows from this resultant, via conversion, that /~. y ;~ as well as a" ~;y=(=l' must also be valid; so that we could also write them out "more fully" in the form

( b + / ~ ) ' y ; ~ + ( a + a ) . ~ ; y = ~ l ' , 6)

while it can still be represented just as well by the s ta tement [Ansatz]:

b" y ; ~ + a" ~;y=(=a'. 62)

This resul tant--as one sees, the first factor on the right in 5)mis now the full one. For, if it is fulfilled by a y, then there also exists an x that makes equat ion 2) true, and, indeed, in the form of x =y itself. This is to be seen in this way. From the converted 61) /~. y ; ~ + a'y;y=fc--l ' it follows that

f ; y:(= d + 1',

Page 603 and thus

y;)7=(=l' +/~, y=(=)Y# (d + 1'), y=~(lq-/~) #.~,

y = {37# (d + 1')} =(=y=~{(lq-~) #y-}, 7)

that is, for x = y, replacing equation 2) in its second form. Equation 2), established for an arbitrary x, must accordingly represent the general root y of subsumption 6), where, in addition to everything else, it must be allowed that, individually or simultaneously, a can be replaced by ~, as well as by a + d, and b by b, as well as by b +/~.


If, to avoid having to write the n u m b e r e d p ropos i t i ons r e p e a t e d l y in

p a r e n t h e s e s , we use the i r n u m b e r s for the n u m b e r e d p ropos i t i ons , t h e n it has so far b e e n p r o v e d tha t

II{2) :(=6)} o r I~2) :(=6), 6) :(= Z 2); t h e r e f o r e 6) = I; 2), x x x x

bu t in add i t ion , 6) - 7), w h e r e b y 6) = 61) - 6z). Now if t h e r e exists an x such tha t the r igh t side of e q u i v a l e n c e 4) is

t rue , t h e n t h e r e also exists a y, name ly the o n e r e p r e s e n t e d by 2), tha t fulfills the r igh t side of equ iva lence 5), and conversely: if t h e r e exists a y tha t satisfies the last cond i t i on , so is t he r e an x satisfying the first, and ,

i n d e e d , at least in the fo rm x equals y. Cond i t i ons 4), 5) t h e r e f o r e imply

each o t h e r rec iproca l ly or are equivalent .

If one wants, one can transform the one into the other by calculation. If, for example, in 5) we replace the first propositional factor after the E, which is y subsumption 6), with the Z 2) that has been proved equivalent to it, then it is

x

possible to refer this E to everything that follows. Then, if, after the E we use

the expression in x equipollent to it in 2) for all instances of y as t~ae more expressive name, then propositional factor 2), as an identity, becomes equal to 1 and can be suppressed; likewise, the previously written 12 is without an object y and invalid because the general term standing behind it is constant with respect to y---namely, has become free of it, and we have equivalence 4).

In o r d e r to show now tha t the two final s u b s u m p t i o n s in 5) have the force o f equa t ions , as was asser ted, we simply have to m a k e use of 6) to der ive the two reverse s u b s u m p t i o n s f rom

b ~ - - y ; a a n d a : ~ - ~ ; b .

Now it follows i m m e d i a t e l y that

~ ; b : ~ - ~ ; y ; a a n d y ; a ~-- y ;~ ;b .

But we shall not succeed without an unusual sleight of hand. While it is true that the predicate in the first subsumption can be transformed into ~;~i;y;1 =

(~;y)~i" 1- -compared to 61)--which now through 61) becomes :(= 1" 1 - 1. Yet going this way, we arrive only at the worthless conclusion/~;y =I~ I. To have the

success we are aiming toward, it is apparently necessary to "commit" ("perpe-

Page 604 trate") a tautology, namely, by writing

~;b=gc-~;ygt;a=(~;y)gt;a :~- l ' ;a=a, therefore y;b=(c--a,

y" a =(c-- y'~D;b= (y'y')D" b a~-- l" b=b, therefore y" a =(eb,

q.e.d. We thus have as a consequence of 6) or 2):

(b =(= y; a) =(= (~; b =(= a), (a =(= ~; b) =(= (y; a :~- b)

and we also have (repeating the premises in the conclusions):

388 SCHRC)DER'S LECTURE XII

(b =~ y ; a)(a =(=37; b) = ( b = y ; a)(a =y; b), 8)

because this equivalence is obvious as a reverse subsumption.

For this and several further results to be understood correctly, it is necessary

not to overlook the following. Because of 7), it is permitted in (4) for x to be

identified with y; only this must not happen. If we do it, we will also commit to a restriction of the mapping rule x with respect to its "external" behavior.

For x, subject solely to the requirements of 1) or (4), such definitions as the

following are still permitted:

E h (h =(=a) E k (k =(=/~) (k =(=x; h) -- ~k (k =(=/~) E h (h =~-a)(k =~--x; h)

= ~ ; x ; a = [t ; s ; [~ = ( x ; a =~ b) = (:~;/~ =~ ti),

Eh(h =~-d) E k (k =(~--b)(k =~--x ; h) = E k (k =~--b)~ h (h =~-d)(k =~--x ; h)

=/~;x;5---a ;:~;b = (x;d 4 /~) = ( :~ ;b4 a),

9)

which will reappear in a further formulation below as well as that which is given

here- - for a, b instead of for a,/; or d, b.

To require the same for the mapping rule y would not work now because a

requirement such as y; a =~ b would obviously come into contradiction with

y; a = b that was proved above.

In y we have in some sense a l ready m a d e a c o m m i t m e n t a b o u t the

e x t e r n a l behav io r of o u r m a p p i n g r u l e - - b u t only in a m a n n e r o f which

we may be sure that the c o m m i t m e n t can be m e t at any t ime, which wou ld no t be cor rec t for the n a r r o w e r c o n c e p t i o n of the m a p p i n g rule

as a "subst i tu t ion," as we saw on page 596 in contex t . As o p p o s e d to a s t ipula t ion of the lat ter sort, we mus t truly w e l c o m e

such res t r ic t ions o f the m a p p i n g rule with respec t to its ex t e rna l behav io r

to the similar systems a, b, the validity of which is g u a r a n t e e d internally. A n d they cou ld secure similar advan tages for us in such th ings as choos-

ing an a p p r o p r i a t e c o o r d i n a t e system for the tasks we have to carry o u t

in analytic geometry . O f course , however (with c o m m i t m e n t s of tha t

sort) we mus t no t allow any th ing to p re jud ice the in te rna l b e h a v i o r o f

Page 605 m a p p i n g rule, tha t is, the ques t ion: be tween which e l e m e n t s of a a n d

which e l e m e n t s of b is the a s s i g n m e n t to per ta in? Only thus will it be

assured tha t the results of o u r invest igat ion can be app l i ed in full

generali ty.

It might be wise to remark in this connection--with G. Cantor, loc. cit., page

242--that, if it is at all possible in some way or other to make invertible assign-

ments [eineindeutige Zuordnung] between all of the elements of a and b, then the

same thing can take place in many other ways. And this issue remains entirely

independent of the "versions" in which wemwith due consideration for the

external behavior of the mapping rule--might formulate the (one specific)

assignment.


In this sense, we must welcome as a cons iderable s implif icat ion of

ou r similarity condi t ion that it can also be r e p r e s e n t e d in a "third version," as follows"

(a,-", b) : E ( z ; Y . + Y.;z:~-l ')(b:~--z;a)(a:,~_~.;b), (10) z

which requires the existence of a relative z that maps a to b absolutely and concerning the f u l l universe of discourse belonging to type a z a 4 of w 30, and which also conversely (as z') maps our b to a.

Proof. (10) :(= (5) and (5) =(= (10) must be verified. Now if there exists a z that fulfills (10), then there also exists a y in the form of y = z that fulfills (5),

because, as we know, b'z;Z=(c--z;Z, and likewise with Z;z=(=l' a fortiori a" Z;z =(= 1' is also given. With (10), (5) is therefore valid.

Conversely, if there exists a y that fulfills (5), then there also exists a z in the form of

z = dby, ~.= ab~

that fulfills (10). For, on the one hand, even [valid] is

( b =~- y" a) - ( b ~ b . y" a) = ( b ~ Sby" a) - ( b ~ z" a),

(a ~.~" b) = (a =go- a " y" b) = (a ~ a~'; b) = (a =#r Y.; b),

and, on the other, we have

11)

(b" y '~+ a" y'y ~= 1') ~ (bb" y 'a~+ a~t" ~;by ~ 1')

= (dby'a~y"+ a~';~by ~ 1') = (z" z~+ Z;z =(= 1')

-- that , because ay =(= ~, b" y" a~ =(e: b" y'y, etc., q.e.d.

It is also possible to arrive directly at version (10) of the similarity condi t ion , if one, in the fo rmula t ion of the doub le r e q u i r e m e n t Xkh---committing to the external behavior of the m a p p i n g rule x in 1) in some r e s p e c t - - d r o p s the restrict ions m~--a and n=~--b (while h =~-a

Page 606 and k ~= b r emain ) , which mus t have the same effect as if one had taken

a = b = 1 (in Xkh ). In o the r words: if, f rom the start, we take the d i f fe rence of the x-images toward various of the objects coming into cons idera t ion , and of the objects toward various of the x-images c o m i n g into consid-

era t ion, no t only for the e l emen t s of b and a, respectively, bu t for the

e l emen t s of the ent i re universe of discourse. In that case, ins tead of 1), we obtain

II,,l(h =(=a) =(=E,(* ~-b)(k~=x ; h)Z,,,llI,{(k~-b)~-Eh(h~-a)(k~-x; h)Z, hl,

12)

where Zkh = II,n{(m 4: h) ~=(k=~=x; m)llI,,i(n ~ k) ~ - ( n ~ x" h)}, and in place of 2), m u c h m o r e simply,

39 ~ SCHRODER'S LECTURE XII

y = (~o ~ l ' ) x ( l ' o~ ~), 13)

the re fo re , in place of (4) as the " four th vers ion" of the similarity

c o n d i t i o n ,

(a,-",b) = ~ { b @ ( ~ l ' ) x ( l ' j - ~ ) ; a } { a @ ( x j - X ' ) ~ , ( l ' j - x) ;b} , (14)

where , however, the x will be a d i f ferent , m o r e res t r ic ted relative than

the x in the ear l ier f o r m u l a s - - i n 12) to 14) this x may by all m e a n s be

iden t i f i ed with o u r z. For now, we will cite the fo rmulas in this way

( t h o u g h t of as having b e e n wri t ten for z ins tead of x), a n d wha t m u s t

be kep t in m ind , namely, is that, j u s t as r e q u i r e m e n t 1 2 ) - - w i t h the E

wri t ten be fo re i t - - c o u l d be rewri t ten equivalent ly as the r igh t side o~f

similari ty c o n d i t i o n (14) or (10), so, conversely, we c a n n o t neg lec t to

der ive the c o n s e q u e n c e s 12) (writ ten in z) by m e a n s of an equ iva l en t

t r ans fo rma t ion .

To discover the connection 11) between y and z, and thus to arrive at version

(10), it is also possible, finally, to move heuristically from (5) by coming upon

certain "external" commitments over y, namely, by setting forth the "adventive"

requirement: that the elements of the universe of discourse falling outside a,

the elements of 4, do not have y-images, and the elements of b, those falling

outside b, should not be y-images. Such is expressed by the two statements

[ Ansiitze]:

IIh{(h =(= 4) =~-(y; h = 0)} = 0 j- 37 ~ a = (y =(= d), 15)

ILl(k ~- D) =(c--(~;k=O)} =O~ ~ b = (y =~-b),

which are easy to justify in terms of P), page 557, with

IIh{4^ @(~exO)^} = II,,(a + ; o~O)h = 0o~ (a +~ o~0) = ~iJ-;ct 0, etc.

If both requirements are posed simultaneously, it therefore follows that

y=(=~b or y=~by=z,

that is, the y that is subject to the indicated condition can be designated as our

Page 607 z. Also valid for this z, of course, is

z=(~--db or z=~tbz, and z ;4=0 , ~;/;=0 16)

- -because the latter, by the first inversion theorem, turns into z =(=0o,-d = d,

~ 0 ct/~=/~, z =(= b. Yet more simply, however, with z =(= 6, az = 0 it can also be

concluded that z; 4 - a z; 1 = 0 ; 1 = 0, etc.

T h e s e re la t ions 16) canno t , i n d e e d , by any m e a n s be de r ived j u s t

f r om the r ight side of (10). It does no t n e e d to fulfill a z tha t satisfies o u r th i rd vers ion (10). Only, if t he re exists any z at all tha t does satisfy

it, t h e n the re also exists, in the fo rm of dbz, a relative such that, w h e n

Page 608


called simply z, fulfills, in addi t ion to 10), the re la t ion 16) as well. T h e r e f o r e we could take this re la t ion 16) a long into similarity c o n d i t i o n (10) as an adventive r e q u i r e m e n t , thus arr iving at the "fifth version" of def in i t ion of similarity ,~32:

(a ,I' b) = E ( z ; i + i ; z =(= l ')(b ~-z ; a)(a =~--~; b)(z ~c--gtb), z

(17)

in which the m a p p i n g rule appears the most restr icted, the mos t nar- rowly conceived, by way of the most comprehens ive , maximal condi t ions : [it appears] as one , to pu t it colloquially, that assigns the e l e me n t s of a and b, and only these e l emen t s to each o t h e r ( s o m e h o w invert ible) . [All conceivable instances of the ass ignments subject to the same lim- i tat ion be tween the e l emen t s of the two systems mus t p r o c e e d f rom this a s s ignmen t t h r o u g h simple p e r m u t a t i o n of the e l e m e n t s o f o n e of them.]

We will call version (17) the normal form of the similarity definition. We distinguish four partial conditions within it, the first of which we will designate the "characteristic" of the mapping rule, while the last represents an "adventive condition" of it; the other two may be called "main conditions."

It is c lear f rom our no rma l form (17) that to the notion of similarity be tween two systems a and b the no t ion of " relativity" (in H o p p e ' s sense) as a result o f be ing taken with respect to a specific universe o f discourse, namely, the fundamental universe of discourse, does no t apply. For every universe o f discourse 11 in which the similarity o f a a nd b is es tabl ished in some way, the e l emen t s of a and b, at the very least, mus t themselves be long to it, and the re fo re systems a and b as well as the i r converses, are also m e m b e r s of the c o r r e s p o n d i n g 12. Now, since in (17) ou r z is comple te ly inc luded in the r ec tangu la r quadri la teral - re la t ive [Augen- quaderrelativ] rib, then, if the re exists a relative z that fulfills the cond i t ions o f (17) in any arbitrary universe of discourse 12 , so also in every such universe o f discourse 12.

Taking the normal f o r m E or occasionally also just the form (10)Efacilitates all of the proofs that we must now present for the propositions.

As a first corol lary to (17), respectively, (10), we have the p ropos i t i on

a,~" a 18)

that is, every system is similar to itself. In o t h e r words, the re la t ionsh ip of similarity be longs to self-relatives.

Proof from (10) by referring to the fact that for b--a, the identity mapping z-- 1' suffices to fulfill the conditions of the definition of similarity, q.e.d.

For the purpose of proof from (17), the "identity" mapping per se (or in the absolute sense) must be replaced by an "identity mapping within a" (namely, in the relative sense) z - a l ' = dl '= ~ial', and then we can check that both a =

Page 609

392 SCHRODER'S LECTURE XlI

d l ' ; a = a l ' ; a = a " l ' ; a = a a = a , as well as ~ i l ' ; a l '= a l ' ; d l ' = a" 1 ' ; 1 " d= al'~i=(=

1' are identical, while the adventive requ i rement ~ial ' :(=da is also obviously

fulfilled.

T h e s e c o n d c o r o l l a r y is t he p r o p o s i t i o n

(a ,~, b) = (b ,~, a), 19)

t h a t is, similarity is a symmetrical or reciprocal relationship. If a is s imi l a r to

b, so m u s t b be s imi la r to a, a n d converse ly . It is this c i r c u m s t a n c e t h a t

first gives us the r i g h t to say t ha t t he sys tem a a n d sys tem b a re " s imi la r

to e a c h o the r . "

For proof, it is enough to see that the right side of (10) or (17), and indeed

directly the proposit ional product , which forms the general term of the sum

over z likewise, is simply mapped to itself if a is exchanged with b and at the

same time z with L With this latter exchange, the first proposi t ional factor

character izing the mapping rule z only maps to itself. The second and third

proposi t ional factors change places. For (17) the fourth (adventive) proposi-

tional factor that comes up characterizing z maps to the reciprocally converted

and thus equivalent one.

Admittedly, however, the z under the I2 turns back to ~? unde r 12. Nevertheless,

this must be irrelevant. For, if there exists one value, one relative, that fulfills

any r equ i remen t stipulated for z, there must also exis twin the form of its con-

ve rsema relative that fulfills precisely this r equ i rement conceived for L [Indeed,

every condi t ion for z can also be r ega rdedmby taking z = rS, ff = w---as a condi t ion

(and therefore for w) for L]

.~33. P r o p o s i t i o n . Similarity is also a transit ive relationship; in other

words, i f a and b are s imilar systems, then every system that is s imilar to b is

also s imilar to a:

(a ,I, b)(b ,-,', c) ~--(a ,.", c).

T h e s imi l a r o f s imi lars is similar.

Proof. T h a t m i n reference to (17)mi t must be that

20)

12 (x" ~ + ~" x=gc--l')(b = x" a)(a= ~c" b)(x.~--db) x

x 12 (y.~ + f ; y =(= l')(c = y" b)(b = ~; c)(y .~c--bc) Y

~(~12 (z" ff + if; z ~ l')(c = z" a)(a - ~.; c)(z ~--ac), z

follows from seeing that unde r the left-hand side condit ions

z = y ' x , s

certainly fulfill requi rements on the right, and indeed the first proposi t ional

factor, as has already been shown implicitly in w 30, but can be shown again

here, however, with


y; x;:~;~7+ ~ ; ~ ; y ; x ~ - y ; 1' ;~7+ ~; l ' ; x = y ; ~ + ~; x=(=l',

the two following propositional factors, with c = y ; x ; a and a=~ ;~ ; c , as well

become evident through substitution, elimination of b from the premise equa-

tions; finally, the last adventive condition, with y ; x =(c- ~c, because it follows from

y .~-c, x .~- d that y ; x ~ - c ; ~t = cd, q.e.d.

The effort of this last observation, as well as the associated s tatement of the

three adventive requirements, could be saved (if desired) by appeal to (10)

instead of (17). m

If, however, we at tempt to base the proof of transitivity on one of the other

versions of tile definition of similarity--including one that will be presented

below (39)rowe will run into major difficulties.

It is, indeed, easy to prove in turn for the mapping rule z, which, as y; x, is

composed of the two mapping rules assumed in our premises, that it fulfills the

two requirements, which in such a case represent the above equations or sub-

sumptions c ~-z; a a n d a -(= i; c. On the other hand, such [a procedure] is usually

not successful for that part of the resulting similarity condit ion which we should

expect to be characteristic for the composed mapping rule and which will take

on the role "o f the requirement A,~A 4 along with the adventive requi rement in

our normal form of the similarity condition."

The explanation for this, however, can only be that the "external behavior" of

z = y ; x with respect to a and c is not of the same type as that assumed for x with

respect to a and b as well as that for y with respect to b and c I a circumstance

that probably deserves more detailed t reatment and complete clarification in

future research.

The proof given here for 20) distinguishes itself, in a not al together inessential

way, from Dedekind's train of thought in his proof of ~ 3 3 on the basis of the

composition of two similar mappings to a third established by proposit ion

,~31mwhere he sets his argument forth regarding the elements. We will return

to this below on page 621, once we have gained a bit more information.

We may now, given tha t the p r emi se s o f 20) have b e e n fulf i l led, also

call the t h r e e systems a, b, a n d c s i m i l a r to one other. A n d by r e p e a t i n g

the conc lus ions , which we are fami l ia r with r e g a r d i n g equa l i t i es , for o u r

s t a t e m e n t s o r asser t ions o f similarity, we easily arr ive at an e x t e n s i o n o f

Page 610 this c o n c e p t o f "s imilar to each o t h e r " f r o m t h r e e to any a r b i t r a r y sets,

i n c l u d i n g even tua l ly an inf in i te system of " s y s t e m s " ~ a s o f a c o n c e p t

tha t is i n d e p e n d e n t of the o r d e r in which they a re e n u m e r a t e d o r

n a m e d .

".~34. Def in i t ion . It is t h e r e f o r e poss ible to s e p a r a t e all systems in to

classes by i n c l u d i n g in a d e t e r m i n a t e class all a n d on ly systems

(a), b, c, ... tha t a re s imi lar to a d e t e r m i n a t e system a, as r e p r e s e n t a t i v e

o f the class; a c c o r d i n g to the p rev ious p r o p o s i t i o n ~ 3 3 , the class does

n o t c h a n g e if any o t h e r system b b e l o n g i n g to it is c h o s e n as

r ep r e sen t a t i ve .

394 SCHRODER'S LECTURE XII

~ 3 5 . P ropos i t ion . If a a n d b are s imilar systems, t h e n every subsys tem of a is also s imilar to a subsys tem of b, every p r o p e r subsys tem of a is s imi lar to a p r o p e r subsys tem of b:

(a ,I, b)(c ~ a) ~ ~ (c ,I, d)(d ~ b), 21)

(a ,1, b)(c C a) =(e ~ (c ,I, d)(d C b). 22)

If, s ince a, b, c, d are to be t h o u g h t o f as systems, we wan t to c o n s i d e r

[ t h e m ] in the formulas , t h e n all we n e e d to do is r ep l ace the i r

n a m e s - - w i t h the e x c e p t i o n o f the d u n d e r the Z- -by , respectively, a ; 1 , b ;1 , c ;1 , d ; 1 .

Proof of the first proposition. Among the assumptions is

(a,-', b) =~ (z;s s =~.;b)(b=z;a)(z =(ectb), z

whereby themunder l inedmadvent ive condition can be disregarded for the mo- ment. If, furthermore, c 4= a, then we can say

z;c - d

and it follows that z; c =(e=z;a; therefore,

d =ge b, b = d + b = d + ~Ib.

Further, it is possible (although basically superfluous) with ~.;d=~.;z;c=ge 1' ; c = c to easily prove the inclusion:

z.; d =~--c.

More important, and not quite so obvious, is the proof of the reverse sub-

sumption, which then indeed has to have the force of an equation. This is done (without argumentation with regard to elements) as follows.

According to rt) of w 30, page 555, the subsumption given with the above

equation

z ; c ~ d is equivalent to s

According to this, however, the last of the most proximate conclusions is Page 611 justified:

c=(r ~; db=(=~; d + ~; d=(=~; d +

and with c=(=s d + g then it is, in fact, justified that

c ~ i ; d .

Putting together the conclusions gained, we have, therefore,

Page 612


d=~-b and ( z ; i+s i.e., c,-',d,

q.e.d. The adventive r equ i rement foreseen for c and d in the normal form

similarity condit ion z :(=[d is not fulfilled by the z we have had until now, which

maps a and b "normally" to each other. If we want to satisfy it as well, we need

merely to in t roduce [dz as a new z.

That for a and b equipol lent systems via similar mappings z, s with

c=~--a and z;c-d always s and d=~--b 23)

as well as conversely*mis given, we would do well to note in part icular as a

corollary of the proposit ion.

For the second proposit ion, 22), the assumption c :/: a is simply added to the

hypothesis of 21) as a proposit ional factor, and likewise the assertion d ~ b to

the thesis after the summat ion sign.

This can be easily proved apagogically [Translator's note: by contradict ion] .

Namely, if d = b, then with z; c = d according to the above, we would also have

c = ~; b = a; therefore, c - a, contradict ing the assumption.

The above proofs also differ essentially from Dedekind 's for ".~35mwhich we

will also discuss in more detail below on page 622.

In what we have covered thus far, however, the most impor tan t proposi t ions

of this au thor about similar mappings and similar sys tems-- those for which the

others served merely as p repa ra t ionmhave been taken over into our discipline

from the parts of his text del imited at the outset and justified, legit imated in

spirit! The re remains nonetheless a whole series of studies of the material to

be produced .

In o r d e r to fac i l i ta te "argumentation regarding the elements" fo r any sim-

i lar m a p p i n g o r to l e g i t i m a t e it for o u r d i sc ip l ine , t h e f o l l o w i n g obser -

v a t i o n s m i n s u p p l e m e n t to 42) o f w 3 0 m m a y be o f i n t e r e s t .

A c c o r d i n g t he a s s u m p t i o n s la id d o w n in (17) ,

II,,k {(k:~--z;h):~--(h:~--a)(k:~--b)(z;h:~--k)(~;k:~--h)} 24)

(h ~ i; k)

a r e valid, w h e r e t he s u b s u m p t i o n in t he s e c o n d l ine s h o u l d m e r e l y be

c a l l e d to m i n d as an e q u i v a l e n t f o r m o f t h a t a p p e a r i n g i m m e d i a t e l y

a b o v e it.

First, we want to prove this (in a fashion) for the subsumpt ion sign.

If h, k are e lements and if k=g~--z;h, then, first of all, h=(c--a must be valid. For

if it were the case that h =(=a, then we would have h-(=fi, and z; h =~z; fi, which

must be :(= 0 according to 16); with that we would arrive at the contradic t ion

k =(=0, q.e.d.

* The converse follows--if we do not want to perform fully analogous inferences--from tile symmetry of the premisses with respect to the relatives b, d, z and a, c, L

39 6 SCHR6DER'S LECTURE XII

Second, it must be that k=(=b;, for otherwise, with k=(=/~, ~;k=(=~;/~=0 would also be valid and thus h0, which is absurd, q.e.d. Finally, it follows no less that

~.;k=(c-~.;z;h~-l';h=h, since z;h=g~-z;Y.;k=gc-l';k=k, q.e.d.

A c c o r d i n g to (17), inclus ions such as

(k~z;h)=(z;h=k), ( h ~ ; k ) = ( ~ ; k = h )

m u s t have the force of equa t ions ; equa t ions , such as the two in

( z ; h = k ) = ( ~ ; k = h )

[are] equ iva len t to o n e other .

As far as the inclusion of the converse form z;h ~ k is concerned, however,

we had, according to 0) of w 30:

(z; h ~ k ) = (z; h =O) + (z; h = k),

likewise (i;k=(=h) = ( i ; k = 0 ) + (~;k=h).

Insofar as h does not belong to a, respectively, k does not belong to b, then the

first alternative on the right canmand must, by virtue of 16)--take effect. In

order, on the other hand, to be able to draw the conclusion for the second

alternative, that is, the last equation, we consequently have to add to the premise the assumption h =~-a, respectively, k =(= b. That is:

To be es tabl i shed f u r t h e r is the p ropos i t i on

IIhk { (Z ; h ~-- k)(h ~-- a) ~_ (k ~-- b)(k ~-- z ; h) }

x {(~.;k~-h)(k ~ b) ~-(h~--a)(h@~.;k)} 25)

- -where the reverse subsumptions already with the preceding 24) appear proved

a fortiori. It is just as clear that, with 25), 24) will also be proved as a reverse

subsumption, therefore an equation.

The proof of 25) can again be strictly conducted as a reductio ad absurdum, as followsmalthough I am not entirely satisfied with it from a methodological point

of view:

If, given h ~ a, then z; h =0, for example, thenmsince, according to 1), for

z as well as for x, (h ~ a) =(c--Ek(k =(c-- b)(k =(c- z; h) must be validmwe get, for the

k fulfilling the right side of the equation [which one is to think of as independent

from that occurring in 25), perhaps vanishing] the conclusion k =(= z; h =(= 0;

therefore, k = 0, in contradiction to the known proposition k g: 0, q.e.d. Likewise,

if ~; k = 0 and k ~(= b, then there must exist an h =~-b that would vanish, which is

impossible, q.e.d.

By the a s sumpt ions h ~ - a a n d k =~ b, or at least by some o n e of t h e m ,


an equivalence must exist between two of the total of six propos i t ions Page 613 o f the following thesis:

II,, k (h ~ a)(k ~-- b) ~- = ( s ; k ~ h) = (h ~ s ; k) = ( s ; k = h) "

This - -24) to 26)---completely formula ted yields a series of distinguished relatives that are recognized as valid (i.e., = 1) . . . .

In his defini t ion .~26 of the similar mapp ing of a (to b), Dedek ind focused on its nature , that which makes it different f rom the single-

Page 615 valued mapping: that different images should always c o r r e s p o n d to different e lements of a, namely, as is apogogically evident, that the equality of the images of two elements h, k of a always also implies the identity of these elements.

By formalizing this, we obtain the proposi t ion:

1-Ihk{(h + k ~- a)(z ; h = z; k) ~-- (h ~ k)}, 27)

therefore , according to ~) of w 30:

YIj, k[ahak{(Z 3- Z)(~.J- s =(= l'hk] = YIhk{dhk "~ (lkh "~- ( Z ; Z'Jff z ; Z)hk -" l't,k}

= 0 J - ( d + a + i ; s l ' ) j - 0 ' - a j - ( s l ' ) j - d

= ( a ; 1 ; d ~ - - i ' i + z ; z + 1').

It appears, therefore , that

0~d ~= s i + z ; z 27~,)

is the most concise expression in our nota t ional system for the require- m e n t p resen ted above. Only with z ; h =(= z ; k as the second premise, even 27) is already valid---cf. X) of w 30--as

0~d~=s ~. 27~)

This relat ion can t h e n - - a s will be seen below regard ing 31 ) - -a l so be c o m b i n e d with 0~6 =(= z ; z which comes out of it via conversion, and, as for a and z with 27 . . . . 27~), so in b and s there be three analogous formulas that are valid.

Since we have made a somewhat different version of the def ini t ion of similarity fundamenta l [to this work], so we must there fore prove propos i t ion 27~) from our own [definition] with which the two previous ones are given a fortiori.

This can be done very easily as follows. From the identi ty 0~6=(=0~z it follows, by substi tut ing a = s 1 = s (~ + z) f rom 25~) on the r ight side,

'The rest of page 613 and page 614, where Schr6der re-proves propositions 24)-26), are omitted here.


0~zd ~ 0 ' �9 ( i ; i + i ; z),

w h e r e now, b e c a u s e ~ ; z : ( = l ' , t h e last t e r m falls away, a n d in t he o n e

r e m a i n i n g o n the r igh t t he f ac to r 0' c an obv ious ly be s u p p r e s s e d - - q . e . d .

O n t h e basis o f 27) , however , we a re n o w also ab le to take ca r e o f

p r o p o s i t i o n s ,~27 t h r o u g h 30 in D e d e k i n d ' s a r g u m e n t a t i o n .

F o r c = c; 1, d = d~ 1 systems, we u n d e r s t a n d

.~27 . ( c + d:~--a)(z; c :~-z; d) :~--(c:~--d), (c + d:~--b)(~.; c:/~-~.; d) :~c--(c~--d),

28)

' ~28 . (c + d ~ a) (z ; c = z ; d) ~-- (c = d) , (c + d ~ b)( ~. ; c = ~. ; d) ~ (c - d) ,

29)

as o u r e x p r e s s i o n o f the two p r o p o s i t i o n s o f D e d e k i n d n a m e d above .

Proof of 28). For if h is an e lement of c and therefore also of a, then z; h is

an e l emen t of z ;c and therefore also of z;d; consequent ly = z ; k , where k is an

e l emen t of d and therefore also of a. Since, however, according to 27) h = k

always follows from z; h = z; k, every e l ement h of c is also an e l emen t of d, as

Page 616 was to be proved.

Consider ing definition (1) of equality, the next proposi t ion ~ 2 8 is thus a

very close corollary of the previous one.

The assumptions c + d =(= a, etc., may not be suppressed here because, since

the z-images vanish for all e lements outside of a (because z; fi =0), there is in

fact no need at all for a p a r t o f c t ha t d o e s n o t b e l o n g to a to be c o n t a i n e d

in d.

The next proposit ion is different because there the z-images of the subsystems

of c, d, cd . . . . fall away in any case:

"~29. z; c d " " = z; c . z ; d . . . , ~.; c d ' " = z ~ ; c ' s 30)

T h e s e e q u a t i o n s a re also f o r w a r d s u b s u m p t i o n s a c c o r d i n g to 5) o f w 6

in any case.

To prove them as reverse [subsumptions], we can follow Dedekind 's consid-

erations:

Every e lement of z ;c" z ; d " ... is in every case conta ined in z ;a; therefore the

image k -- z; h of an e lement h is conta ined in a. Since, however, z; h is a c o m m o n

e lemen t of z ;c and z ;d "'", so, according to 28), must h be a c o m m o n e lement

of c and d ' " . Consequently, every e l ement k of z; c" z; d" ... is the z-image of

an e l emen t h of cd"" and therefore an e lement of z ;c" z ; d " . . . - -q .e .d .

No doubt it would be possible to produce, for the last three proposi t ions with

II prefixed to c, d . . . . . o ther analytical proofs besides these, which would opera te

Page 618


more computationally [rechnerisch], rather than arguing with regard to the elements.

With 30), a remark made on page 354 appears to be justified.

T h e p r o p o s i t i o n ",~30 mere ly es tabl ishes tha t the ident ica l m a p p i n g of a system a is also a s imilar m a p p i n g of i t - - a n d for us is no less self- evident .

Wi th in b, however, the s ame is possible only insofar as a ~= b . . . . 2

We still have to b e c o m e a c q u a i n t e d with o n e last ( e x c l u d i n g the "ex-

plicit" one ) vers ion of the def in i t ion of similarity:

It is also possible to f o r m u l a t e ind iv idual ly the four c o n d i t i o n s of which

the r e q u i r e m e n t of the s imilar m a p p i n g of a to b consists (ana logous ly to the A~ t o A 4 of w 30) and then s u m m a r i z e it in r e t rospec t . It would m e a n :

3'1 = For every e l e m e n t h of a t he re exists at least o n e e l e m e n t k of b such tha t k =(=x; h,

3'2 = For every e l e m e n t h of a t he re exists at most o n e e l e m e n t k of b such that k ~ - ; h ,

3"3 = For every e l e m e n t k o f b t he r e exists at least o n e e l e m e n t h o f a such tha t k =~- x; h,

3'4 = For every e l e m e n t k o f b t he r e exists at most o n e e l e m e n t h of a

such that k =(=x; h. Thus

3', = IIh{(h ~--a) =~--~k(k ~-b)(k =~--x; h)},

3'2 = II,,k,,{(h=~--a)(k ~--b)(k=~--x; h)(n ~-b)(n r k) =~--(n~- x; h)},

3"~ = Ilk{(k=(c--b) @~h(h=~-a)(k=~--x ; h)},

3"4 = 1-Ihk,,,{(k:(c--b)(h @a)(k=(ex ; h ) (m~-a) (m r h) ~--(k ~- x; m)},

32)

We then f ind (see c o n t e x t f u r t h e r below)

3', =/~; xo ~ d =a j- ~; b = ( a = ~ ; b),

3'2 =/~ o~{~+ l'o~ (/~+ ~ ) } j - d = ( d b x " 0 ' ; b x = 0 )

= a a~ {(x +/~1 a ~ 1' + x } a~/~= (~/~;0' �9 ~/~a = 01,

3"~ = ~ j- x; a = (b=~--x; a),

3"4 =/~ ~ { ( J ~ ' ~ - a ) ~" 1' + ~}j-d = (xd ;0 ' �9 x~b = 0).

33)

Yet ju s t as quickly it is possible to give ")'2 a n d 3'4 in the s i m p l e r fo rms

of express ion :

A list of formulas that Schr6der gives for the student's convenience (pp. 616-617) are omitted here.

4 0 0 SCHR()DER'S LECTURE XII

/ T z = 1 ;{1'~- (/~+ x - ) l~ -d=ld (O~ .O ' ;bx ) =0}

= a ~- { (x +/~) 0~ 1'} ; 1 = {(;/~; 0 ' 0~ 0 )a = 0},

"y4 - /~ o ~ { ( ; + a )~-1'} ; 1 = { (xd ;0 'o~0lb =0},

34)

o r a lso

{ T 2 = a 0 ~ l x o ~ ( l ' + / ~ ) l ; l = l ( ; ; 0 ' b o ~ 0 ) a = 0 } ,

3'4 =/~ ~ - I ; # (1' + d)} ;1 = { ( x ; 0 ~ e 0 ) b = 0}. 35)

Justification. These [values] must be given independen t ly only for 3'1 and

3'2 because, a m o n g the expressive forms requi red for these two condi t ions and

proposi t ions, those for 7:~ and 3'4 are p roduced by exchanging a, h m, x with,

respectively, b, k, n, ;, as can be seen from a look at 32), as soon as we think,

for y:~ and 3"4, of the proposit ions k =(c-x;h and k=~ x ' m as having been rewritten in the equivalent forms h=(=;" k and m=(~ :~" k.

According to 32), we now have the coefficients:

3'1 = IIh(ah=~-F-,kbkxkh) =Ilhldh + F, kbkxkh) =/I;h(d;h + (1 " bx)ih}

=II ;h(a+ 1 "bx);^ = 0 # (c~+ l 'bx) j -0= ( a + 1 "bx) ~0

=/~; xj- 6 = a o ~ ~" b = (1 :(= ad- s b) = (a" 1 :(= ;" b) - (a=(=s b),

"}"2 = 1-I^k,,(a^bkxk^b,,O',,k:~-;,,^) =I-I^k(d^ +/~k + ;k^ + FI,,(/~, + 1~,,, + ;,,h)]

=IIhk[dhk + (['+ ;)k^ + II '~ (/~+ X)]k^]

=IIkhla+ /~+ ; + 1',~ (/~+ ;)}k^

=0d-{(a+ b + ; + l'j- (/~+ K)}aO =~ o~{;+ 1'~ (/~+ :~)} j- 6

=a ,~ {(x+/~) j- 1' +x}}~/~= (1 =(=idem)

: {a" 1"/~= a/~=(=(x+/~)# 1' +xl - - - ( ; " /~ ;0 ' " ;"/~; a =0)

Page 619 q.e.d., that is: this constitutes p roof of the s tatements in 33).

In o rder now to get two of these for forms 34), 35), we state the lemma:

( a ' 0 " a ~ b " 1) : ( a ' 0 ' ~ 0 : ~ - b" 1),

�9 ~ 0 / , ( a ' 0 ' a=~- l 'b) (0~ ;a=(=l 'b)

(b ' l : ( = a o ~ + a ) I b ' l : ( = ( a j - 1 ) ' l }

( l ' b = ( = a + 1' = �9 ' . j-a) {l'b=(=l (1 j-a)}

36)

What is to be presented here are a, an arbitrary relative, and b; 1, jus t any

arbitrary system that can thus also be represented by b a-0.

M t h o u g h for a---log33'0 we have a ; 0 " a = la300, on the o ther hand, a;0'o~

0 = 11100, according to which the two subjects in 36) differ: this proposi t ion

must be valid, as can be seen by comput ing the rows [zeilenrechnerisch]. It can

be proved more elegantly as L -- R thusly. Because a; 0' �9 a =(= a; 0'0'- 0, L follows

from R, that is, R =~- L is valid afortiori. Conversely, (a ;0 ' )a ; 1 :(= b; 1 also follows


from L, which, because (a ;0 ' ) a ; 1 :(= a ;0 ' j - 0 , cf. 30) o fw 15, page 216, goes over

to R, that is, L =(= R is also valid, q.e.d.

According to this schema 36), ~4 from 33), for example, written as

x6; 0' �9 x~i =(=/~(=/~; 1), is easily t ransformed to x6; 0',t- 0 :(=/~, that is, into 3 4 ) h a n d

cor responding conjugates ")'2 from 33) into 34), as well as conversely, q.e.d.

However, 35) is only an obvious transformation of 34) according to known

proposi t ions about systems.

It is very m u c h w o r t h r e m a r k i n g that , whi le , r e g a r d e d as a r e l a t i o n

b e t w e e n t he sys tems a a n d b, t he r e l a t i o n s h i p s ~ a n d 3':t a re transitive,

such is by no means the case with "Y2 a n d "g4.

T h a t can be i m m e d i a t e l y p r o v e d ana ly t ica l ly wi th t he c o n d i t i o n s a

fortiori valid:

( a ~ ; b ) ( b ~ ; c ) ~ ( a = ~ ; ) ; ; c ) = ( a = ( = i ; c ) for z = y ; x ,

( b : ~ - x ; a ) ( c @ y ; b ) @ ( c : ~ - y ; x ; a ) = (c:~c--z;a) fo r z = y ; x ,

j u s t as it is also f u r t h e r i l l u m i n a t i n g tha t , if fo r eve ry a ( a c c o r d i n g to

m a p p i n g ru l e x) t h e r e is at least one b b e l o n g i n g to it, a n d for eve ry b

( a c c o r d i n g to a d i f f e r e n t m a p p i n g ru l e y) at least one c b e l o n g s , t h e n ,

for eve ry a ( a c c o r d i n g to b o t h m a p p i n g ru les t o g e t h e r ) , at least one c m u s t b e l o n g to it.

If, th roughout , we replace the words "at least" here with "at most," it may be

that for some the proposit ion will be every bit as i l luminating. Nevertheless, the

rhetorical evidence would simply have led to er ror (namely, to such an a to

which--as "at most one"ran0 b belongs, any n u m b e r of c could be long much

more directly!). It is also possible to show that an inference from

(x;O'a,l-O:l~-D)(y;O'bj-O:~-g) to z;0~z~0:(::g

can be compulsory nei ther with z =y; x nor with any o ther z, in that the con-

clusion in quest ion would indeed have to have been a resultant of the el iminat ion

of b from the premisses. No such [elimination] is at hand, however, because the

premisses have always proved satisfied for b--0. The conclusion, accordingly,

must signify no th ing [nichtssagend] as a relation; in o ther words, it would have

to be valid as a general formula for arbitrary a, c, and z, which is easily recognized

Page 620 as absurd.

Things are different for a = b = c = 1. The re it really is that

( x ; 0 ' ~ 0 -- 0)(y; 0'a~ 0 : 0) :(:: (y; x;0'~t 0 = 0) 37)

- - a s we see by row computa t ion of the first of the equivalent forms:

(a;0 ' j- 0 = 0 ) : (a ~(= da~ 1') = (gt;a =~- 1') 38)

---or also by deriving it from 36) with the assumption b - - 0 - - m o s t conveniently,

then, as shown on page 567. If, in fact, according to a first mapp ing rule, once

Page 621

402 SCHRI~)DER'S LECTURE Xll

again there belongs to every e lement of the universe of discourse at most one

e l emen t from this entire universe of discourse, likewise according to a second

mapp ing rule, then also according to the composite mapping rules.

W i t h resu l t s 32) to 35) , t he fo l l owing new ve r s ion o f t he s imi la r i ty

r e q u i r e m e n t for a a n d b is g a i n e d i n d e p e n d e n t l y : (a ~, b) - I2 3'~3"23'33'4,

t h a t is, sys tems a a n d b will be ca l l ed s imi l a r if a n d on ly if, if t[aere exists

a m a p p i n g ru l e x, wh ich with r e s p e c t to it fulfills all f o u r r e q u i r e m e n t s

3'1 to 3'4 s i m u l t a n e o u s l y . We have, t h e r e f o r e , as the "sixth ve r s ion" :

( a ~ b) =12(b@x;a)(a@~'c;b){(x;O'a~O)b+ ( ~ ; 0 ' b ~ 0 ) a =0}. (39) x

If at first, in 1), we had formulated the similarity condi t ion in pr incip le as

follows: for every e lement h of a there should be within b (at least) one e l emen t

k, which is "uniquely an x-image of it and of it alone" (and converse ly--us ing

"s instead of x-image), then it is already clear to c o m m o n sense that

there cannot also be a second e lement k' of b that would be "severally" (that is,

a m o n g others) an x-image of it (that h), or, in addition, of it and of o ther

e lements of a. And one feels, or believes to feel, that the earlier version must

essentially coincide with the one formulated now, a l though the latter poses the

relevant requi rement , not as a d e p e n d e n t o n e - - i n a relative p ropos i t i on - -bu t

independent ly . Only, we must not content ourselves with such an intuit ion in

this case, but must prove analytically the equivalence of the latest version with

at least one of the earlier versions.

We can do this by reducing (39) to (10) or (17) as follows. Because

xdb;O'= xb;O'a ~ x;O~ and xdb ~ b, with (x ;0~ j -0 )b=0 ,

it also follows a fortiori that (xdb;O'j-O)xdb--O, and likewise with (~;O'bj-O)a- 0 also that (s163 Thus, if we take

xdb = z,

the last condit ion in (39) teaches us that

(z;O'd-O.)z=O and (z?; 0' j- 0) s - 0.

According to 32), page 216, however, it is generally the case that ( a ; 0 ' j - 0 ) a =

a; 0 " a, and consequent ly it must be valid that

z ; 0 " z + s s

which is easily t ransformed to z;27+ ~.;z=lc-l' because, e.g., it follows from the

vanishing of the first term that z =(= s 1', 27;z=(=l', etc. Moreover, that, in the

above context, the first two requi rements in (39) go over directly into the last

two from (10) we already showed on page 605 (with the difference that our

present xwas represented then by y). Therefore (10) follows from (39), that is,

if there exists an x that fulfills r equ i rement (39) with respect to given systems

FROM PEIRCE TO SKOLEM 4 o 3

a and b, then there will also exist a z (= glbx) that, with respec t to the same

systems, fulfills r e q u i r e m e n t (10) and even (17).

Conversely: if any z satisfies the condi t ion (10), then x = z must also satisfy the

condi t ion (39), which is obvious with respect to the first two r equ i r emen t s , but

can be shown with respect to the last ones as follows.

We have z; 0 ' �9 z = 0, and the re fore z; 0 ~ �9 z~-/~, as well. Acco rd ing to o u r prop-

osition 36), this is equivalent to

z;0'0~0=(:/~, and f rom the re it follows a f o r t i o r i :

z;0~j-0:~-/~, i.e., ( z ; 0 ~ 0 ) b = 0 .

Page 622

And jus t like that, it is shown that ( i ;0 'b0~0)a =0, q.e.d.

A c c o r d i n g to this, t h e m a p p i n g r u l e z t h a t sa t isf ies t h e normal s imi l a r i t y

c o n d i t i o n (17) a l so sat isf ies, in a n y case , t h e c o n d i t i o n s e x p r e s s e d in

a n y o n e o f t h e o t h e r o f o u r v e r s i o n s o f t h e m a p p i n g r u l e (x o r y ) m b u t

n o t c o n v e r s e l y . m

W e a r e n o w in a p o s i t i o n to t ake u p D e d e k i n d ' s p r o o f s o f , ~ 3 1 , 33,

35 wi th t h e i r argumentation regarding the elements.

Proposition ~ 3 1 , which conforms m o r e closely to the s t anda rd t e rmino logy of

ou r discipline, says:

If x is a similar m a p p i n g of a to b and y is a similar m a p p i n g of b to c, then

the m a p p i n g z =y;x c o m p o s e d of x and y is also a similar m a p p i n g of a to c.

Proof. For the distinct e l ements h and h' of a c o r r e s p o n d as e l emen t s of b

to d i f fe ren t images k = x; h, k ' = x; h ~ and these again as e l emen t s of c to d i f ferent

images l - y ; k = y ; x ; h , l ~ =y;k ~ =y;x;h ~, t he re fo re z-- y ;x i s a similar m a p p i n g of

a t o c.

Moreover , every e l e m e n t l of c goes over by f to an e l e m e n t 37; l = k of b, and

this by ~ to an e l e m e n t ~;~; l = ~; k = h of a, so that ~= ~;37 is s imul taneous ly a

similar m a p p i n g of c to a. Various e l emen t s of c in fact must c o r r e s p o n d in this

way as images to various e lements of b, or, as the case may be, of a, for o therwise

the reverse conclus ions would necessarily lead to cont rad ic t ion .

T h e conclusiveness of this p r o o f rests essentially, as we see, on the justif ication:

to express the re la t ionship be tween image e lements and object e l emen t s by

means of a m a p p i n g rule in the form of an equationmwhich we m a d e sure of

analytically u n d e r 26).

Now the transitivity of the similarity re la t ionship be tween the systems a, b,

and c stated in ,~33 is also obvious. For, if there exists a s imilar m a p p i n g x

be tween the systems a and b, and one y of b to c, then acco rd ing to ~ 3 1 the re

also exists, in the form y;x, a similar m a p p i n g z of a to c, q.e.d.

Somewha t m o r e c u m b e r s o m e is the p r o o f of ~ 3 5 , page 610.

To this end, we might well recall the def ini t ion of similarity in the version

f rom 1) and 4):

4 0 4 SCHRC)DER'S LECTURE XII

(a ,I, b) -

E Il^l(h=(=a) "4~----~-~=Y.,k(z'h=k=gc-b)Zkh}Ilkl(k-~E---b ) =gc--Y,^(~.;k=h=~--a)Zk^}, 40)

where, for Zhk = H,,{(m r h)=(=(k r z; m)}H,,{(n r k)=(e(n r z; h)}, we may some-

what more conveniently, as in 1), take the express ion given in 12) for Z^k, and

where replac ing the inclusion sign and its negat ion by the equality sign and its

nega t ion from 26) already appears justif ied for the theses (or proposi t ional

subsumpt ion predicates, such as k=(= z; h).

If now, accord ing to the right side of 40), which has been raised to the level

of a p recondi t ion , z maps system a similarly to system b, therefore , b - -z ;a ,

~; b - a is valid, and c = c; 1 :(= a represents a subsystem of a, then it also follows

that z; c =(= z; a, and, if z; c = d is taken, d =(=b. What is to be shown now* is that

this system d - z ;c must be similar to c, and consequent ly that

IIh{(h =(=c) =(=E~(z; h = k =~-d)Zkh}Hk{(k =(~-d) =~-F.,h( s k = h =(~--c)Zkh}

must be valid. Now since, with h =(v-c, respectively k=~-d, it also follows that

h =~=a, respectively k=~-b, so it appears in fact that all of the subs ta tements of our

thesis, inc lud ing Zkh, are immedia te ly con ta ined in 40), with the excep t ion of

these two:

that k =(= d would be on the left, and h ~(= c on the right.

[Already in t roduced into l-I^ and II k by various p recondi t ions are h and k, and

they the re fore have different meanings a priori and no th ing to do with each

other. If, now, for the first h, which :(= c=g~--a, it is immedia te ly establ ished that

there exists a k=(~-b such that z; h = k, then we have still not achieved our goal

because of course it remains to be proved that this k is even :(=d. Etc.] Now

(k =(c-d) + (k ~ = ~ , (h =(= c) + (h =(= c-) hold, and likewise for the e l e m e n t k that

has already been proved to be a m e m b e r of system b:

( k =g~-- b d - d ) + ( k -~-- b d )

and for the h be long ing to a: (h =~-ac = c) + (h=~-ac-), whereby the two possibilities

Page 623 are mutual ly exclusive.

If, now, k=~-d, where d - z; c, were on the left, then it would follow that

k=~2?a-g ergo s k=(=g, and given z ;h = k, ergo k=gc-z; h

as well as h ~--~.; k, a fortiori: h ~ g, in cont radic t ion to the assumpt ion h =(= c.

If h ~= g were on the right instead, we would not be able to draw that con-

* For this, unfortunately, Mr. Dedekind neglected to offer a justification, in that, on page 10, he introduces the system z;c simultaneously with the predicate attribute as the "system that is similar to c." This, of course, is obvious, as is the entire proposition ~35. from the intuition of the invertible assignment. Nevertheless, the assertion, as provable, should not be deprived of proof, and that it requtres it from the perspective of our discipline, will be shown in the considerations to follow.

F R O M P E I R C E T O S K O L E M 4o5

clusion so quickly, because ~; d as the representation of c is not yet available

here--unless , that is, that it is first proved as we did on page 610--a much better

way to arrive at our goal, if we want to stay with an a rgument regarding the

elements, will be thus.

Given k 4= d = z; c and E; k = h, it follows that s k =(= E; z; c =~- 1'; c = c; therefore,

h =ge c also, by a direct proof.

Likewise for the previous assertion, a direct proof in place of the apagogic

one, would also be productive-- insofar as we want to make use here or there

of the characteristic A2A 4 o r z;z~ + z; z =(=1' of the mapping rule. And this appears,

at least in the last case, to be unavoidable, q.e.d.

T h e c o n d i t i o n for the s imilar i ty o r e q u i p o l l e n c e o f two systems a a n d

b is to be r e g a r d e d as a pu re ly logical r e l a t i on b e t w e e n the two, for the

a d e q u a t e e x p r e s s i o n o f which o u r d i sc ip l ine has the means . It a p p e a r s

as the r e s u l t a n t o f the e l i m i n a t i o n o f the m a p p i n g ru le x, o r y o r z, as

the case may be, f r o m the r e q u i r e m e n t s o f o u r d e f i n i t i o n o f s imilar i ty

in any o n e o f the versions.

As l o n g as the e l i m i n a t i o n is n o t ac tual ly a c c o m p l i s h e d , the ~ o r l-I,

i n d e e d , with the h e l p o f which the resu l t an t s may be r e p r e s e n t e d for

g e n e r a l p r o p o s i t i o n s , a re n o t e v a l u a t e d - - i n shor t : as l o n g as the n a m e

o f the m a p p i n g ru le c o n t i n u e s to f igure in the e x p r e s s i o n o f the r e s u l t a n t

as an i nde f in i t e b ina ry r e l a t i v e - - w e m i g h t still call t he d e f i n i t i o n o f

s imi lar i ty an impl ic i t one .

To prepare for the elimination, we will, for example, form the combined null

equation of all the partial conditions of our definition of similarity: f(z) --0,

whereby, under a number of the expressions for its polynomial f(z) we will still

have a choice, even if we take a definite version, such as (10) or (17), as the

basis. The characteristic z;E+ E; z=(el' is probably best put in the form

z;O'ctO + O,,O';z =0

since, by doing so, we gain the advantage that in every term off(z) the name

of the eliminant z only appears once. Moreover, we can regard the two main

conditions in (10), etc., merely as subsumptions, or also as equations, whereby

the former seems simpler since we end up with (two) fewer terms for f (z). The

Page 624 adventive condition in (17) can also be taken into account or suppressed. Thus,

as the simplest expression off(z), we may offer

f(z) =z;0 'a -0 + 0 j - 0 ' ; z + a(za-/~) + b(s 41)

The following, however, may be added:

a . ~ ? ; b + / J . z ; a + ( a + [ , ) z , O'. z;~+O'. ~?;z

as well as others. Then

406 SCHRIDDER'S LECTURE XII

( a v , b) =E{f(z) = 0 } = [ l = E { 0 0 ~f(z) ~O}]=[II1; f (z);1-O]. 42) z z

Or (av, b)--(L=0), where L = I I U a n d U=l;f(u);1, which, by 41), with the conversion of the third term, yields

U= 1 ; (u ;0 ' j -0) + (0at0 ' ;u) ;1 + (/~j-ti) ;a+/~; (~iat d)

and to which can also be added

+b;u;d+~;u;a+~;u;1 + l;u;d

of which, obviously, the first two terms go into the two last ones. However, we can also combine that with the two last terms of U, according to the corollary

to 38), page 449, yielding

(/~;u + 0at ~i);a+/~; ( u ; d + fiat0)

so that we could also take

U= 1 ; (u;0 ' j -0) + (0 . t0 ' ;u) ; 1 +/~; (fiat0) + (0at ti) ; a+ /~ ;u; 1 + 1 ; u; d,

and others as well.

Now, c o n c e r n i n g the explicit similari ty cond i t ions , to beg in with, certain (two) part ial o r subresu l t an t s can be listed. Moreover , we are in a pos i t ion , for the lowest universe o f d i s c o u r s e m i n d e e d , theoret ical ly , for every finite universe o f d iscourse , to p r e s e n t the c o n d i t i o n of similari ty "explicitly," in a truly c losed form.

To this end , all we need , in fact, is j u s t to write d o w n the c o n d i t i o n

f(z) = 0 in the coeff ic ients as

II0[If(z)}ij = 0] o r E0lf(z)} 0 = 0

e l imina t ing , by the m e t h o d s es tabl i shed, f rom this c o m b i n e d null equa-

t ion, in which the sum e x t e n d s over all o f the suffixes ij of o u r (finite) un iverse o f d iscourse , the coeff ic ients Zhk of the m a p p i n g ru le o r eli- m i n a n t s as a whole (if necessary, successively).

I wan t nex t to offer this expl ici t similarity c o n d i t i o n as such for the Page 625 four lowest universes of d iscourse . It is:

(a , l , b) is e q u i v a l e n t to 43)

Page 626


u n d e r 11 (1 ; a = 1 ; b), w h i c h c o i n c i d e s h e r e w i t h (0 ~t a = 0 ~t b)

u n d e r 1 ~ 2 (1 ; a - 1 ;b ) (O#a=O#b)

u n d e r 11 ( 1 ; a = l ; b ) ( O # O " , a = O # O " , b ) ( O # a = O # b )

o r ( 1 ; a = l ; b ) { 1 ; ( l ' # a ) = l ; ( l ' # b ) } ( O # a = O # b )

u n d e r 11 O' ' 4 ( 1 ; a = l ; b ) ( O # ; a = O # O ;b)

{ 1 ; ( l ' # a ) = l ; ( l ' # b ) } ( 0 # a = 0 f b )

But I do not dare say "etc.," because how it will go on remains s h r o u d e d in

would jo in to the four previous requ i rements as yet a fifth one in darkness. 1.~

the middle, and discovering it promises to be very instructive. Al though a possible

way to do it has already been indicated, it remains a quest ion worthy of being

posed by an academy for a prize; for, lacking truly except ional skill, no one is

likely to discover the trick for e l iminat ing 25 (double suffixed) unknowns Zll,

Zl2 . . . . . Z55 that is requi red here---or, as the case may be, with 20 of them, insofar

as the coefficients of the individual self-relatives of z (here five in n u m b e r ) would

also be general ly subject to e l iminat ion in the case of unqual i f ied [vorausset- zungslos] universes of discourse.

It is possible to avoid the calculations to this point (up to and inc luding 1~4),

which quickly build to a terrible cumber some crescendo, by cons ider ing a col-

u m n schema t i c - - a l t hough , admittedly, not without appea l ing to a c o m m o n un-

ders tanding "that can count to four (or at least to more than one) ." I want to

give the comple te in t roduct ion for the final case here.

As systems conta in ing the "same number" of e lements , a and b consist of

e i ther 0, 1, 2, 3 or of all 4 e lements , which, however- -wi th the except ion of the

two ex t reme cases---can be completely different for b than for a. At the same

time, 5 and /~, respectively, must contain (all) 4, 3, 2, 1 e lements .

The five possibilities that could occur for a can be rep resen ted by the equa-

tions [Ansdtze]: a = 0 , a=i, a = i + j = h + k, a = i + j + h=fq a = i + j + h + k = l , with the unde r s t and ing that the e l emen t letters i,j, h, kshou ld all signify different

elements . We abbreviate this ad hoc a = 0 i - 0 , a = li=i, a=2i, a--3i, a =4i - -1 .

Now each one of these five possibilities with respect to system ais character ized

by a different value system of the four dis t inguished relatives

1 ;a , 0 # 0 ' ; a , 1 ; ( l ' # a ) , 0 # a , namely with a =0 by 0, 0 0, 0 a=i by 1, 0 0, 0 a--2i by 1, 1 0, 0 a=3i by 1, 1 1, 0 a--4i by 1, 1 1, 1,

because for a=klc~/3~,0 we have 1 ; a = l l l l 0 , 0 # 0 ' ; a = l l l 0 0 , 1 ; ( l ' # a ) =

11000 , 0 # a -- 10000 .

Now if a consists of one full row, so does every co lumn of a, as an occupied

one, a m e m b e r of co lumn category % and it will be cast off in the relative 0

0 ' ; a , which is why it then vanishes. But if a consists of more than one full row,


then will every column of this system (and all columns of such a system have

to be congruent) , as a severally occupied one, belong to one of the first three

column categories 1, ct, 3, which transform the relative 0 ~ 0'; a into full columns,

and the latter, consequently must =1.

If a consists precisely of two full rows, then, in our universe of discourse 11 4,

it will have just as many empty rows; its columns belong then to category/3 of

the severally vacant, severally occupied ones, will be transformed in 00 ~ 0'; a into

full columns, while, in contrast, being cast off in 1 ;(1 '$ a), just as in the latter

the occupied columns of a will be cast off; therefore, while the former =1, the

latter must vanish.

If, finally, a consists of three full rows (and, therefore, an empty row), then

the columns of a, as having one empty row, belong to category ct and will be

transformed into full columns in the relative 1 ; ( l 'd-a) , and, in contrast, into

0 j-a, which retains only the full columns of a (that exist only in the case of

a = 1), yet cast off as was apparent directly. [In the last case, it would also have

been possible, instead of considering the problem anew, to apply the earlier

consideration to the relative d and take the contrapositive of its results.

O u r table is t h e r e f o r e jus t i f ied . the value systems of the above table, Conversely, on the basis of 43) under 14 ,

as one also valid for b, again characterize for its part the composition or mode

of formation of b as 0, l j, 2j, 3j, respectively, 4)'. And it is thus clear that in fact

the four conditions taken together make up the necessary and sufficient ex-

pression of the equinumerabili ty [Gleichzahligkeit] of the elements in a and b

under 1' 4"

T h e first 1 ; a - 1 ; b o f these c o n d i t i o n s is e q u i v a l e n t to 1 ; a ; 1 --

1 ; b ; 1 a n d t h e r e f o r e c o m e s o u t as

(a = 0) -- (b = 0) c o n s e q u e n t l y (a 4: 0) = (b :# 0) 44)

t ha t is, it always r equ i r e s s i m u l t a n e o u s van i sh ing o r n o n v a n i s h i n g in the

s imi la r systems a a n d b.

T h e s e c o n d c o n d i t i o n , 0 o ~ 0 ' ; a = 0 0 ~ 0 ' ; b, expresses , solely for itself,

t ha t a a n d b m u s t always s i m u l t a n e o u s l y c o n t a i n m o r e t h a n o n e e l e m e n t ,

or, respect ively, n o t m o r e t h a n o n e e l e m e n t ( insofar as they a re s u p p o s e d

to be s imi lar ) . In c o m b i n a t i o n with the first c o n d i t i o n , however , it is

also the suff ic ient e x p r e s s i o n for the r e q u i r e m e n t that , as s o o n as system

a consis ts o f exact ly o n e e l e m e n t (accordingly , is i tself an e l e m e n t ) , t h e n

b m u s t also consis t o f exact ly o n e e l e m e n t (likewise b e i n g s o m e ele-

m e n t ) , as well as conversely.

Now these two c o n d i t i o n s are (obviously) valid for every un ive r se o f

d i scourse , i n c l u d i n g the u n q u a l i f i e d un iverse of d i scourse , a n d la te r we

will also have to der ive it analyt ical ly f rom o u r de f i n i t i on o f s imi lar i ty

Page 627 ( | 7 ) . I in 43). It is o the rwi se with the last two c o n d i t i o n s u n d e r 14


The last of these 0 a- a = 0 a- b, which is equivalent to 0 a- a a 0 = 0 a" b a" 0, comes

out to

( a = l ) - ( b = l ) , therefore, also (a#: 1 ) = ( b # : 1)

and says that if of two similar systems, the one encompasses all of the e lements

of the universe of discourse, the same must be true of the other, and if not

there, then not here either.

In combinat ion with this last one, the second to the last condi t ion expresses

1 ; (l'a- a) = 1 ; (l'a-b) or 0a-0' ; d =0a,-0';/~, that, fur thermore , if a contains all of

the e lements of the universe of discourse except for one, then b must also contain

all e lements with the except ion of some one of them, and otherwise

notmlikewise conversely.

In fact, however, these two condit ions will be valid only for a "finitg' universe

of discourse, and it will be valid for all of them. They could not possibly be valid

for an infinite universe because it is known of such [a universe] that it can also

be bijectively [eineindeutig] or similarly mapped to p rope r parts of itself (page

596). Just as little, therefore, will these two "last" condi t ions be able to claim

validity for the unqualif ied universe of discourse; they could not at all, for ex-

ample, be necessary conclusions from our definition of similarity and they cannot

in fact be proved as such.

In general , a cut has been put roughly in the middle of the s ta tement 43)

with the dots "..." mentally inserted into the opening, to be made about 14 ,

which can be thought of in the higher universes of discourse as being replaced by

an always increasing n u m b e r of fur ther interpolat ions and previously unknown

conditions, while the first two condit ions (as partial resultants) must exist and

go on existing for every universe of discourse. Leaping past the points ... now, the one before last and the last of the four

condit ions presented unde r 14 ~ must retain their validity in every finite universe

of discourse and conclude the series of partial resultants.

In contrast, it will be necessary to imagine that, if for a possibly infinite universe

of discourse along the path being followed here, the explicit definit ion of similarity

can ever be ex tended (to a full resultant), which it still lacks, by adding fur ther

and fur ther condit ions (as partial resultants) to the first two [condit ions] unde r

141 in 43), then the two "last" condit ions will "never come" from t h e r e . m

T h e first c o n d i t i o n a n d partial resultant:

(a,-'- b) =(= (1 ; a = 1 ;b) = ( t i ; a =/~;b) = etc . 45)

o t h e r f o r m s o f w h i c h we a l r e a d y i n t r o d u c e d n s t a t e d m i n t he first two

l ines u n d e r t h e ru l e in 31) , as we said: Similar systems can only vanish Page 628 simultaneously and must otherwise be altogether different from O. It c an be

i n f e r r e d easily a n d in a var ie ty o f ways f r o m o u r d e f i n i t i o n o f similari ty.

First, we have, for the two main condit ions of (17) a =(c-z; b, b 4= z; a, the single

resultants (the el imination of z): a=(=l ;b, b=~=l ;a. Since, however, according to

4 1 0 SCHRODER'S LECTURE XII

49), page 453: (a=(=l ;b)=~-(1 ; a ~ l ;b) and likewise (b=(e:l ;a)=(=(1 ;b=(=l ;a),

then the combinat ion of these two yield precisely the equat ion 1 ; a - 1 ;b, q.e.d.

But a=(=l ;b could also be rewritten in a = a " 1 ; b - a ; b , so that

( a - a ; b ) ( b = b ; a )

represents ano ther way of writing the same resultant. However, the two main

condit ions are also valid as equations: a - ~; b, b - z; a, and, as such, yield the

single resultants: a - (a ~/~)" b, b = (b,t- a ) - - cf. w 19. Since our a j-/~ = a o'- 0 ,t-/~ =

ad-0 + 0 ,t-/~- a +/~, and/~; b = 1 ;/Jb --0, so, moreover, is the identity o f these latter

two with the previous a - a; b immediately evident.

If we convert the first equation, then in d- /~; z substitute b - z ; a, we conclude

b ; b - b ; z ; a - - d ; a , thereby gaining 6;a--/~;b.

[It would be possible to reach the same conclus ion- - ins tead of in zmwith

the values of y in 13), whereby the x that occurs merely in the c o m p o u n d

~k(x) = (~c,j- l ' ) x ( l ' j - ~) in similarity condit ion (14) appears to be completely elim-

inated toge ther with ~b(x)=y. However, we already had the oppor tuni ty in a

footnote on page 287 to stress what we here find confirmed: that when x appears

merely as an element , an a rgument of an expression ~b(x), the resultant of the

el iminat ion of q~(x) does not represent by a long shot the resultant of x, but

merely a subresultant of it. For the condit ion d; a =/~; b is far from sufficient to

guaran tee the similarity of a and b. Rather, it is merely equivalent to 44).]

In fact, d; a = 1 ; a a - 1 ; a, and consequent ly the s ta tement 6; a--/~; b of

1 ;a = 1 ;b, also does not differ essentially.

We obtain the resultant d; 1 ; a = b; 1;b which was also presented in 31) by

multiplying the last [resultant] by the converted [resultant] on both sides, and

the same also by shifting it to the form d ; l ' ; a - b ; l ' ; b , which has yet to be

justified.

As the second c o n d i t i o n a n d part ial resultant was

( a ~ b ) ~ - ( 0 ' 0 ~ 0 ' ; a = 0 ' 0 ~ 0 ' ; b ) = ( 6 ; 0 ' ; a = / ~ ; 0 ' ; b ) = etc. 46)

a l o n g with m a n y o t h e r f o r m s o f e x p r e s s i o n t h a t have a l r e a d y b e e n l i s ted

t o w a r d the e n d o f 31). W h a t it s t ipu la t e s we a l r e a d y d i s cus sed in t h a t

Page 629 c o n t e x t . To p rove it f r o m 1 7 ) n w h i c h is w h a t we a re o b l i g e d to d o

n o w n w a s n o t all t ha t easy.

I will give three proofs, and also make ment ion of a failed a t tempt that was

instructive insofar as it discloses numerous new expressive forms and relations.

The first thing to note is, according to the most well-known proposit ions

about systems, that c~;0';a = 1 ;a(O';a) - - 0 ' d - 0 ' ; a m t h e latter according to 30),

page 916. The two forms presented in 46) therefore completely converge to

each other, and with 0 " a = 0 ' d " 1 we gain the additional form" l ' 0 ~ d ' l =

1 ;0'b/~; 1. According to this, (0~zd=0)= (0'b/~=0) would be the only thing to

prove, which can easily be traced back to the forms in the final lines of 31).

Since, fur thermore , 0~z, as well as a" 0'" a (= aO'" a), is a system, the dist inguished

Page 630


relative O' O' 0'- �9 a would then not be 0 only when O~'z - 1, which is why the assertion

also comes out to (1 =(=O';a) = (1 =(=O'b); and fur ther the dis t inguished relative

1 ;a(O' ;a) will not be 1 when a(O';a) =0, which is why the same ones [expres-

sions] must also be equivalent with (a" O ' ; a = O ) = (b" O' ;b=O). With this, the

forms presented unde r 31), insofar as z does not occur in them, appear to be

traced back to each other. Because b = z ; a , a=ff;b, O';z =I(= s and s etc.,

we can conclude:

O' O' O' Oj- "b=Oj- " z" a=(c-O j- s a = O j- i " ~" b=(c-O a- " b,

0 / t Oj-O"a=O0,- " Y . ; b = ( c - O j - z ; b = O , t - z ; ~ . ; a = g c - O j - O "a. 47)

Because the initial subject and final predicate coincide here, all in te rmedia te

termini must be equal to them and among themselves, whereby, when we also

take into considerat ion that z; a = z; 1 and ~; b = ~; 1, it yields a long list of o ther

forms of expression presented in 31) in addition, as soon as our assertion 46)

is proved in some o ther way. It cannot itself be gained in this way, unless we

should manage first to demonst ra te equality between some one expression of

the one and the o ther line of 47 )msuch as, for example, the equat ion 0j-

i; a =0, t-0 ' ; a.

Now, by means of the inferences

s ; a = s ; ~. " b=~- O " b, O " b = O " z " a=~-- s a ,

it is not difficult to prove that

s and analogously z ; b = 0 ' ; a 48)

must be valid. Only with that would the last assertion a m o u n t to a pe t i t i o pn 'nc ip i i .

Meanwhile, with 48) a n d proposi t ion 46), the rest of the expressive forms in

31) are securedmwith the except ion of the first two formulae of the fourth to

last line.

[Therefore, the main thing now would be to prove the equivalence (1:4=

i; a) - (1 :4=0 ' ; a). Concern ing failed attempts, it will be noted that it is indeed

possible to show

( l : ( = 0 ' ; a ) = ( z ~-0 ' ; a ) , ( l : (=s =(a=(=0';a), 49)

but with that appears the real difficulty of br inging the two s ta tements closer

together.

The first thus:

(1 :(=0'" a) = (1" 1 : (=0"a" 1) = (1 ' : (=0" a" 1,t-0 = 0 " a " 1 = 0 " a ) = (1' : ~ : 0 " a) = (1' :4=0" a + 0 3- z) = (1' :4=0" a, i -0 + 0o'- z) = (1' 4 : 0 " a j - 0 j- z) = (1' : (=0" a j- z) = (1' ;z=(=0" a) = (z =(=0'" a)

because 0 a- z = 0---cf. 31 ) m a n d 0" a is a system. The second thus:

4 x 2 SCHRODER'S LECTURE XII

(1 :~-s ={1 :(= (s a=~-s a}=(=(1 =(=s a)

= (27' 1 =(=0" a).

On the basis of 48) it is certainly valid for a,-,',b that (1 :~-0' ;b) =(1 :(=i; a).

To say that, because a,-",a, for b = a , (1 : (=0 ' ;a) --(1 : (=i ;a) must also be valid

would be a mistake, however, because, while there would indeed exist a ~, re-

spectively z of that sort, as m a p p i n g a comple te ly identically to itself, it would

be a different o n e m Z - - t h a n the z that maps our a to b (and possibly even to

a). There fore , this is not the way that the goal is to be reached.]

To first prove proposi t ion 46) in the form (1 :(=0'; a) --(1 :~:0 ' ; b), it suffices,

because of symmetry, to merely show that

(1 :(=0'" a):(=(1 :(=0" b) i.e., n,(1 :,l~--r, tOlta ,) ~r l ; (1 :(=I],Oi, b,).

In o rde r for Et01tal = a a + alj + "'" (without a;) to be equal to 1, every l there

must be more than one I for which a t equals 1. For if only one a~ma~ say--were

equal to 1, then for this i the sum would vanish, and if there were no a t equal

to 1 at all, then it would vanish for every i. The re exist there fore at least two

values h and m, where h ~ m, of I for which a t = 1, that is, we have a h = 1 and

a,,, = 1 or, in o the r words, both h=(~-a as [well as] m=(~--a with h ~: m.

Accord ing to the original version of the concep t of similar m a p p i n g z there

exist now a k ~=b and an n ~=b such that k -- z" h and n = z" m, and indeed , because

h ~: m, it must be that k ~ n as well. T h e r e are therefore for l at least two b t,

namely, b~ and b,, that are equal to 1, and thus Et01tb ~ is also equal to 1 for every

i, q.e.d. And conversely.

This proof, a l though binding, is not satisfying methodologica l ly because it is

based on reasoning with regard to the e lements and the dist inction between

the s ingular and plural a m o n g them.

Analytically, a second p roo f of our assertion 46) succeeds in the form

( 0 ~ d = 0 ) = (0'b/~=0) as follows. Let 0~zd=0; thus 0 'b /~=0" z ' a " ~;~?=

0 " z ' a ' d ; ~ ? = 0 " z" ad ;~?=0" z" ( l ~ d + 0~a') "~?=0" z" l~d '~? : (=0" z" l " i =

0 " z ' i ~ 0 ' �9 1'= 0, therefore 0'b/~=0 as well, q.e.d. The same conversely.

A third p roof is to be given directly for the form d" 0 " a = b ' 0 " b. Because

a = 2?; 1, b = z" 1 is d" 0" a = 1 �9 z" 0 " 27; 1, b" 0 " b = 1 �9 ~; 0 " z" 1, consequen t ly show-

Page 631 ing the equality of the two right sides, or to also t ransform the first of the four

express ions into the last one. We also have: d ' 0 " a - 1 "0~d" 1 = 1" 0'(~?; 1 "z)" 1.

Breaking down the middle 1 he re into ( 0 ' + 1'), then, because ~; l"z=Y.'z=(~--

1', the last part has to fall away, leaving d ' O " a = l ' O ' ( ~ . ' O " z ) ' l . But with

0 " z =~-s = s 0', it also follows that ~?; 0 " z ~= 0', therefore 0'(~?; 0 " z) = ~?; 0 " z itself.

T h e r e thus remains d ' 0 " a = 1 "27;0"z" 1 =/~;0"b , q.e.d.

The relat ion used here a long the way

z . 0 ' . ~ + i; 0' �9 z=(=0' 50)


can also be justified for 0';z=(=s ergo i ; 0 ' ; z =~-~?;s =(=0', etc., whereby now

formula 31) has also been completely proved.

To justify the equation 1 ; z ;0 ' ; i ; 1 = 1 ;~?;0'; z; 1 directly, incidentally, for

which proving it either as a forward or reverse subsumption, that is, proving

~?;0~;z=(=l ;z;0';~?; 1 suffices, we can also call upon the coefficient evidence:

!

E^kzhiO^kzkj~C--'Eh .... kZh,,O' mnZkn �9

Now if i ~: j, then every term Zh,Zkj (where h ~: k) on the left side will also appear I on the right, and indeed for m = i, n =j, because in that case 0~,,,, = 0O= 1. If, on

the contrary, j = i, then, the terms on the left side zhizki (where k ~ h) will not appear on the right, but then the sum of the latter must vanish: it must be that

~hkZh iOhkZk i = ( z ' O t ' g ) i i " - ( l " ~ ' 0 " z ) o =0, because of 50). q.e.d.

W h a t was p r e s e n t e d above a b o u t the "explicit" r e p r e s e n t a t i o n of the

f u n d a m e n t a l cond i t i ons of similarity or cardinality, incidental ly , is s imply

what o c c u r r e d to me on the first a p p r o a c h to the p r o b l e m , a n d it is

no t yet necessary to give up h o p e that with a d e e p e r t r e a t m e n t of the

e l im ina t i on p r o b l e m as that cha rac t e r i zed on page 624---fol lowing the

m e t h o d s at the e n d of w 29 which shou ld be d e v e l o p e d f u r t h e r m i t will

also be possible to ob ta in a concise express ion for the expl ic i t c o n d i t i o n

in genera l .

This would, of course, be a great triumph for our discipline: we can, pro-

paedeutically, give an explicit formulation of the concept of the equipollence

[G~'chzahligkeit] (and even equal cardinality [G~'chmiichtigkeit]) of two systems

without the concept of number and quantity. The possibility of accomplishing

such an ideal seems to me already to have been proved by the realization of

the implicit version of this concept. For, if we can eliminate each z-coefficient

independently of the others, then it should be possible, after all, to eliminate

all of the z-coefficients! An [even greater] prospect opens up: learning one day

to express every possible pair of equipollent manifolds by two arbitrary parameters u and v by "solving" an imagined similarity condition for the unknown variables a and b.

Now that, in the fo rego ing , we have also dea l t f r om the s t a n d p o i n t

o f o u r t heo ry with eve ry th ing tha t is said in D e d e k i n d ' s text c o n c e r n i n g

Page 632 "similar" systems a n d "similar" or inver t ible mapp ings , we wan t to tu rn

to the p ropos i t i ons in the text that refer to the s ingle-valued m a p p i n g

in possibly o n e d i r ec t ion only.

The latter precedes the former in Dedekind's text, and to the extent that the

latter may be regarded as merely propaedeutic to the former (the former as the

final goal of the latter), it appears that in our arrangement it has actually been

rendered dispensable.

~21 provides the "definition" of a "mapping"---or, as it necessary here for us

to say more completely, the "single-valued mapping" of a system a = a; 1 of ele-

ments h by a relative x.

414

To be quoted from the outset, mutatis mutandis:.

SCHRODER'S LECTURE XII

By a single-valued mapping x of a system a = a;1 we unders tand a "law" (binary relative) according to which to every well-determined [ bestimmte] element h of a there belongs a well-determined "thing" (for us, once again, "element" k of the universe of discouse 1), called the x-image of h and denoted x; h; we also say that (k =)x; h corresponds to the element h, that x; h arises or is produced from the mapping x, that h through the mapping x goes over to x; h. Now, if c (= c; 1 :(= a) is any subsystem of a, then the mapping x; a "contains" at the same time a well-determined "mapping" of c because x; c~-x; a, which, "for the sake of simplicity," we may denote by the same sign x, the content of which is that the same image x; i corresponds to every e lement i of system c, which i possesses as an e lement of a; at the same time the system, which consists of all images x; i, should be called the x- image of c, denoted x; c, whereby the meaning of x; a is also explained.

Giving its elements determinate names or symbols is itself to be regarded as an example of a mapping of a system. The simplest mapping of a system is that through which each of its elements goes over to itself; it will be called the identical mapping of the system.

Page 633

We wan t now to fo rmal ize tha t which is ca l led for in the f o r e g o i n g

c o n c e r n i n g the s ingle-va lued m a p p i n g of a system a in a way tha t is

pa ra l l e l to wha t has b e e n said a b o u t s imi lar m a p p i n g s - - - o f several types

a n d o n c e aga in relative with r e spec t to a d e t e r m i n a t e s e c o n d system b

as the r ec ip ien t , r ece iver o f the x- images of a.

H e r e [it is i m p o r t a n t ] n o t to con fuse a ( s ingle-va lued) m a p p i n g by

m e a n s o f x "of a to b" with o n e "of a into b." For the f o r m e r e a c h e l e m e n t

o f b ( a n d also converse ly) wou ld have to be an x- image of e l e m e n t s o f

a. For the la t te r it is necessa ry only for every e l e m e n t o f a also to have

an x- image ins ide b, a n d it is this latter, as the less n a r r o w c o n d i t i o n ,

tha t is o f i n t e re s t to us next .

As the m i n i m a l o r m o s t b r o a d l y c o n c e i v e d c o n d i t i o n such tha t x single- valuedly maps system a to system b a p p e a r s the fol lowing:

Fo r every e l e m e n t h o f a t h e r e s h o u l d exist at least o n e e l e m e n t k o f

b tha t is its x-image, while at the same t ime the c o n d i t i o n Xkh is fulf i l led

tha t every e l e m e n t n o f b tha t differs f rom k is n o t its x-image. In o t h e r

words: for every e l e m e n t h o f a ins ide o f b t h e r e s h o u l d be o n e a n d

on ly o n e e l e m e n t k which is :(= x; h.

This by i tself will in no way p r e j u d i c e the e x t e r n a l b e h a v i o r o f x toward

a a n d b . - F r o m the above, we have now

IIh{ (h :(=a) :~--F~k(k:~--b)(k:~--x; h)Xkh}, 51)

w h e r e Xkh=II,,{(n:~-b)(n :/: k):~--(n4~x;h)}; thus IIh(a- h + ZkbkXkhXkh),


w h e r e Xkh = II,,(D,, + lk,, + Y,,h = {1'0 ~ (/~ + X)}kh, and we let xX = y, thus yielding:

I-Ih(d h + Ekbkykh) = 0 ~ (a + 1 ; by) j- 0 : [J; y j- d = (a d" y; b) = (a ~--~; b).

O u r resul t is thus:

a = ~ - l x o ~ ( l ' + / ~ ) } ~ ; b or a0 ~{x0 ~ ( l ' + / ~ ) } ~ ; b o r a=(=)7;b, 52)

w h e r e y = x{l'0 ~ (/) + x-)} = x{l' +/~) o ~ ~)}. As the r e su l t an t of the e l imina t ion o f x, a a(=l ;b m u s t be valid; that

is, b may not vanish without a. We want to s t ipula te f r o m h e r e on tha t this is fulfilled. It will b e c o m e ev iden t that it was the full resu l tan t , that

is, tha t it will always be the case that for every [ somehow] given a and

b, t he r e will exist a m a p p i n g x tha t satisfies the cond i t ion : Every system a can be mapped single-valuedly to a nonvanishing system b.

From the equa t ion for y, it is now possible to e l imina te x as follows:

y=(=l ' j - ( /~+ ;), y~--x, O;y~(=/~+ ;, b" O';y=(=;=(=)7;

t he r e fo re , b y ' O ' ; y = O [or also b x ~ l ' # ~ , t h e r e f o r e afortion]:

b y :~-- l' j- ~, b y ; y:~c-- l', b;y;y~(=l ' ,

a n d this r e su l t an t that comes in two forms is the full one , for, if it is fulfilled, t hen x - y also satisfies the e q u a t i o n for y.

It is thus also possible for 52) to be r ep l aced equiva len t ly by

(b" y ; ~ @ l ' ) ( a @ ~ ; b ) or (by. O';y=O)(a=(=y;b). 52,~)

F r o m this, however, it follows with y; a@b, namely: y; a ~ - y ; f ; b = (y;y')D;b:~--l ' ;b= b, because convers ion also yields b'y;~:~c--l'.

We would thus like to no te o u r resul t in the "full" f o rm as well:

Page 634 {(b+/~) " y ; ~ - - l ' } ( a ~ - f ; b)(y; a@b) . 53)

Staying with the assumption b--1, as only relative with respect to system a, we have the two conditions:

A2 = (y;)7=(=l') and a=(=37; 1, 54)

the first of which characterizes y as never being a many-valued mapping, while the second--as equivalent to IIh{(h=gc--a)=(c--F,k(k=(ey;h)}= 1 ;y,l-d=a a-f; 1--guarantees that the elements of a at least have images or can really be mapped,

namely, that for every h=(ea there exists a ksuch that ha@-~;k, k=~--y;h.- Since

y = x( l ' j - ; ) is the general root of A2, so, naturally, as is also obvious from 52) for b = 1, 54) can also be represented by


a=(=(xd-l'):~'l, which splits into (d=(=l "x){d=~l �9 (l'd-s 54,,)

according to 29), page 215--which proposition is only a special case of 60) given below, g

If we let dby = z, where if there is

z~db , z ; d = O , ~;/~=0,

then there will similarly be, as before (cf. page 605):

(a:~-~; b) - (a:~--a" ~; b) = (a:~--ai~; b) = (a=(=~; b),

(y; a:~--b) :~--(b " y ; a:~--b) - (byd ; a:~--b) = (z; a:~--b),

{(b +/~. y ;y:(=l'} :(= (b/~. y ;37:(=1')

= (by; [ry':(c-- 1') :(= ( ~ y ; a/~':(= 1') = (z ; i ~ 1'),

and it follows that

(z ; s :~--s163 55)

and this z, put in the place of y, satisfies a for t i o r i the earlier conditions 53), 54).

That the second subsumption in 55) is also valid as an equation follows from the fact that, for i:~--a, also s b:~--a ; b = a �9 1 ; b, and consequently ~; b:~--a must also be valid.

Let us call 55) above the " n o r m a l condit ion for the single-valued mapping by means of z of system a to bb."

It is also possible to get to it by the following essentially different paths and, in part, with new forms of expression.

First, by combining from the outset two of the four conditions 3, formulated on page 617,

Tf't'2 = {(~; 0'b~t 0)a = 0}(a~::~; b). 56)

This approach is also an expression for the fact that system a is mapped single-valuedly to b by means of x. Yet the external behavior of the present x with respect to a and b will possibly be different from that

Page 635 of the earlier x, y, z in 51) to 55). If we continue taking [tbx = z here as well, we likewise arrive at normal

form 55). With respect to the last factor on the right and the adventive condit ion

this has been shown repeatedly (as immediately above, but in y instead of x), and, as for the first factor, it has also been shown, on page 620.

Further, we can also present the condition independently, that for every e lement h of a there would be an element k of b such that the x-image of h would be equal to k. With that, the availability of o ther


e l e m e n t s k' of bb, w h e t h e r as x- images o f h, o r w h e t h e r a lso as m e r e l y

in t he r a n g e x; h as such , is e x c l u d e d , b e c a u s e in t h e s e cases (k'=(=k

t h e r e f o r e ) k' = k w o u l d have to follow. T h e n e c e s s a r y a n d su f f i c i en t con-

d i t i o n for a to be s ing le -va lued ly m a p p e d to b by m e a n s o f x m u s t

t h e r e f o r e also be

II,,{(h @a) :~-Ek(k =(=b)(x ; h = k)}. 57)

D e p e n d i n g o n w h e t h e r we use the s c h e m a 0) o r ~j) o f w 30 fo r t he

e x p r e s s i o n o f t he final thesis factor , we o b t a i n q u i t e d i f f e r e n t e x p r e s s i o n s

for this c o n d i t i o n , a n d it is w o r t h w h i l e to have a l o o k at b o t h o f t h e m .

W i t h the first o n e , we ge t

/ ~ ; x ( l ' ~ ~) ~ d = act ( x ~ l ' ) ~ ; b

= { a @ ( x ~ 1 ' )~ ; b} = (a:~--~;b){a:~--(x~ 1 ' ) ;b} , 58)

a n d with the s e c o n d , in con t r a s t :

a ~ l '{(xff 1') ;xb} ; 1 = {l~z~-(x~t 1') ;xb}. 59)

This must first of all be proved and then traced back to each other, which is

not a l together easy to do but is instructive. For 58) we have

IIh[d + E,bk{x(l' j- X-)}kh] = IIhklaWl" bx(l'd- :~},h ={a+/~" x(l'd- x)} d- 0,

because a- can be suppressed before a system converse. With this we obtain the

first form that converts itself to the second, and then, posi t ioned as a predicate

to 1, also maps to the third, according to the first inversion theorem. The

equivalence of this to the last subsumption and fourth form, however, rests upon

a general proposit ion:

(ci,j- l ' )a" b = (dd- 1') �9 b" a" b,

a b ( l ' d - /~ )+a b a ( ' /~) . . . . l a - ,

(a'O' + fi),j-b=a'O',j-b+ dd-b,

ad-(D+O"b) =ad-/~+ aj-O"b, 60)

in contrast to which, however, an analogous proposi t ion for a(l'a- d) ; b does not have to be valid. It is proved with

Lij = ~hIIk(ctik + l'kh)aihbhj, Rq = ~tIIk(dik + lkt)btjF-,haihbhj = ~hlIIk(~tik "b lkl)aihbhjblj,

consider ing that all terms in which l ~ h must drop out of the latter double

sum because if k ~: /, then aih becomes an effective factor of the rl, toge ther

with a~h; and if we now let l--h, then Rq coincides completely with L O, q.e.d.

Page 636 This proposi t ion obviously combines with certain proposi t ions given in w page

525ff.

For 59) we have, first,

4 1 8 SCHR()DER'S LECTURE XlI

IIh{d h + E~,bj. f ~ ' x ' h . (/~j-s h)} =IIh{6 h + F.qfi;bx'h. (]~j-~" h)},

because, namely, b k = b,h--k" b" h as well as bl, xk^ - (bx)kh, etc.

Acco rd ing to this, ou r next step is the auxil iary task of do ing the s u m m a t i o n

in the last term, that is, conceived somewha t m o r e generally, to carry ou t in

closed form a sum of the fol lowing form

z = X j ; a" ( i # b)

for any a,b. For purposes of solving it, we form its genera l coefficients to the

suffix hk:

z^k = ~ ( i; a)h,( ij- b)hk = ~tithatkII, ,(i , ,h + b,,,h ) ! I I I

= ~;J;taaII,,,(lim + b,, k) =~;a;k(1 j- b)ik) = { l ' a ( 1 j-b)lh,,

whereby z has been found. Thus it is worth no t i ng the propos i t ion :

{ ~ , i ; a " (i'j-b) =1 ; a ( l ' j - b ) , I I ; ( i ; a + ~ ' b ) = 0 o , - ( a + 0 " b ) ,

Zia ' i ( b ~ i ) = a ( b ~ l ' ) , I I i ( a ' i + b ' z ) = b ' O ' + a ) j - O . 61)

Acco rd ing to the first of these schemata , ou r ~, is equal to

1 "(bx" h)(l' j- s h) =/~" bx" (l'j- x')" h =/~; (x~ 1')" xb" h

- -c f . 9) of w p. 444 and 27) of w p. 419.

[Accord ing to this, in par t icu lar for b = 1, we may note the schema:

Ek(x" h = k) =/~" (xj- 1') "x" h = {(xj- 1') �9 x}^^.]

With this, ou r cond i t ion becomes

IIh{(d + (xj- 1')"xb}hh=a j- l '{(xj- 1')"xb}" 1.

62)

For if, for a m o m e n t , we call the con ten t s of the braces c, and the second

t e rm in t h e m d, then, first,

Hhchh = Hhk(l~; 1)^j, = O,r 1~" 1,

because the j-0 at the end can be suppressed . Moreover , 1~:' 1 = 1'5" 1 + 1~/" 1

breaks down and, since he re 6 is a system, we have 1'6; 1 = 6" 1'; 1 = 6, f rom

which arises 0 j- ~ + 1~/; 1) =a j- 1~/; 1, which is what was to be shown at the start.

O u r cond i t ion thus comes out as a=~l~/; 1. Now, it is worth n o t i n g that such

a cond i t ion for any d, if a is a system, mus t be equiva len t to the s imple r 1~=(=

d, because, then

(l~=(=d) --(a" 1 : ( = d + 0') : {a=(=(0' + d) j - 0 : 1~/" 1}.

Accordingly, it is possible to note the proposi t ion"

Page 637

FROM PEIRCE TO SKOLEM 4 x9

(a;1 :(= l'b; 1) : (1" a; 1 :(=b) 63)

and according to this we finally obtain, out of the one we found last, the second

form 59) of our condi t ion, q.e.d.

In o rde r to derive the two forms 58) and 59) directly f rom each other, we

can first go with the dist ingished relative and second go with the subsumpt ion

of our condi t ion.

In regard to the first, it is easy to prove for every a, bb, c from the coefficient

evidence of the proposi t ion that

1" a;cb=l" a(;b 64)

---which, indeed, simply comes out to (c")u=c w Accordingly, 1" ( x j - 1 ' ) ; x b =

1" (x3- l ' )~ ;b is already by itself, and since the last relative p roduc t is a system

because b -- b" 1, and if we call that system e, then 1~" 1 = e, thereby t ransforming

the dist ingished relative 59) into 58) without fur ther ado, q.e.d.

In regard to the latter, the proposi t ion

(l~z; 1 :(= l'b; 1) : (l~z~(=l'b) = (1 ; a l ' :(:: 1 ; bl'),

(1~- 1 = l'b- 1) = (1~ - l'b) - ( l ' a l ' - l ' b l ' ) 65)

is to be established as valid for any relatives a, bb Of these equivalences, only

the first one needs to be p r o v e d ~ p r o v e d , indeed, as a forward subsumpt ion ,

since it is self-evident as a reverse one. To be proved here is

L=(l'a,~-l'b;1j-O) = (l~=(=l'b) as =(=(1 ;a=(=l'bl) =R.

This can be done with

L = L(l~z=(=l') = (l~@-l 'b; 1 �9 1 ' - l'b) = R.

In o rder now to obtain from the first subsumpt ion 58) in le t us call it L - - t h e

last one 59)- -ca l l it R m a n d conversely, we conclude by using the above abbre-

viations d = (xd- 1 ' ) ;xb and e = (xd- l ' )~ ;b as follows: L=(a~(~-e)~(l'a;1 :(~1~; 1),

where now, according to 64), 1~= 1~/ must be valid, therefore L~(=(I~;1 :~:

l ~ / : l ) = ( l ~ ( ~ l ~ / ) = ( l ~ ( = d ) = R , q.e.d. And conversely: R = ( l ~ d ) = ( l h ~ ( ~

1~/= It) = (l~t~(~e) ~(=(1~ ; 1 :(~e; 1), which, since a a n d eare systems, e i ther is with

(a~-e) =L. Proved thereby is L~-R and Rd~-L, therefore L =R, q.e.d.

Given with the thus proved equivalence of the subsumpt ions in 58), 59) is

a l so- - for a = b = l m t h e reference back of the two ex t reme forms of the char-

acteristic of AlA 2 in 17) of w 30, page 587, and thus a heuristic derivation of

the latter of these, revealing the chain of t hough t by which I found it.

Now that we have taken care of these surely instructive details of the derivation,

let us take a closer look at the results.

T h e in i t ia l e q u a t i o n [Ansatz] 58) o r 59) is l ikewise a n e x p r e s s i o n fo r

t h e c o n d i t i o n t h a t a sys tem a can be s i ng l e -va lued ly m a p p e d to b by x.

Th i s x obv ious ly n e e d n o t e v e n be a m a p p i n g in t h e s e n s e o f w 30, fo r


the c o n d i t i o n co inc ides with the charac te r i s t i c o f n o t o n e o f o u r f i f teen types. However , this c o n d i t i o n is essent ial ly d i f f e r en t for all p rev ious 51), 52), 53), 55), which can be u n d e r s t o o d in tha t it o n c e aga in allows

Page 638 a n o t h e r e x t e r n a l re la t ion o f x with r e spec t to a a n d b.

If, however , we let x ( l ' 0 ~ ;) =y, t h e n we get

a=(=37; b, w h e r e y = x ( l ' ~ ; ) o r y ;~=(=1', 66)

a n d can ge t f rom h e r e back to o u r n o r m a l fo rm 55) by app ly ing the

e q u a t i o n [Ansatz] dby = z. It is also possible to show with fo rm 59) that , if an x satisfies the

c o n d i t i o n , t h e n it mu s t also be satisfied by z = dbx, a n d converse ly (where the conve r se is i m m e d i a t e l y obvious for x - z). To be shown, t h e r e f o r e ,

on the basis o f 59), is tha t for o u r z, l~z=(=(zct 1') ;zb(= R) m u s t also be

valid. In fact, R = {(d +/~ + x ) ~t 1'} ; ~bx = {d + (/~ + x ) 0 ~ 1'} ;bx" gt. Con- sequent ly , the asser t ion breaks d o wn in to l~z=(=d, which is obvious be-

cause l~z = 1'6, a n d

l~z~=ld ; bx + {(/~ + x) 0 ~ 1'} ; bx,

which , because of the c o n t a i n m e n t [Einordnung] of l~z in the the un-

d e r l i n e d pa r t o f the r i g h t - h a n d side, is a l ready valid a fortiori by vir tue o f 59).

Now we have a c o u p l e m o r e p r o p o s i t i o n s to prove.

To p r o p o s i t i o n .~35 for s imilar [mapp ings ] c o r r e s p o n d s for the m e r e l y s ingle-valued m a p p i n g which was n o t explici t ly s ta ted by Dede-

k ind as m u c h as taken up inc iden ta l ly in ,~21: P ropos i t i on . If a system a ma p s s ingle-valuedly to b by x, or, respec-

tively y o r z, t h e n by the same m e a n s will every subsys tem c o f a also

m a p s ingle-valuedly to b.

Proof. This follows (for c=c ;1) with c=(~--a from a=~-f;b and y;a=gc-b in 53) a

fortiori as c=g~--f;b and y;c=g~-y;a=f~b, while the characteristic of y in reference

there only to b for the subsystem of a remains the same as for a---q.e.d.

Something similar is also true for the "normal" single-valued mapping z of a

to b in 55)Dwith, however, one exception. If, namely, c is a proper subsystem of

a, then by no means will z; ~=0, the adventive condition as z a~-[b, namely, the

partial condition z=gc-[ of the same, will not be valid, and thus in general does

not need to be valid. From z=(=6 and [ ~ d such a conclusion simply cannot be

drawn. Rather, we get the following breakdown: z; g = z; (~ + ag) = z ; a? [which

~: 0. because every element of a has a genuine image], because c -ac , ~= (t + ~? = d + a~? and z;d = 0 was valid.

A normal similar mapping with respect to a system is consequently a similar

but not normal similar mapping in respect to a (proper) [echtes] partial system


of the former. However, it would naturally be possible to derive one such [mapping] from it again in the form of ~z.

D e d e k i n d ' s "Def in i t ion a n d P ro p o s i t i on" ~ 2 5 has to do with the "Zu- sammensetzung," c o m p o s i t i o n o f two s ingle-valued m a p p i n g s in to a th i rd , as well as the associative law g o v e r n i n g such c o m p o s i t i o n s .

De Morgan-Peirce 's associative law 6) of w 6, which has already been proved

Page 639 for the multiplication of all binary relatives, makes it superfluous to accentuate

the same for the special case of mappings. Nor does composition or relative

multiplication require any further explanation from our perspective.

Thus , for us, the asser t ion of the transitivity of the s ingle-va lued map-

p ing r e m a i n s as the co re o f the p r o p o s i t i o n m w h i c h c o r r e s p o n d s to

p r o p o s i t i o n s ~ 3 1 , 33 r e g a r d i n g s imilar m a p p i n g s .

In so far as the "single-valued mapping .... in the absolute sense" is understood

as referring to the entire universe of discourse, that is, that the name is taken

to be synonymous with "function," this question has also already been taken

care of by our general proposition on page 567. It is otherwise if the single-

valued mapping is understood in merely "relative" terms: as such from one well- determined system to another. To be established here is:

P ropos i t i on . If a system a is m a p p e d by a relat ive x s ingle-valuedly to a n o t h e r system b a n d the la t ter is m a p p e d s ingle-valuedly by y to a system

c, t h e n system a will be s ingle-valuedly m a p p e d to c by the relat ive c o m p o s e d o f the two t o g e t h e r (z =)y; x.

This is valid in fact for the m a p p i n g s x, y, a n d z, c h a r a c t e r i z e d as

"normal" s ingle-valued m a p p i n g s a c c o r d i n g to vers ion 55), w h e r e for

z - - y ; x we have the f o r m u l a

(x; ~=(=l')(a =(=~; b)(x; a=g~-b)(x=(e:~b)(y; ~=(= l')(b =(=~; c)(y; b=(~--c)(y=(~-bc)

-(= (z ; ~?=(= l')(a =(=z?; c)(z; a =~-c)(z ~-dc)

and the three first parts of the assertion with

y ; x ; ~ ; ~ = l ' , a=g~-~;f;c, y;x;a=~-y;b~-c

as appeared before to be easy to prove, but it is also possible with x.~-d, x=geb, y=g~--b, y=g~-c, to conclude in regard to the adventive condition that

y ; x.~-y ; d =~-c; d = cd, therefore z =(ed and z =(=c,

q.e.d. The proposition, however, once again is not valid for the mappings we

defined as single-valued in our other, further versions. Rather, what is (con-

spicuously) required for its validity would be that the external behavior of the

mapping rules is limited with respect to a, b, c, as we have just seen for the normal version of 55).

By tak ing b = 1 in 55), we can also conce ive of this de f i n i t i on o f single-


va lued m a p p i n g as o n e tha t is me re l y relat ive with r e spec t to the ob jec t o f the s ame [not, however, also, like 55), with r e spec t to the i m a g e o r the r e c i p i e n t o f the la t ter] , a n d i n d e e d in the form:

(z; i:(=l')(a:~-~?; 1)(z:(=a') = (z ; i=(=l')(z:(=6 = 1 ;z) 67)

which is the n o r m a l fo rm for 54).

With this version we can also easily prove the p ropos i t ion : If system a

Page 640 is m a p p e d single-valuedly by x, whose image x; a is m a p p e d single-valuedly

by y, so will a be m a p p e d single-valuedly by (z =)y ; x. T h a t is,

(x; :~=~-l')(x.(=6 = 1 ; x ) ( y ;~ - - l ' ) ( y~- -6 ;~= 1 ;y)

~e(z; i ~ l ' ) ( z ~ = ~ = 1 ;z)

for z = y ; x . In fact, it follows both with x=~-d, therefore x ; 6 = 0 also y ; x ; d - O therefore y ; x ~ - d , as well as with the other preconditions: 1 ; y ; x = d ; ~ ; x = 1 ;x; ~; x - 1 ;~; x = 1 ;x -- d according to 26), page 447. q.e.d.

T h e p r o o f of the last p r o p o s i t i o n can also be de l ive red very nicely in

o u r n o t a t i o n with a r g u m e n t s r e g a r d i n g the e l e m e n t s in p rec i se con-

n e c t i o n to D e d e k i n d ' s r ea son ing , in tha t we p u t vers ion 57) for b = 1

at the basis. We have then:

IIhl(h :~--a) :~--Ek(X ; h = k :~--x ; a)}IIk{(k :~--x ; a) :~--Et(Y ; k = l:~--y ; x ; a)} ~=

IIh{(h:~:a ) :(=Et(y ; x ; h = l :~--y;x; a)},

w h e r e the u n d e r l i n e d expres s ions are s u p p o s e d m e r e l y to be r e m a r k s tha t cou ld also be suppres sed ; w h e n p resen t , however , they m a k e it

poss ib le to r ecogn ize tha t t h r o u g h the thesis o f the first p r e m i s e (be fo re

Ilk) , the hypothes i s o f the s e c o n d o n e (af ter the Ilk) is s i m u l t a n e o u s l y

ce r t a in to be fulfilled. To speak general ly:

If for every element h of system a there exists one (and only* one) e lement

k (in the universe of discourse) that is its x-image (consequently because

x;h~E~--x; a will also be contained in the x-image of a), and if for every element

k of the system x; a there exists one (and only one) element l (in the universe

of discourse) that is its y-image, so must there also exist for every e lement h of

a one (and only one) element I that is the y-image of its x-image, that is, is its

y; x-image, q.e.d.

It appears to be not at all easy, on the other hand, in justifying version 59),

for b = 1, for example, to draw directly from the premises

* The statement after the I]k can eo ipso be fulfilled only tbr one k; for if it were fulfilled by a second, k', then, from x;h -- k and x;h -- k' would follow k' = k.


l~z=(=~; ( l ' j - x-) and 1" x;a=r (l'j-37)

the conclusion l~=~-~;f; ( l ' j - 3~0,-~).

Tha t task is hereby r e c o m m e n d e d to researchers, and a similar task could also

be connec ted to version 58) for b = 1. In regard to the latter, one part of the

assertion, namely,

(d=(=l ; x)(d; ~=(=1 ;y)~(=(d=~-I ;y;x)

can easily be proved as follows: It follows from the first premise that ~=

d" 1 ;x, and from the second that d;~;x=(=l ;y;x. According to 20), page 254,

however, ~" 1 ;x=(=~; ~; xmwhereby the conclusion now follows afortiori. The

o the r part of the assertion:

{d=(=l ; (1'3- ~)}{d; ~=(=1 ; (1'3-)~)} ergo{~=(=l ; (I'3-)Yj- Y)}

appears not to be so easy and may be impossible to prove analytically, which

indeed must be possible only if, on the left, we retain the premises from the

previous assertion.

I f a sys tem a s i ng l e -va lued ly m a p s to a sys tem b, t h e n t h e r e m u s t ex is t

Page 641 b e t w e e n a a n d b a c e r t a i n r e l a t i o n t h a t is g iven by t h e e l i m i n a t i o n o f x

o r z, respec t ive ly , f r o m t h e v e r s i o n o f o u r re la t ive m a p p i n g d e f i n i t i o n .

Th i s is

1 ; a=(=l ;b 68)

a n d says sole ly t h a t b m a y n o t van i sh w i t h o u t a.

If, in fact, b c o n t a i n s o n l y o n e e l e m e n t , t h e n t h e r e is n o t h i n g to s t op

us f r o m u s i n g it as t he i m a g e o f every e l e m e n t o f a.

Now, if we a s s u m e t h a t this r e l a t i o n is fu l f i l led , it is a lso p o s s i b l e to

o b t a i n t he m o s t g e n e r a l re la t ive x t h a t s i ng l e -va lued ly m a p s a to b, a n d

bes t s u i t e d for t h a t is v e r s i o n 59) in t h a t in t he c o e f f i c i e n t s it r e q u i r e s

a,~-Y-',,Ilkb,,,x,,,(l',k + s

- - - w h e r e b y we have t a k e n t he r i g h t s ide c o n v e r t e d to/~:~; ( 1 ' ~ :~). F i g u r i n g

in h e r e as a n u n k n o w n is on ly t he coe f f i c i en t s o f t h e / th c o l u m n o f x.

For every i, where ai; = a i --0, this xhi = Uha remains complete ly u n d e t e r m i n e d .

The system converse 1 ;dl ' therefore receives an arbitrary defini t ion in x, or

1 ; 41' �9 u must be a c o m p o n e n t of the relative x for which we are searching.

For such an i, however, where a , = a i = 1, at least one term bh,x^fiafiB,... (with-

out ~^;) of the E^ must be equal to 1 and thus b^; =b h = 1, xhi = 1, XA;-Xt~ ;-- ' '"

(without xh; ) . . . . 0. If there was such an i, then 1 ; a = 1 and by virtue of 68)

also 1 ;b = 1, that is, there exists a certain h for which in fact bh~ = b h = 1, while

for o the r [values of] h it could also be that bh; = b h =0. T h e n we need merely to

keep an bullt out in this co lumn i" for x, on any one of the places where it is

Page 642

424 S C H R I ~ D E R ' S L E C T U R E X I I

intersected by the full rows of system b, while all o ther places in this column

for x must remain empty so that the column i for x will be occupied.

Inspite of the transparency of the structure of x, there does not seem to be

a general expression for x represent ing every root of proposi t ion 59). Only for

the case b = 1 this seems to be possible without considerable addit ional effort.

The situation in this case is, namely, that the columns of the system converse

1 �9 al' at x can only be simply occupied; that is, we have

x = l ' d l " u + l ' a l " f

if f represents the most general "function " that is, a relative with all occupied

columns. The (pasigraphic) expression of this f w e gave in 27), page 589. Since,

however, a and d are systems, then it is possible to simplify

l " d l ' = a , l " a l ' = gL,

in that, e.g., 1 "a l '= 1 "51'= 1 "(1 "d)l '= 1 �9 1" 1 "d-- 1 �9 d, and thus

x = a u + a[u(l 'a- ti) + {O,t- ( a + 0'" u)}l'] 69)

- ( a u + 1' - d- zi)u + al'{O o'- (*i + O' u)}

is the general root of the condit ion

l~z=(=)~. (1' . . . . cl-:~) or a=(=l x 1 (1' :?) for a a 1 ,J- = . 70)

It is r e c o m m e n d e d to researchers that they follow out the case of b ~ 1.

As with the c o n c e p t o f the " s ing le -va lued" m a p p i n g , it is also pos s ib l e

to c o n c e i v e o f the "identicalential r e c i p i e n t o f t he i m a g e o f a" m a p p i n g

as m e r e l y " r e l a t i v e " - - a s o p p o s e d to "abso lu t e , " as it has b e e n u p to this

p o i n t as l ' - - -namely , g iven t h a t r e f e r e n c e is m a d e to a d e t e r m i n a t e sys tem

a as t h a t o f t he subs t r a t e (of the o b j e c t as well as t he i m a g e ) o f t he

m a p p i n g in the un ive r s e o f d i scour se .

In doing this, system b, as a potential recipient of the image of a, can be

disregarded; for if a is not =(= b, then the task is impossible, and if a.~-b, then

it [the problem] takes care of itself as soon as we map a in any way identically

(to universe of discourse 1).

A re la t ive x will be ca l l ed an i d e n t i c a l m a p p i n g with respect to a sys tem

a = a ; 1 - - n o t on ly in the e n t i r e x ; a = a, b u t r a t h e r i f - - t o eve ry e l e m e n t

h o f a, h is its own x - i m a g e - - w h e r e the e l e m e n t s o f d can be m a p p e d

a n y w h e r e , if at all.

This condit ion can be formulated

Ilhl(h~gc-a) ~-(x" h = h)} = IIh[~i^ + Ix(I'd- ~?)}h ] 71)

according to 7r) of w 30, for which, however, IIhld + x^h(/~ d- ~" h)} can also be

taken. The last two II split into I I ^ ( d + x ) h h - O d - ( d + 1%'1) and I I^(d+ l'd-


x-),h = 0,t-{d + l'(l ',t- s 1)}, respectively, II,(d h + [z 3- s =0j - (1' + x-) j- d, which is

obta ined according to a general easy-to-define schema:

II;(i'; a + ia-b;i) =0a- (1' + b) j-a. 72)

If, however, we now place the factor as the predicate of 1, then we obtain, first

of all, 1 =(= a0,- l'x; 1, or a ; 1 = a = l~z ; 1 =(= l'x; 1, which, according to 65) comes

out to 1}z=(=x, a n d , second, 1 =(= aa- l ' ( l 'a- x-) ; 1 or a = 1}z; 1 =(= l ' ( l ' j - :~) ;1, that is,

likewise 1}z=(=l'j-s l 'd~- l ' j - :~ , 0';x=(=0' + a , x=(=l ' j-(0' + a ) = l ' a -0 ' + a =1' +

a, respectively (shorter): 1 :~ (1' + ~) j-d, 1 ;d = ~i=(=l' + s ax=(=l', which comes

out to the same thing. Both times, therefore, we have al together:

l'd~(=x~(= 1' + a, 73)

from which, according to the rules of the identity calculus, we can calculate

x = l ' ~ + u ( l ' + a ) o r :

x = 1'ci + u(~ 74)

Page 643 as the most general relative that identically maps system a, and 73) is the characteristic

of such a relative.

It is evident at first glance that (~30) the latter is identically fulfilled by

x--1' for every a.

It is also possible to use the general root 74) of the salne to verify that our

x must also fulfill the characteristic of the similar mapping (for b = a )mas is

clear a priori. It is instructive, meanwhile, that the latter succeeds only if it is

based on the "first" version 4) of the definition of similarity, in which there is

no th ing at all to prejudice the external behavior of mapping rule x. With o ther

versions, in contrast--- such as (10), for examp le - -whe re there is some degree

of prejudice, it will certainly not succeed (as is easy to see).

If, in o r d e r to o b t a i n the " n o r m a l f o r m " o f t he re la t ive ly i d e n t i c a l

m a p p i n g o f a, we a d d the c o n d i t i o n x; d = 0 (as an a d v e n t i v e c o n d i t i o n )

to t he c o n d i t i o n s we h a d thus far, t h e n it m u s t be t h a t u a ~ -g , t h e r e f o r e

u a ~= ~a = 0, a n d

x = a l ' 75)

r e m a i n s as an e x p r e s s i o n for t he fully determined m a p p i n g t h a t m e r e l y

m a p s a ident ica l ly . T h e s a m e w o u l d also m a p every p r o p e r s u b s y s t e m

( b u t n o t " n o r m a l l y iden t i ca l ly" ) . O b v i o u s l y it also satisfies t he s imi la r i ty

c o n d i t i o n t a k e n for b = a in its " n o r m a l " ve r s i on (17) . I

I n s o f a r as p a r t o f the goa l we h a d in m i n d was the e m b o d i m e n t in

o u r d i s c ip l i ne o f t he de f in i t i ons , p r o p o s i t i o n s , a n d c o n c l u s i o n s o f De-

d e k i n d ' s t ex t u p to t he p o i n t i n d i c a t e d on p a g e 597, t h a t is, u p to ~ 6 4 ,

we c o m e h e r e to the end, a n d it will be f o u n d t h a t t he p r o p o s i t i o n s

4 2 6 SCHRODER'S LECTURE XII

21, 25, 26, 27, 28 29,30 31, 32, 33, 34, 35

on page 638, 639ff, 615, 616, 621, 598607 622, 610(622) ,

are taken up, r ep resen ted , and deal t with, with the modif ica t ions necessary f rom the s t andpo in t of ou r discipline.

To convey an idea, however, of the multiplicity o f the cond i t ions that impose themselves, at the same t ime provid ing the s tuden t with m o r e c o m p r e h e n s i v e pract ice m a t e r i a l - - w h e t h e r in word ing cond i t ions in the fo rm of affirmative or also nega t ed subsumpt ions as well as of distin- gu i shed relatives; w h e t h e r in the in t e rp re ta t ion of the l a t t e r - - w h a t we want to do next is p r o c e e d t h r o u g h a n u m b e r of the mos t no tab le condi t ions , p re sen t ing them fo rmu la t ed in the symbolic l anguage of

Page 644 o u r discipline. Since it is always the case that 0 =(= x; a, it is meaning less to r equ i re

that "one" x-image of a disappear. Rather, this has m e a n i n g only if it is r e q u i r e d in the form x ; a = 0 of "the" x-image of a.

Now, if a = a ; 1 and b = b;1 are systems, and if in e n c o d i n g the condi t ions we make use of the des igna t ing m a p p i n g rules given in the con t ex t o f page 597, then what follows immedia te ly is an overview of obvious possibilities for the condi t ions and their fo rmula t ion .

T h e x-image of every e l e m e n t o f a vanishes:

c~ = IIh{(h :(=a) :(= (x; h =0)} = 0 ~ s d = (x; a =0) = (x:(=a).

T h e r e exist e l ements of a whose x-image does no t vanish:

&~ = F,h(h:~-a)(x; h =/= 0)} = 1 ;x ; a = (x; a ~= 0) = (x=~ a).

T h e x-image of no e lements of a vanishes:

c~,, = IIh{(h=~--a) =~--(x; h =/= 0)}= 1 ; x j - d = (a:~--~,;1).

T h e r e exist e l ements of a whose x-image does vanish:

&2 = F , h ( h ~ a ) ( x ; h = O ) } - (0 j - ~ ) ; a = ( a ~ ; 1).

To be no t ed is that the universal j u d g m e n t s o~ and og 2 are also valid for a = 0, that is, when there exists no e l e m e n t of a at all. For, in the spirit o f ou r a lgebra of logic of "every" e l e m e n t of a is valid, as we know, for every th ing conceivable: bo th that it vanishes as well as that it does no t vanish. In that case, or, and og 2 b e c o m e consis tent (meaning : that the x- image of each e l e m e n t of a does no t vanish) . . . . :~

The rest of the conditions (pp. 644-648), closely analogous to his first example, are omitted here.

Page 649


In order once more to characterize the similar m a p p i n g of a system a (= a ;1 ) to itself, we a p p l y n t o the c o n c l u s i o n n t h e normal form (17) of the similarity condit ion to a system b that is similar to a and is conceived of as =(=a.

It would be poss ib lemunders tanding u(= u ;1 ) as an unde t e rmined system~simply to enter b = ua in that formula, having only to associate with the previously written E another E. Better, however, if we add to this the condit ion b ~ a and do comp~'etely away with the name b by replacing it wherever it appears with the equivalent z ; a .

Then we have:

(a~'- a subsystem of itself)

= I] z maps a similarly to itself 76)

=~ (z ;~+ ~z=~-l ' )(a= i ; z ; a)(z; a=~-a)(z:~--d . z" a), z

where the under l ined factor expresses the adventive condit ion and as such could also be suppressed. The same, however, could be replaced yet more completely by

(z:~--a6 " z ; a)

because, with z:~--z; a and z; a:~--a it also follows that z:gc--a

The equivalence of the general terms in both ~, which is supposed to be given implicitly here, will likewise be frequently needed as such.

Formula 76) now forms the point of depar ture for fur ther impor tant considerations that make up the second part of the volume.

There we will see how easily, with the scant designatory resources of our discipline, it is possible to formulate, as it were, pasigraphically, quite likely the majority of fundamental number theoretical as well as arithmetical concep t s~ inc lud ing "being ordered," "being discrete," "being dense," and "constancy," e tc .mof an amount , and how the goals of reasoning and proof can be advanced by such representat ion.

Appendix 8" Norbert Wiener's Thesis


One of the brightest lights in American mathemat ics of the first half of the twentieth century was Norber t Wiener, founder of cybernetics, the modern theory of Brownian motion, and of the Paley-Wiener Theory, among many other accomplishments. Wiener started his career with a thesis at Harvard (at the age of 18) in mathemat ical logic, and subsequently traveled to Cambridge, England, to study unde r Russell. Wie- ner 's thesis turns out to be a careful examinat ion of Schr6der ' s algebra of relatives vis-fi-vis Russell's t rea tment of relatives in Principia Mathe- matica. Wiener quotes Russell's disparaging remarks about Schr6der, and then with meticulous detail makes the case that the o rder of propositions and proofs regarding relatives in Schr6der was essentially copied into Principia Mathematica with an adjustment of basis from classes to propositions, a distinction without much essential difference. In o ther words, he is saying that his men to r Russell did not give credit where credit was due. He also makes a case, as L6wenheim did later, that what Russell did with the theory of types Schr6der could do as algebra, different algebras at different types. He also points out that Schr6der, accused of confusing unit sets with their elements, essentially only uses unit sets, never their elements, a point we have made before. This is natural if we think of the Boolean algebra order ing of the subsets of a power set unde r inclusion as the basic datum. The accusation of confusion was made by Alessandro Padoa and implied by Russell, and Wie- ner refutes it. However, Wiener also notes that Schr6der misread Peano so far as the e relation goes.

Here we give the introduct ion and last chapter of Wiener ' s thesis.

429

43 ~ NORBERT WIENER'S THESIS

Introduction to Norbert Wiener's Ph.D. Thesis (1913)

C. S. Peirce was the first au thor to begin a serious and detailed study of the Algebra of Relatives, but his work was so hampered by an inor- dinately complex and awkward symbolism that it thereby lost much of the value which it might have had to other workers in this field. Schroe- der took up Peirce's work, and developed it in a form at once more manageable and more sequential. After Schroeder had shown the possibilities of the algebra of relatives on the basis of the old Boolian symbolic logic, Russell tried, both in his Principles of Mathematics, and later in the Principia Mathematica, ~ written in collaboration with Whitehead, to construct a new algebra of relatives on the basis of Peano's work on classes and propositions. Peirce, Schroeder, and Russell stand almost alone in the work they have done on this subject. Since Russell claims that his work is done on a foundat ion different from that of Schroeder, and since their symbolisms are almost totally dissimilar, the question of their connect ion and the translatability of the formulae of the one into the terms of the other becomes of the first importance.

Russell says 2 of Schroeder, "Peirce and Schroeder have realized the great impor tance of the subject of the Algebra of Relatives, but unfortunately their methods, being based, not on Peano, but on the older Symbolic Logic derived (with modifications) from Boole, are so cumbrous and difficult that most of the applications which ought to be made are practically not feasible. In addition to the defects of the old symbolic logic, their method suffers (whether philosophically or not I do not at present discuss) from the fact that they regard a relation as essentially a class of couples, thus requiring elaborate formulae of summat ion for dealing with single relations. This view is derived, I think, probably unconsciously, from a philosophical error: it has always been customary to suppose relational propositions less ultimate than class propositions (or subject-predicate propositions, with which class propositions are habitually confused), and this has led to the desire to treat relations as a kind of classes. However this may be, it was certainly from the opposite philosophical belief, which I derived from my friend, Mr. G. E. Moore, that I was led to a different formal t rea tment of relations. This t reatment , whether more philosophically correct or not, is certainly far more convenient and far more powerful as an engine of discovery in actual mathematics."

Without any desire to belittle in any degree the magnif icent work of Russell, I would like to raise the question whether the advances which he has made in the Algebra of Relatives are of so sweeping a nature

'I shall refer to tile l'rincipia as 'Russell'. "~ Principles of Mathmatics, p. 24.


and mark such a radical d e p a r t u r e f rom the d i rec t ion of work p o i n t e d

ou t by S c h r o e d e r as he then s e e m e d to think. It is perfect ly t rue that m u c h of Sch roede r ' s work is c u m b r o u s and difficult, :~ and that, on the whole, Russell 's work in the Principia Mathematica is c o m p a c t and simple, bu t even he re it is an o p e n ques t ion to me whether , in genera l , when S c h r o e d e r and Russell t reat of the same subject, S c h r o e d e r is so m u c h

b e h i n d Russell after all. As to Sch roede r ' s r ega rd ing a re la t ion as a class

of couples , Russell explicitly affirms this very s t a t e m e n t in his Principia Mathematica. It is t rue that S c h r o e d e r regards a relative as a sum of

relatives which are of such a na tu re that they hold be tween the two terms of a uni t couple , whereas Russell regards a relative as a class whose

members are uni t couples , but to me no vast d i f fe rence in the complex i ty

of ~(xRy) and EoR!~(i" j) is apparen t . In fact, as I shall show, no t only are these two express ions equivalent , but any o p e r a t i o n on the o n e can

be carr ied ou t on the o t h e r with exactly the same ease in an exactly parallel manner.4

As to Sch roede r ' s t endency to r educe relative p ropos i t ions to class

propos i t ions , he is perfect ly willing on occasion to r educe class propositions to relative proposi t ions , as in his whole t r e a t m e n t of systems, 5

and, as a ma t t e r of fact, it seems to be mere ly for the sake of conven ience ,

ent i re ly apar t f rom any metaphysical cons idera t ions , that S c h r o e d e r de-

velops his t r e a t m e n t of relatives f rom that of classes and propos i t ions .

S c h r o e d e r tells us why he starts with subject -predicate propos i t ions . He says: '~ "Beg innen wir sonach damit , die Urteile in 's Auge zu fassen, wie

sie die Wor t sp rache als Saetze formulir t ! Es muss sich uns h ierbe i emp-

fehlen , u n t e r Beisei te lassung de r z u s a m m e n g e s e t z t e r e n , zunaechs t uns an die e in fachs ten Arten der Urtei le zu hal ten. Als solche e r s c h e i n e n die s o g e n a n n t e n ,,kategorischen" Urteile, welche sich dars te l len in Fo rm

eines Satzes, de r mit e i n e m "Subject" ein "Praedikat" verknuepf t . " In o t h e r words, the re is no metaphysica l ques t ion involved he re at

all. As S c h r o e d e r finds predicat ive p ropos i t ions easier to deal with than

th'e o t h e r types of p ropos i t ion which he considers , he p r o c e e d s in an

ent i re ly just i f iable m a n n e r to analyze all o t h e r p ropos i t ions in te rms of

them. It may well be that S c h r o e d e r was mis taken as to the simplicity

and the conven i ence of this m e t h o d of p r o c e d u r e , and that it is actually

an easier and a c learer course to prove the p rope r t i e s of classes and of

relatives f rom those of propos i t ions , but the re is no "phi losophica l e r ror"

here , for the re is no metaphysical purpose .

T h e r e migh t seem to be some tacit metaphysical p u r p o s e in Schroe-

~ In certain points, such as the treatment of the classes determined by a given relation. Schroeder's treatment is far more compact and simple than Russell's.

4 However, i and j do not correspond precisely to x and y. " Algebra der Logik, III,w 27, etc. ~ A. der L., I., p. 126.

432 NORBERT WIENER'S THESIS

der ' s Der Operationskreis des Logikkalkuls, for the re he treats his universe o f discourse as all-inclusive, but this view is explicitly r e p u d i a t e d in the Algebra der Log~k. 7 With Russell, however, as he h imsel f admi t t ed in the Principles of Mathematics, the case is different . He a t t e m p t e d to cons t ruc t a sort o f universal g r a m m a r of thinking, a definit ive set of postula tes for all mathemat ics , never to be superseded . He said: ~ "I h o l d - - a n d it is an i m p o r t a n t par t of my p u r p o s e to p r o v e - - t h a t all pu re ma the ma t i c s ( inc lud ing Geome t ry and even rat ional Dynamics) conta ins only one set of indef inables , namely the f u n d a m e n t a l logical concep ts discussed in pa r t I. W h e n the various logical constants have been e n u m e r a t e d , it is somewha t arbi trary which of t hem we regard as indef inable , t h o u g h the re are apparen t ly some which must be indef inab le on any theory." This last clause indicates the fatal weakness of Russell 's en t i re posi t ion. It is, as a ma t t e r of fact, entirely arbi t rary which p ropos i t ion a m o n g those t rue in a given system we take as a postula te for that system, p rov ided we make a r ight select ion of o t h e r postulates to go with it. In any system, we may e i the r d e d u c e all the propos i t ions t rue within that system f rom a given set o f postulates for the system, or that set of postula tes f rom the r e m a i n i n g propos i t ions t rue within the system. If it can be shown that o n e can select f rom Sch roede r ' s t r e a t m e n t a set of postula tes ad- equa t e for the Algebra of Relatives, then, however m u c h we may call his t r e a t m e n t of the subject c rude or c u m b e r s o m e or i ncomple t e , we canno t , as Russell does, brush it aside as based on a false ph i losophica l theory.

I am glad to be able to say, however, that Russell 's t r e a t m e n t of the na tu re of postulates in the Principia Mathematica indicates that he has b e e n a p p r o x i m a t i n g m o r e and m o r e closely to a cor rec t posi t ion. He says:" "Some propos i t ions must be assumed wi thout proof , since all in- f e rence p roceeds f rom propos i t ions previously asser ted ... these, like the primit ive ideas, are to some ex ten t a ma t t e r of arbi t rary choice; t hough , as in the previous case, a logical system grows in i m p o r t a n c e acco rd ing as the primitive propos i t ions are few and simple ... T h e p r o o f of a logical system is its adequacy and its cohe rence . " This is surely a d e p a r t u r e f rom the posi t ion that ma themat i ca l postulates are metaphysical ult imates.

In a compar i son of two systems of ma themat i c s or ma themat i ca l logic, the natura l way to begin is by a compa r i son of their postulates. Now, as the initial postulates of Sch roede r ' s a lgebra of relatives are those which def ine the na tu re of classes, whereas Russell begins with propos i t ions , and since it is their t r ea tments o f relatives which we wish to c o m p a r e , we c a n n o t follow all the stages by which they develop their discussion

v A. der L., I, w 9, p. 245. Throughout this thesis we shall take the Algebra der Logik as the definitive expression of Schroeder's views.

"E 112. "E 13.


of relatives. Therefore we must be content with making, as it were, cross- sections of their methods of procedure, and compar ing these. In o ther words, we shall compare Schroeder 's postulates for classes, not with Russell's postulates for propositions, but with a set of theorems concerning classes which Russell claims are sufficient to define all their formal properties. Similarly, we shall compare Russell's postulates for propositions with certain formulas of Schroeder from which we may deduce all the laws to which propositions are subject.

As both Schroeder and Russell treat the calculus of relatives as derivative from one or the other of these two calculi, and as it is part of my purpose in this thesis to show how each of the two authors proceeds to construct it, it is not really essential that I should try to discover an i ndependen t set of postulates for the Algebra of Relatives from among the formulas of either. However, it is an exceedingly impor tan t fact that the Algebra of Relatives possesses every formal law of the Algebra of Classes, and only differs from the latter in having other formal laws also. I intend to devote the first chapter of my thesis to a proof that the sets of postulates for classes and propositions given by Schroeder and by Russell respectively are equivalent, and that, in so far as the laws of the Algebra of Relatives coincide with those of the Calculus of Classes, their t reatments of the Algebra of Relatives are also equivalent.

After I have proved these things, I propose to devote the second chapter to a comparative study of the methods by which Schroeder and Russell make the transition from the calculi of classes and proposit ions to the calculus of relatives. I shall first discuss the two formulae which Schroeder and Russell respectively make the basis of their t r ea tment of the subject,

a = ~ o a o ( i ' j ) and R = 5c~(xRy),

and show that Schroeder 's formula is equivalent to a proposi t ion true in the system of Russell (al though I cannot prove that Russell's formula is true in the system of Schroeder, because, as I shall show, it cannot be expressed in his language).

Afterwards I shall take up Schroeder 's definitions of the various operations which can be pe r fo rmed among relatives and the relations of subsumption, equality, and inequality which hold between them. To do this, I shall demonst ra te that the a + b, ab, a j-b, and a ; b of Schroeder are equivalent to the a ~ b , a ~ b , - ( - a ] - b ) , and alb of Russell; that 6 and d correspond to - a and ~ (or cnv 'a); that :(= corresponds to C and = to =.; that 1 --V, 0 = A, 1'= I, and 0' =J. I shall also show that the relative i of Schroeder is equal to i 1 V, where i is a unit class, and that ( i ' j ) = u $ v, where i = t ' u and j =t 'v , or that it is equal to i r~j, where i and j are taken as relatives in Schroeder 's sense. On the basis


of these facts I shall establish a general method whereby I can translate into the language of Russell any proposition whatever of Schroeder, provided that it is obtained simply by 'adding' , 'multiplying', and negating proposit ions formed by the combinat ion of relatives by means of the symbols shown above, or propositions of the form a 0, or both. It will also enable me to render into Schroeder 's terminology any similar combination of Russell's propositions concerning relatives with one another or with expressions of the form xRy, provided that the x and y are only "apparent variables." I shall also compare the g' and p' of Russell with the E and II as applied to relatives.

The following chapter will be devoted to a comparison of the ways in which Schroeder and Russell respectively approach the problem of the connections between classes and relatives. It will be shown that, a l though the D'R of Russell is a class, whereas R; 1 is a relative, they have precisely similar formal properties, so that the one can be substituted in any formula for the other, if the necessary changes are made. The class i corresponds to the relative i ment ioned by me in the discussion of the previous chapter. CI'R will become in Schroeder 's language /~;1 , and C'R will become R; 1 + /~ ; 1, or, what is the same, (R + R) ; 1. I shall illustrate these translations by render ing various propositions which Russell proves concerning these classes into the terminology of Schroeder, and proving them from Schroeder 's theorems. It will be shown that oil R is equivalent to otR, where the relative ot is formed by adding all the couples whose foreterms belong to a. Similarly, R I a will be translated as 6lR, and oL 1 R ~/3 as c~/3R, c~ 1' 13 will be shown to be equivalent to c~/3, and to be what Schroeder calls an "Augenquad- errelativ", or an expression of the form C0{II h (aihbhj) }(i : j) . On the other hand, I shall show that - ( a T/3) is a "Lfickenquaderrelativ", or an expression of the form Co{Eh(a~h + bhj)}(i:j). On the basis of our translation of ot 1'/3, our render ing of ( i ' j ) as ij" will be verified. We shall also show that the relative which Russell call R"[3 is simply R;/3 in Schroe- der 's terminology.

Although there can be rmthing in Schroeder 's system corresponding to the classes of one-many, many-one, and one-one relations which are represented by Russell as 1 ~ cls, cls ~ 1 and 1 ~ 1 respectively, since Schroeder is unable to treat of any classes of relatives, nevertheless R �9 1 ~ cls, R e cls ~ 1, and R �9 1 ~ 1 can be translated into certain proposit ions of Schroeder which concern R. It will be shown that

(R e 1 ~ cls) = (R;/~ ~: 1'),

(R �9 cls ~ 1) = ( /~ ;R =(= 1'), and

(R �9 1 -~ 1)= (R;R + /~ ;R:~-I ') .

Both Schroeder and Russell treat of the similarity of two classes. For


"a is similar to b" Schroede r writes a ~ b, and Russell writes a s m b. a ~' b is de f ined as

( z ; ~ + ~; z :(= l')(b = z; a ) (a= ~.;b)(z=~-[tb), z

whereas an equa t ion sufficient to def ine c~ sm 13 is

k : c~sm/3.-=: (3R) .R e 1 - - * I . a = D ' R . ~ = C I ' R . .73.1

I shall show that these two not ions of similarity are equivalent . I shall also show that R e a sm b, or "R is a relat ion which puts a and b in one- to-one co r re spondence" , can be t ranslated by

( R ; R + k; R=~--l')(a = R;1)(b = R;1).

From these examples I shall be able to show that I can translate into Schroeder ' s t e rminology without any al terat ion of its truth-value, every propos i t ion of Russell which deals solely with relatives of the same "type", and classes of the type of their domains and converse-domains (which must be of the same type).

Closely c o n n e c t e d with class-relatives or "systems" (relatives of the form a; 1) are the "ausgezeichnete Relativ" of Schroeder , that is, relatives which can be r educed to the form 1 ; a ; 1. These have only two possible v a l u e s - - l , if a ~ : 0, and 0, if a = 0 . Now, since a - l = a = a " i and a �9 0 = 0-- a" 0, b(1 ; a ; 1) is equal to b (a ~: 0), or b(Eoao). This makes it possible in an express ion which involves the p r o d u c t of a propos i t ion by a class or relative to replace the proposi t ion by an "ausgezeichnetes Relativ". As the a o in Zoao(i:j) is a proposi t ion, and is mul t ip l ied by the relative (i: j ) , we may replace it by an "ausgezeichnetes Relativ." Sch roede r gives us the relative i; a;j as this "ausgezeichnetes Relativ." Tha t it is an "ausgezeichnetes Relativ" is shown by the fo rmula

(i; a;j) = (i; 1 ) ; a ; ( j ; 1) = (1 ; i ) ; a ; ( j ; 1) = 1 ; ( i ; a;j) ;1.

This gives us a n o t h e r way by which we migh t have p r o c e e d e d f rom the calculi of classes and proposi t ions to the calculus of relatives. Tha t this is a valid way I shall show.

To conc lude the chapter , I shall try to show that 1 ; a ; 1 ; a ; 1, a ; 1, and a form a regular sequence , and that the step f rom class or system to relative is closely analogous to that f rom propos i t ion or "ausgezeichnetes Relativ" to class. In general , the steps f rom binary to ternary, f rom ternary to quaternary, f rom qua te rna ry to quinary, and, in general , f rom n-ary to (n + 1)-ary relatives, are of the same nature . F rom this I shall a rgue that the Algebra of Logic may be t reated m o r e symmetr ical ly by start ing with proposi t ions ra ther than with classes, since proposi t ions are 0-ary relatives, and hence form the lower end of the scale of relatives.


This does not tend in the least, however, to disprove that a perfectly cogent, rigorous, and altogether logically satisfactory t reatment of the Algebra of Logic may be obtained by starting with classes rather than with propositions.

In the next chapter I shall discuss the e-relation and its absence in the t rea tment of the Algebra of Logic given by Schroeder. I shall show that the statement made by Padoa and implied by Russell to the effect that Schroeder confuses the e-relation and the C-relation is totally false, and that the instances which Padoa gives of the alleged fallaciousness of Schroeder 's method of procedure can be dealt with by him with perfect rigor and correctness. I shall also show that Schroeder 's symbolism involves the t reatment of none of the notions which the e - relation is designed to embody, and that, therefore, he nei ther needs nor can express any hierarchy of "types" by his formulae, nor deal with relatives of different types. Therefore, the :(= and = relations as applied to relatives cannot be treated by him as relatives in any formulae together with the relatives which they connect.

I shall also show, however, that Schroeder was fully aware both of the possibility of a theory of types and of an e-relation, and that, as a matter of fact, we must consider him as one of the discoverers of both. Nevertheless, he misunders tood the e-relation of Peano in a most peculiar way.

I believe that I have been able to prove the following theorems, though I have not included them in this thesis ... [A list of theorems has been omit ted here.]

The total result to which I come, then, is this: In so far as the subjects which they treat are identical, Schroeder and

Russell are able, each on his own basis, to give equally accurate and rigorous accounts of them, which may always be translated step for step from the language of Schroeder into that of Russell. In very many cases a perfectly parallel translation may be made in the reverse direction, al though certain of the ideas involved in the formulae of Russell must be paraphrased before they can be expressed in Schroeder 's terminology. The sole essential point of difference between their algebras of relatives lies in the fact that Schroeder conscientiously limits himself within the confines of what Russell calls a single type, and so is forced to do without many of the formulae with which Russell finds himself able to deal.

Before closing this introduct ion I should like to call at tention to the difference between the aims of Schroeder and of Russell respectively, and the differences of methods which this entails. Russell's purpose in the Principia Mathematica is, as the name implies, to give an orderly development of the whole of mathematics from a few simple logical postulates. Accordingly, the theorems are arranged in a clear-cut, def-


inite order, with the one purpose of arriving as swiftly as possible at the ordinary theorems of Arithmetic and Algebra. Whenever a group of symbols keeps constantly recurring, Russell, in order to prevent his formulae from becoming unwieldy, promptly proceeds in a quite justifiable manner to abbreviate them into a single symbol by means of a definition. As a result, an appearance of great simplicity and obviousness is given to many of his formulae, which is, however, largely specious.

Schroeder, on the other hand, as the name of his book similarly implies, is interested in developing a depar tment of Mathematics which shall represent in a symbolic fashion certain of the facts of ordinary logic, and which shall discover new facts of the same general nature. It is only of incidental concern to him what particular applications his work may have in other fields of Mathematics. Therefore, he strikes out in many directions at once, and the various chapters of his work do not, in general, each form a definite advance on the preceding one, but rather each one opens up some new field of work which is often hardly more than indicated. Since the main purpose of Schroeder 's work is to show the manifoldness and the significance of the conclusions which follow from his postulates, he tries to retain his initial symbolism in all his formulae, wherever possible, and so is little inclined to be as lavish as Russell in making new definitions. Because Schroeder 's purpose is analogous to that of ordinary Algebra, in that he wishes to develop a self-contained science, his technic [sic] and symbolism become assimi- lated thereto; the equation becomes of great interest to him, as do also the so-called inversion-problems and elimination-problems. These latter were probably suggested to him both by the solution-problems of ordinary Algebra and the elimination-problems of the Logic of Classes, namely, the syllogisms. The problem of the nature of the relatives obtained by certain transformations interests him, much as the nature of the terms satisfying a given equation interests the algebraist. As one consequence of this, he invents his "fuenfziffriges Rechnen. ''~~ While it is perhaps true that in certain points Schroeder's t reatment has been more largely influence by the analogy of ordinary Algebra than has suited the nature of his problems, I cannot but regard as unjustifiable the statement of Russell ]] to the effect that "His [Peano's] merit consists not so much in his definite logical discoveries, nor in the detail of his notations (excellent as both are) as in the fact that he first showed how symbolic logic was to be freed from its undue obsession with the forms of ordinary algebra, and thereby made it a suitable instrument for research." I shall proceed in what follows to refute, to a large degree, this imputation against Schroeder, and to show that within the algebra of

Io A. der L., III, p. 209. ii Principia Mathematica, p. viii.


relatives, at any rate, Schroeder is not only able to treat of everything with which Russell deals, but to treat of it in a way which is in many cases at least as simple, if not simpler. I shall leave it for the reader to j u d g e how much Schroeder ' s "obsession with the forms of ordinary algebra" has h a m p e r e d him.

Chapter 4 of Norbert Wiener's Ph.D. Thesis (1913): On the ~-Relation and Allied Notions in the Pr/nc/p/a Mathematica

It has been repeatedly asserted in the course of this thesis that Schr6der does not treat of individuals, in Russell's sense, xRy canno t be translated into the language of Schr6der unless x a n d y are both appa ren t variables, or else unless we know some proposi t ion such as i = t ' x . j = t'y. Tha t is, we can translate (~ 'i)R(t'~/'), but not merely xRy. Similarly, we saw, we may find Schr6der ' s equivalent for (~'i) ~ o~, but not merely for x c~. The z-relat ion is untranslatable into the language of Schr6der simply because it has no coun te rpar t in the actual system of Schr6der. The quota t ion from the Algebra tier Logik which we gave in the beg inn ing of chap te r II showed us that the ' Individuen ' of Schr6der are genu ine unit- classes, quite equivalent to the unit-classes of Russell, and are connec t ed with o the r classes by the relation of subsumpt ion ra ther than the ~- relation. Not a single symbol in the Algebra tier Logik which possesses a m e a n i n g in isolation from all o the r symbols ~2 is a symbol of anything but a proposi t ion, a class, or a relative; and Schr6der explicitly refuses to deal with classes involving the existence of o the r classes, ~:~ as such, and, as a consequence , with classes of relatives or of proposi t ions, for he treats of relatives as a part of classes, and all his proposi t ions concern themselves with classes or with relatives. He says, TM "Und damit auch in der ursprf inglichen Mannigfaltigkeit dis Subsumtion (2+) aufrecht ero hal ten werden k6nne, ist von vornhere in erforder l ich (und hinrei- chend) , dass unter ihren als "Individuen" gegegebenen Elementen sich keine Klassen befinden, welche ihrerseits Elemente derselben Mannigfaltigkeit als In- dividuen unter sich begreifen."

This clearly excludes any formulae in which classes of classes or relatives occur together with the classes or relatives which they classify, connec t ed with them by any of the operat ions and copulae of the calculi of classes and of relatives. As for classes of proposit ions, it should be r e m e m b e r e d that there are, from the s tandpoin t of the calculus of

J~ i.e., all symbols not representing operat ions or copulae. Is There is an important exception to this in I, w 23, p. 481-3. I shall treat of this later. ~4 A. der L., I, p. 248. All of w 9, 'Pure Manifolds', in which this passage occurs, is invaluable

for a p roper unders tanding of Russell's theory of types, which is simply a device to insure that his manifolds shall be 'pure ' : i.e., shall have the existence of no part condi t ioned by another.


proposi t ions , only two proposi t ions, i and (). Treat ing i a n d () as unit- classes, then, there are only four classes of proposi t ions: 0, 1, 0, 1, where 1 contains both 1 and 0. Since 0 =(= 1, we might write these as 0, (0 =(= 1), (1 :~-0), 1. Since, in our system, (0=(= 1)(1 : ~ 0 ) , 1 =0 . This makes our system dwindle down to 0 alone, and shows its absurdity. ~ Such a class would not be a pure manifold, and could not be r e p r e s e n t e d by 1, for its e l ements canno t coexist. To quote Schr6der ' s words, ~6 "Als eine erste A n f o r d e r u n g haben wir schon in w 7 un t e r Postulat ((1+)) die n a m h a f t gemacht : dass die E lemen te der Mannigfal t igkei t sS.mtlich ver- einbar, m i t e i n a n d e r "vertr~iglich" sein mhssen. Nur in diesem Falle bezel chnen wir die Mannifaltigkeit mit 1." This excludes any possibility of Schr6der ' s t rea t ing of classes of proposi t ions as such, and comple tes the p roo f that he canno t treat of a class toge ther with its member s , as he virtually admits.

Sch r6de r actually never does unreservedly treat of a class toge the r with its m e m b e r s in the same formula. As the e - re la t ion is the relat ion between a te rm and the class to which it belongs, he makes no genera l use of it no r of any equivalent symbol, nor need he do so. It is an utterly inexcusable mistake for Padoa to say ~7 "I1 est donc 5. p ropos de r appe le r ici que M. S c h r 6 d e r - - q u i s 'etait mis fi l 'oeuvre avant M. Peano et qui en 1877 avait d6jfi publi6 son Operationskreis des Logikkalkuls---n'a pas r6ussi/ l nous laisser une ideograph ie logique satisfaisante; et cela, prin- c ipa lement , parce qu'i l n ' a pas dist inqu6 les appartenances des inclusions et par suite il les repr6senta par un seul symbol. Et m 6 m e e n s u i t e - - d a n s ses trois gros et lourds volumes sur l'Algebra der Logik, d o n t le p r e m i e r suivait d6j~t les Arithmetices principia de M. Peano---il ne voulut pas re- conna i t re la n6cessit6 de cette distinction.

"Mais je crois que deux exemples suffr i ront / t vous 6claircir la diff~rente signification des deux symbols " e " et "D",ls que d 'a i l leurs les logiciens scholast iques d is t inquaien t en "sensus compositf' et "sensus divisi," touffois sans d o n n e r ~ cette dist inction l ' impor t ance que j u s t e m e n t lui d o n n a M. Peano.

"Voici les deux exemples: d ' u n c6t6 vous avez les inclusions

genevois D suisse suisee D e u r o p 6 e n

desquel les on tire

genevois D europ6en ,

et d ' u n aut re c6t6 vous avec les appartenances

Pierre ~ ap6tres ap6tres e douza ine

15 Cf. A. der L., I, w 9, p. 245. '" A. cLer L., I, w 9, p. 247. 17 Rev. de Met. et de Mor., 19, 1911, pp. 852, 853. i~ Padoa, following Peano, uses D for both implication and subsumption.

44 ~ NORBERT WIE NE R ' S THESIS

( q u ' o n peu t lire "Pierre fut un des ap6tres" et "les ap6tres 6 ta ient une

douza ine" ) desquel les on ne peu t par t irer

Pierre ~ douza ine ."

I can say definitely that I know of no case where the =(= of S c h r 6 d e r

definitively represen ts any th ing but the D of Peano, or its equivalents ,

D, C , and C in the system of Russell. W h e n i:~--a represen ts x ~ a, we

have, as we have shown repeatedly, i - L'x, no t i = x. It is perfect ly t rue

that S c h r 6 d e r would translate "Peter ~ apostles" into his own termi-

no logy as "Peter =(= apostles", but Schr6de r ' s 'Peter ' would be the class

o f m e n of which Padoa ' s 'Pe ter ' is the sole member . If, however,

S c h r 6 d e r should translate "Peter ~ apostles" in this manne r , he would

no t r e n d e r "apostles ~ dozen" as "apostles :(= dozen," for this would

involve a fo rmula dea l ing with a class t oge t he r with a n o t h e r class o f

which the first is a member , which, by the pr inciples laid down in A.

der L., I, w 9, he is unab le to discuss. He would translate "Apostles

dozen" as "num. apostles = 12"; that is to say, "the n u m b e r of the class,

apostles, is 12." Now, to say that the n u m b e r of a class is equal to a given finite integer,

n m t h a t is, that the class has n m e m b e r s - - , is a s t a t emen t which may

be m a d e entirely in the l anguage of the calculus o f classes. I:' As

n u m . a = 12, however, would be qui te a lengthy express ion to e xpa nd ,

let us translate into the l anguage of S c h r 6 d e r a n o t h e r pair of express ions

qui te ana logous to those of Padoa, but having ' coup le ' in place of

' dozen ' .

Let us suppose , then, that we have the two z -propos i t ions , '[John is

one of the Jones twins", and "the Jones twins are a couple" , f rom which we c a n n o t infer that J o h n is a couple . Let x s tand for J o h n , ol for the class consis t ing of the J o n e s twins, and 2 for the class o f all couples . Russell would write the two propos i t ions as x ~ ol and c~ ~ 2, respectively, while Sch r6de r would write t hem as i=(=ol and num. ot - 2, i s t and ing for the t'x of Russell. Now, num. ol = 2 is de f ined as TM

J"Jy (x ~ y ) (a = x + y) . x, y

This gives us:

~" A. der L., II, 1, pp. 348, 349. See also Monist, IX, 1898-9, On Pasigraphy, pp. 54, 55. This latter article contains many valuable suggestions concern ing number , series, etc., which, as far as I know, have never been followed out fur ther by Schr6der.

~~ der I.., II, 1, p. 349.


(Num. c~ - 2). =. E J X J Y ( x e: y)(o~ = x + y ) x,y

=" ( 3 t ~ , v ) " ( 3 u , v ) . g = L ' u . v = t ' v " Ix ~ v . c~ = g u v

=" (3g, t,) �9 (3u, v)./x = t ' u . 1,, = t ' v . u ~ v . o~ = t ' u u t ' v

~. (3u, v). u :r v. o~ = t ' u t J t ' v

~ . 0 / e 2.

This is precisely Russell 's fo rm of s ta tement , and any objec t ion against

Sch r6de r ' s posi t ion here would tell equally against Russell or Peano.

Russell seems to be guilty of the same e r ro r which we have jus t ex-

h ib i ted in the case of Padoa, a l t hough he is not qui te so explici t as to the object of his attacks. He says, 21 "Before Peano and Frege, the re la t ion

of m e m b e r s h i p ( e ) was r ega rded as mere ly a par t icu la r case of the

re la t ion of inclusion ( C ) . For this reason, the t radi t ional formal logic

t rea ted such propos i t ions as "Socrates is a man" as ins tances of the universal affirmative A, "All S is P," which is what we express by "c~ C 13." This involved a confus ion of fundamen ta l ly d i f fe ren t kinds of prop-

ositions, which greatly h i n d e r e d the d e v e l o p m e n t and usefulness of

symbolic logic."

Since S c h r 6 d e r does no t deal with the e - re la t ion , all his classes are

of o n e ' type' , for Russell 's types are re la ted to one a n o t h e r by the e- relat ion, or some rela t ion der ived f rom the e- re la t ion . 22 T h e same holds

t rue of relatives. 2:~ All the express ions of Russell which involve classes

and relatives of d i f fe ren t types are wi thout d i rec t equivalents in the

system of Schr6der , for if an express ion involves classes and relatives of d i f fe ren t types, it ul t imately involves a given class, and a class of classes of which the given class is a member .

Schr6der , then , is unab le to express objects of d i f fe ren t types by his

formulae . W h e n he wishes to make a s ta tement , for example , c o n c e r n i n g the fact that a cer ta in class is a couple , he must, as we have seen, express this as the af f i rmat ion of a cer tain formal p rope r ty of this class e n t i r e l y

w i t h i n the c a l c u l u s o f classes. Yet, for all this, the re is a cer ta in i m p o r t a n t

analogy be tween the p r o c e d u r e s of Sch r6de r and of Russell in, for ex-

ample , thei r discussions of the no t ion of a couple , or in gene ra l of a

number . It will be r e m e m b e r e d that Russell 's oL e 2 was shown to be

equiva len t to Schr6de r ' s num. c~ = 2. Russell 's 2 is a class of classes. As

a class, it is an i n c o m p l e t e symbolZ4--it has no m e a n i n g excep t in a

p ropos i t ion such as o~ e 2, which stands as an abbrevia t ion for a cer ta in

~ Cf..63.02-.051, 103.5.51.52. ~2 Cf..63.02-.051, 103.5.51.52. u:~ Cf..64.01-041.3-34. ~1 Principia Mathematica, pp. 75-84, inclusive.

44 2 NORBERT WIENER'S THESIS

proposi t ional funct ion of ~. Similarly, the 2 in num. ot = 2 is an incomplete symbol, and has no m e a n i n g except in some express ion such as num. c~ = 2. Now, jus t as Russell defines u , r C, 2~ etc., by means of combina t ions of the proposi t ional funct ions d e t e r m i n i n g the classes which they connect , in terms of the opera t ions and copulae of the propos i t iona l calculus, so Schr6der defines +, x , =,26 etc., be tween

n u m b e r s by means of a proposi t ional funct ion of classes of the given n u m b e r of terms, expressed by the opera t ions and copulae of the calculi

of classes, proposi t ions, and relatives. T h e r e is a certain paral lel ism be tween

and Schr6der ' s definit ion: 27

Def. ( .22.01)

(num. a = num. b)

= ( ~ ; a =/~" b)((~; 0'" a = b '0 '" b){~" 0'(0'" aO')'a =/~" 0' (0 ' �9 b0') "b} , , , , , , , , , �9

In o t h e r words, there is an analogy, which seems to me m o r e than trivial,

be tween Schr6der ' s step f rom a to num. a, and Russell 's step f rom in-

dividual to class, or, in general , f rom type to type. A poin t which seems to me worthy of at least passing not ice is that

Russell can t reat =, C, and C as relatives, while their equivalents, = and =(=, c anno t be relatives for Schr6der. The reason for this is c l e a r m t h e y

are of a type h igher than the classes or relatives which they relate. Suppose that R C S. Then , by .64.201, t 'R = t'S, or R and S are of the same type. By .55.3, R C S may be read as (R ~ S ) C ( C ) , where the first C is one type h igher than the second. By .64.31 and .63 .16, t ' (R $ S) = t ' ( t 'R $ t 'S) = t ' ( t 'R "~ t 'R) . By this we get, by means of .64 .11 , t ' (R ,[ S) = t "R . Now .64.201 tells us that R C S . D . t 'R = t'S, so that, since (R $ S ) C ( C ) , the type of the second C is that of R ~ S, so that

we get, t ' ( C ) = t"'R, where t " R stands for a cer tain type above that of

P~ There fo re , :(= canno t be t rea ted as a relative by Schr6der.

Similarly, he canno t treat = as a relative. (i =j ) = 1' 0 seems to be an !

excep t ion to this, but it really is not, for 1,)does not co r r e spond to the

iIj of Russell, but to the xIy, where i = i'x and j = i'y, a l though it is t rue

in this case that F x I y . - , iIj, by .51.23, .50.1. The genera l equali ty relat ion, a = b, where a and b are not ' Individuen ' , is not equivalent to

any relative coefficient. We have now show that, in general , Schr6der ' s symbolism has no

place for objects of d i f ferent types, and have exhibi ted the consequences

2~ , 22.01-03. '~" On Pasigraphy, Monist, IX, 1898-9, p. 54. ~7 Ibid.


which follow this perfectly justif iable l imitation. It would be utterly un- just , however, to claim that Schr6der was unaware of the possibility of fo rming a system involving a h ierarchy of types. He discusses at some length the h ierarchy of classes, zs classes of classes, etc., and is very careful to keep these various grades of classes sharply separa te f rom one another, d is t inguishing them as the "unspr / ingl iche Mannigfal t igkei t ," the

"erste abgele i te te Mannigfalt igkeit ," the "zweite abge le i te te Mannigfal- tigkeit,"etc. These notions, t hough they have an i m p o r t a n t place in his thought , have no place, in general , in his symbolism.

T h e r e is, as I have said in a note, one highly significant passage in the Algebra der Logik, where Schr6der breaks his genera l rule, and tries

to r ep re sen t objects of d i f ferent types th rough his symbolism. ~~ He wishes to express the fact that the 'quot ien t ' of two classes may have many values by the formula, x=~--a :: b, where x is a par t icu lar value of

the 'quot ien t ' , and a :: b is the class of all values of the 'quo t ien t ' . To give his own words, "Auch diese Subsumt ionsze ichen [the =(= in x4= a + b :~0 or x =(=a] :: b w/iren aber als solche der abgeleiteten Mannigfal t igkei t

zu in te rp re t i e ren , und nicht als solche der ursprf ingl ichen. Die Sub- sumt ion besagte hier nicht, das Gebie t x sei als Teil en tha l t en in e i n e m

rechts a n g e f ~ h r t e n Gebiete , sonde rn nur, es sei als Individuum en tha l t en

in der rechts s t e h e n d e n Klasse von Gebie ten .

"Gerade in j e n e n Grenzf/illen aber, wo die Klasse a + b rechts selbst nur ein Gebie t umfasst, mfisste das Subsumt ionsze ichen Missverstand- nisse nahe legen, i n d e m es E i n o r d n u n g (als Teil) mitzuzulassen scheint ,

wo, wie erw/ihnt, nur Gle ichhei t gel ten kann. Zur V e r m e i d e n solcher

( u n d / i h n l i c h e r schon in w 9 un t e r ~b) charac ter iz i r te r MisstS.nde mfisste man eigenl ich zweierlei Subsumtionszeichen verwenden f/ir die urspriingliche und die abgeleitete Mannigfalt igkeit ."

In o the r words, Schr6der wishes to cons t ruc t a symbol to signify that x is a "Individuum," not a part, of the class a + b. Now, since Schr6de r

has no separa te symbol for x as an ord inary class and x as a unit-class ( the c~ and t'o~ of Russell, respectively) the word, " Individuum," in this passage must have the same m e a n i n g as the word "individual" used by Russell. Since this is so, we see clearly that the new "Subsumtionszeichen" suggested by Schr6der is precisely the ~ of Frege, Peano, and Russell. This no t ion

could hardly have been bor rowed f rom any of the early works of Frege

or Peano, for Schr6de r makes no re fe rence to this po in t in his discussion of Frege 's work, :~ whereas he says that he b e c a m e acqua in t ed with the

writings of Peano too late to make use of t hem in his first volume. :~

A. der L., I, w 9, pp. 247, 248. *' A. der L., I, w 23, p. 482. :~~ + b is the 'difference' between the classes a and b. :~l A. der L., I., pp. 703, 704. ~ A. der L., I, pp. 709, 710.


Schr6der therefore has full claims to be considered as one of the discoverers of the theory of types and the z-relation.

Before we close our discussion of the z-relation, I wish to call the at tent ion of the reader to an extremely curious passage in the Algebra der Logik. In the posthumous second part of the second volume of this work, which Schr6der never finally revised, the following s ta tement occurs: :~:~ "Statt unseres Subsumtionszeichen, wendet Her r Peano---was sicher kein Vorzug is tBmeis t zweierlei Zeichen an, n~mlich zwischen Klassen ein ~ als den Anfangsbuchstaben von 'eorL, "ist," zwischen Aus- sagen dagagen ein umgekehr tes C, also D, was er innern soll an con- cluditur (6 contenutao) . Ich glaube unwiderleglich dargethan zu haben, dass ein besonderes Zeichen ffir die letztere Beziehung entbehrl ich, m.a.W, dass die Kopula der kategorischen Urteile auch ffir die hypoth- estischen verwendbar ist ... BAl lerd ings bin ich auch fiberzeugt, und habe es schon gelegentlich (Bd. 1, S. 482) ausgesprochen, dass man unter Umstanden noch einer besonderen zweiten Art von Subsumtion- szeichen bedarf, nfimlichen neben einem solchen ffir die ursprfingliche noch eines anderen ffir die abgeleitete Mannigfaltigkeit. So kann, wenn wie gew6hnlich, a~--b die E inordnung eines Gebeites a in ein Gebiet b ausdrfickt, der Satz dass da Gebiet a zur Klasse J d e r Individuen geh6re, oder dass a ein Punkt sie, nicht zugliech durch a:rc--J dargestellt wero d e n . m D o c h ist hiervon bei Peano nicht die Rede."

In o ther words, Schr6der mistakes the z-relation of Peano for the general relation of inclusion between classes, and misunders tands D, conceiving it to apply only to propositions. He then criticizes Peano for distinguishing these two signs, at the same time reiterating a suggestion which amounts to the construction of an z-relation~ Of this suggestion, however, he says, there is no question in Peano. It is peculiar how the excellence of Schr6der 's reasoning shows itself even in his blunders.

p. 461.

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Index

Absolute terms, 24, 26-27, 32, 34, 35-38

Ackermann, Wilhelm, 196; model of, 257. See Hilbert, David: Orundziige de, theoretischen Logik

Addition, 15, 25, 100, 116; and disjunction, 15-16, 29-30, 52; arithmetical, 100-102, logical, 15-16, 25, 29-30, 52, 91, 100-101, 150; relative, 92, 1{)2-103, 150

Adjointness condition, 61-62 Algebra: associative, 25, 95-98; Bool-

ean, 14-22, 116-121, 223; of the copula, 52, 64-73; of propositions, 52, 64-73, 116-126; of relations, 11-14, 23-49, 74, 98-112, 143-165, 258, comparison of Schr6der's and Russell's treatments of, 165-167, as matrix theory, 98-110; Schr6der's exposition of the algebra of binary relatives, 210-427. See also Boolean algebra; Calculus of relations

Algebraic closure, 194 Antisymmetry, 27 Aristotelian propositional forms, 63,

80 Aristotelian syllogisms, 21, 46-47, 51,

66-70, 104, 110, 149 Aristotle. 21, 110 Arithmetic, theory of: Dedekind's, 14,

157-158. 257: Frege's, 156-158, 160; G6dei's, 158; Grassn~ann's, 14, 155-156; Peano's, 14, 157-158:

Peirce's, 10-11, 13--14, 159; Schr6der's, 158

Associativity, 32 Aufl6sung, 5, 7, 152-155, 195,257-258 Automated reasoning, 1, 85, 131 Axioms: for equality, 133; for lattices,

101, 144-148; for natural numbers, 11, 13-14, 101; for propositional logic, 123-124; for set theory, 133-138: coxnplement, 136; extensionality, 133; infinity, 295, replacement, 135; pairing, 138; successor, 136; union, 136

Bernays, Paul, 195 Binary relatives: Schr6der's exposi-

tion of the algebra of, 210-427. See also Relatives

Binomial theorem for logic, 39-40 Birkhoff, Garrett, 120, 145, 146 Bolzano-Weierstrass theorem, 4, 12,

191-192, 204 Boole, George, 5, 9, 14, 91, 104, 110,

112; and existentials, 46-48, 62-64, 91, 115, 117; influence on Peirce, 18-20; and particular propositions, 1 6-17, 46-48, 62, 91; and probability, 15n, 18-19

Boolean algebra. 9-10. 14-20. 144-149. 207-208; axiomatic definition of, 146; lattice-theoretic treatment of, 9, 10, 52-53; Mitch- ell's quantification for, 77-94; in

461

462

Peirce's algebra of logic, 107-108, 116; Peirce's improvement on, 14-17; of truth functions, 125, 147-149

Boolean operations, 15-17, 80-81, 100

Boolean polynomials, 79-82, 147 Bracket operator, 27 Burch, Robert W., 34n

Calculus of relatives (relations), 4-5, 9, 11-14, 23-49, 98--106, 111-112, 207; axiomatization of, 165; expressiveness of, 4, 46-49, 166-172; as a foundation for logic and mathematics, 4-5, 12; logical interpretation of, 26-46; scope of, 23-24; syntax vs. semantics in, 25. See also

Algebra of relatives (relations) Cantor, Georg, 8, 28, 99n, 133, 140,

141, 157; set theory of, 157 Cardinal equivalence, 379; Schr6der's

exposition of the theory of, 38O-427

Cardinality, 28, 137-138 Carnap, Rudolf, 163 Cayley, Arthur, 5, 11, 95 Chain theory, 12, 155-160, 295-296;

Schr6der's exposition of Dede- kind's, 297-338

Characteristic function, 223, 251 Choice, axiom of, 3, 4, 14, 191, 198,

203 Choice function, 199 Church, Alonzo, 6, 17, 251 Closure argument, 194, 199-200, 203 Coefficient evidence, 150-152 Comma operator, 35-38 Compactness theorem, 202 Complementation, 16, 25, 103 Completeness theorem, 2, 3, 163,

203-204 Conjugative terms, 24, 26-27, 32-35,

42-44 Consistency, 203. 204 Contradiction, 198 Converse, 25, 53, 103, 150 Copi (Copilowish), Irving M., 20n

Cup quantifier, 62-64, 69-70 Curry, Haskell, 60 Cylindric algebras, 48, 207

INDEX

Decision method, 17, 85, 89, 195,207 Dedekind, Richard, 6, 10, 11, 17, 27,

53, 133, 141, 145-146, 149; chain theory of, 155-160, 257; and the development of set theory, 157; finite and infinite collections in, 14, 140-141, 157; inductive definitions of, 14, 157, 160; lattice theory of, 53, 145

Deduction theorem, 57, 61, 64-69 De Morgan, Augustus, 5, 9, 60, 81,

104, 112, 138-139; and the calculus of relations, 21; influence on Peirce, 21; law of, 103, 118,207; and the syllogism of transposed quantity, 138-140

Disjunctive normal form, 80 Distributivity, 31-32, 107, 146, 191,

207 Distributive law, 31-32, 103, 107-108,

151, 176-177, 191, 193, 195, 207, 339

Dreben, Burton, In Duality, 16, 145

Elementary relatives, 44-46, 96 Elimination of quantifiers, 41-42,

163-164 Elimination problem, 147, 257-258 Elimination theory, 5, 152, 257-258;

Schr6der's exposition of, 259-294 Ellis, Leslie, 91 Equality, 27-29, 133-134; derivation

rules for, 118; inference rules for, 207

Equational theory: of Boolean algebra, 114-119; of calculus of relations, 207

Existential graphs, 10, 13 Exponential, 20. 39-44. 46-47, 101"

quantification in, 40-44 Expressiveness, 4, 16-17, 111-112,

127, 133, 166-172; of the calculus of relatives, 4, 169-172, 194


Extensionality, 26-28, 31, 32

Finite collections, 14, 138 First intentional logic, 127-132, 170 First-order domains, 339 First-order logic, 1-7, 13, 48-49,

127-132, 151-152, 169-171, 194-195; development of, 1-8; influence of Hilbert on development of, 7; influence of Peirce on development of, 1, 11-14

Fleeing equation, 175, 183, 194 Fraenkel, A. A., 157 Frege, Gottlob, 2, 7, 10, 24, 55, 133;

Begriffsschrift, 6, 91, 104, 156; inductive definitions, 156-158, 160; review of Schr6der's Die Algebra der Lo- gik, 150; universe in, 60, 104

Freyd, Peter, 12 Function: propositional, 106; truth,

116, 125, 147-148

General relatives, 98-99 General solution to relative equation

(Aufl&ung), 5, 7, 152-155, 195, 257-258; Peirce's criticism of, 153-154; as precursor of Skolem function, 160-163, 258

Gentzen, Gerhard, 58 Girard, Yves, 10, 52, 158 G6del, Kurt, 1-4, 17, 163, 173,

203-204; conlpleteness theorem, 191, 203-204; L6wenheim's influence on, 2-3; Skolem's influence on, 3, 203; theory of arithmetic of, 158

Grassmann, Hermann, 11, 101, 157; theory of arithmetic of, 14, 155-156

Grassmann, Robert, 91 Greatest lower bound, 52, 106-108,

111, 144, 223, 251; of Boolean algebra of truth functions, 148-149, 2O8, 223

Habit, 54-55 Hailperin, Theodore, 17n tlalmos, Paul, 48 Hanf number, 192-193

463

Heine, Eduard, 157 Herbrand, Jacques, 1, 4, 163, 204 Herbrand universe, 204 Hilbert, David, In, 2, 7, 13, 133, 195 ~Grundziige der theoretischen Logik, 1,

2, 13, 63, 89, 173, 195-196

Icons, 121-125, 128, 135-138 Identity calculus, 207 Identity product and sum, 145, 207 Illation, 6, 56, 121-122, 124 Implication, 6, 10, 51-53, 56-62,207;

introduction and elimination rules for, 6, 57-58, 65, 123-124; negation of, 60-64, 69-70, 73

Implicative propositional logic, 10, 51-52, 52-53, system of, 64-73

Inclusion, 27-29, 91, 145, 167-168 Index notation, in Mitchell, 73-74; in

Peirce 20, 34-35, 79, 92, 132 Individual relatives, 53, 98 Individual variables, 73-74, 79, 92,

111 Individuals, 44-45, 100, 133-136 Induction (mathematical), 140,

155-160 Inductive definitions of addition and

multiplication, 156-160, 295 Infdrence, 52, 56; rules of, 56-59,

75-77, 107-108, 145; De Morgan's syllogism of transposed quantity, 138-140

Infinitary propositional logic, 107-108; 111, 127-128, 149, 172, 192-193, 195, 202-203

Infinite collections, 138 Intensionality, 26-27 Involution, 25, 39-44

Jevons, w. Stanley, 9, 91, 147

Kleene, Stephen C., 156, 163, 173-174 K6nig infinity lemma. 4. 172.

191-192, 200, 203 Korselt, Alwin, 49, counterexample to

Schr6der, 170, 171, 193-194, 251 Kronecker, 157, 194, 257

464

Ladd-Franklin, Christine, 90-91, 149; notation ot, 91

Landau, Edmund, 158 Lattice, distributive, 101, 146 Lattice theory, 145-146, 208; Dede-

kind's, 53, 145; Peirce's, 6, 9, 10, 52-53, 119-120; Schr6der's, 53, 121, 145-149

Lawvere, E W., 10n, 20 Least upper bound, 52, 106-108, 111,

144, 223, 251; of Boolean algebra of truth fi~nctions, 148-149, 208, 223

Leibniz, Gottfried Wilhelm, 115, 134 Lewis, C. I., 13; Peirce's influence on,

13 Lipton, James, 13 Linear associative algebra: of B.

Peirce, 23, 95; in Peirce's logic of relations, 99

l,ogic, first-order, 1-13, 127-132, 151-152, 169-174, 194-195; modal, 13; origin of, 54-55; second-order, 1, 13, 103, 133-140, 173-174, 192-194; use of algebraic notation in, 25-26, 49-50

Logical terms: absolute, relative, conjugative, 24, 26

L6wenheim, Leopold, 1-9, 141, 159, 251; on the elimination problem, 258; normal form reduction of, 175-179; "On possibilities in the calculus of relatives," 1, 2, 4, 169-196; Schr6der's influence on, 2, 4-5, 12, 155, 163-164, 168, 169-171, 173, 195, 258, 339-340; Steinitz's influence on, 194-195; theorem on first-order logic and its proof, 172-191; theorems on expressibility, 171-172; tree argument, 180-191, 192-193, 203

L6wenheim-Skolem theorem, 1, 2-5, 12, 13, 143, 173-174, 175-191" 1,6w- enheim's proof of, 175-191, 191-193; Skolem's proot~ of, 3-4, 169, 173-174, 192, 198-203

Lukasiewicz, Jan, 123 Lusin, Nicholas, 163, 174

INDEX

Mac Coll, Hugh, 90-91, 147, 149 MacFarlane, Alexander, 91 Mac Lane, Saunders, 20 Mappings, theory of" one-one, 379;

Schr6der's exposition of the theory of, 380-427

Martin-Lof, Per, 171 Matrix theory, 99-104 Membership relation, 134, 167-168 Metaphysical Club, 75 Mitchell, Oscar Howard, 1-9, 48, 74,

112, 115,126-127, 143, 149; algebra of quantifiers in, 83-84; disjunctive normal form in, 80-84; inference rules of, 75-77, 82-84; negation in, 79-80; "On a new algebra of logic," 75-94; Peirce's intluence on, 75, 87, 88-89, 90-94; Peirce on, 90-94, 128-129; quantifier notation of, 79-84, 86-88; theory of quantification of, 6, 10, 79-94; two-quantifier forms of, 86-87; on universes, 78

Modal logic, 13 Model theory, 1,172; development of,

1-8; origins of, 1, 172 Model, 1, 105-106 Modus ponens, 122, 124, 207 Monk, James D., 251 Morley, Michael, 171 Multiplication, 25, 30-38, 52, 116; ar-

ithmetical, 100-102; and conjunction, 15; logical, 15, 30, 52,100, 150; notation for logical multiplication, 15, 36; relative, 30-32, 96-97, 101-102, 150, ofa conjugative and a relative term, 32-35, of a relative and an absolute term, 35-38; relative product as matrix multiplication, 102

Murphy, J. j., 91

Natural deduction system, 6, 10, 51, 52, 61, 64-73, 124-125. 207

Natural numbers: axiom systems for, 11, 13-14, 101

Negation, 150: in Boolean algebra, 15-16; in implicational logic,


60-61, 67; in propositional logic, 122; in relational algebra, 42, 150

Normal form, 59, 147, 175-176 Notation: for absolute terms, 26-27;

indeterminate v of Boole, 62, 91; for conjugative terms, 26-27; for individuals, 20, 92; for involution, 39; of Ladd-Franklin, 91; logical disjunction, 15-16, 29-30; logical multiplication, 15, 36; of Mitchell, 79-84, 94; for negating class inclusion, 15; for negating relatives, 103; H and E, 92, 106, 128, 147, 150, 160-165, 223, 251; relative product, 4, 30-38; for relative terms, 26-27; relative sum, 4, 92, 95: and symmetry, 95

One-one correspondence, 138, 140-141

Operations, 96-98; of the calculus of binary relatives, 234-235. See also Addition, Converse, Exponentia- tion, Multiplication, Negation, Subtraction

Ordered pairs, 97, 98-100, 149, 208

Padoa, Alessandro, 168, 429 Partial order, 27-29, 52-53, 101, 115,

119-120; axiomatic treatment of, 145; greatest element in, 145; least element in, 145; least upper bound, greatest lower bound in, 145

Particularity, 16-17, 46-48; quantification for, 1 6-17, 46-48, 62

Peano, Giuseppe, 11, 14, 429; axiom system for natural numbers, 11, 157-158

Peirce, Benjamin, 5, 9, 11, 20, 23, 95; linear associative algebra of, 23, 95

Peirce, Charles Sanders, 1-9, Benja- min Peirce's influence on, 5, 11,20, 23, 48; Boole's influence on, 18-21, 23-24, 115-116; De Morgan's influence on, 21; Mitchell's influence on, 94; Schr6der on, 143; Schr6der's Die Algebra der Lo~k re- viewed by, 7, 151-152, 153, 154

mAddition in, 15-16, 91; analogues to

465

arithmetic in, 20, 25-26; analogLles to algebra in, 40, 43, 49; attempts to reconcile Aristotle and Boole in, 17, 21-22, 46-48; Boolean algebra of, 9-10, 11, 14-17; calculus of relatives (relations) of, 5, 9, 11-14, 23-49, 98-106; existential graphs of, 10, 13; exponentiation in, 39-42; first-order logic in, 127-132, 169-170; implicative propositional logic of, 6, 9-10, 51-53, with negation, 64-73; lattice-theoretic Boolean algebra of, 6, 10, 52-53; linear associative algebra in, 95-98, 99; mathematical systems of, 9; multiplication in, 15, 30-37, 39-44; quantification in, 5-6, 16-17, 46-48, 62-64, 73-74, 92-94, 104; quantifier logic of, 6, 10, 106-111, 127-132; relational operations, 30-42; relative product in, 30-38, 101-102; second-order logic in, 133-142; syntax vs. semantics ill, 25, 109-110; theory of arithmetic of, 9, 10, 13-14; universe in, 60, 104

--Writings of Peirce: "Associative algebras," 95-98 "Description of a notation for the

logic of relatives," 5, 23-49 "Exact logic," 151-152, 153, 154 "On an improvement in Boole's cal-

culus of logic," 1 4-17 "On the algebra oflogic," 10, 51-74 "On tile algebra of logic: A contri-

bution to the philosophy of notation," 92, 11 3-142

"On the logic of number," 11, 138 "The logic of relatives," 92, 95-112

Peirce's law, 123 Polyadic algebras, 207 Power set axiom, 105 Prawitz, Dag, 6, 10, 53, 125 Prenex form, 6, 10, 93 Prenex predicate logic, 106-109.113.

127, 129-132 Primitive recursion, definitions for,

295-296 Prior, Arthur N., 123-125

466

Probability: relation to logic in Boole, 18--19

Product symbol (l-I) 92, 106, 128, 147, 150, 160-165, 223, 251

Proof theory, 24 Propositional function, 6, 106, 117,

207 Propositional logic, 6, 107, 147-149,

151 Propositions, logical values for,

116--117; and negation, 122

Quantification, 5-6; algebraic properties of, 41-42; in Boole's algebra of logic, 46, 62, 91;in the calculus of relations, 5-6, 31-34, 38--39, 41-44, 46-48, 104-110; first-order, 127-132, 163; as least upper bounds, greatest lower bounds, 106-108, 111, 144, 150, 223, 251; Mitchell's theory of, 6, 79-90; Peirce's early attempts to express "some," 16-17, 46-48; second-orde~, 104, 133-142, 160-164, 339

Quantified propositional functions, 6, 106, 127

Quantified propositional logic, 9, 10 Quantifiers, 112, 132; algebraic treat-

ment of, 143-144, 151-152, 207; elimination of, 163-165; in first- and higher order domains, 14, 144, 160-165, 169; Mitchell's, 48, 79-80, 86-88; in Peirce 1870, 38-39, 42-43; in Peirce 1880, 62-64, 73-74; in Peirce 1883, 92, 95, 106-108, 110-111; in Peiroe 1885, 127-140; rules for, 83-84, 88, 94, 107-108, 131 ; Schr6der's, 5, 148-149, 160-165, 223, 251; Schr6der's exposition of quantifier rules, 340-377

Recursion: definition by, 155-160, 295-296

Reflexivity, 27-28, 145; Relational programming, 13 Relative operations: addition, 92,

102-103; product, 20, 25, 30-38,

INDEX

48-49, 101-103; Boolean product as, 36-38; quantification in, 31, 48

Relative terms, 24, 26-27, 30-35 Relatives, binary, in Schr6der,

218-291; dual, 98; individual, 98; elementary, 44-46, 96; general, 98--99; triple, 87, 89n, 108. See also Binary relative

Robinson, J. Alan, 1, 76 Rules of inference: Mitchell's, 75-77,

82-84; Peirce's, 56--59, for prenex predicate logic, 130

Russell, Bertrand, ln, 2, 6-7, 12, 13, 133, 156, 165-168, 169; in Wiener, 429; logic of relations, 12, 165-168; on Peirce, 7n, 12; on Schr6der, 7, 12, 165. See also Whitehead, Alfred North: Principia Mathematica

Scedrov, Andre, 12 Schr6der, Ernst, 1-9; Aufldsung, 5,

144; abstract Boolean algebra ill, 120-121, 146--150, 208, 223; calculus of relations in, 4-5, 143-144, 149-165, 207-208; Dedekind chain theory in, 141, 155-160; method of elimination of quantifiers in, 163-165, 169; foundation of mathematics in relational calculus in, 144, 159; first-order theory of relations in, 170; identity calculus in, 207; on implication, 145; on inclusion (subsumption), 145; lattice theory in, 5, 12, 144, 145-149, 208; partial order in, 120, 144, 145; Peirce's influence on, 2, 5, 11-12, 95, 143, 145; second-order theory of relations in, 5, 104; set theory in, 207-208; E and 1-I in, 147, 150, 160-165, 223, 251, rules for 339; Russell on, 165, 429; types in, 168

--Writings of Schr6der: Die Algebra der Log~k, vol. 1, 5,

144-147 Die Algebra der Logik, vol. 2, 5,

147-149 Die Algebra der Logik, vol. 3, 149-165;

Schr6der's Lecture I (Appendix


1), 207-221; Schr6der's Lecture II (Appendix 2), 223-249; Schr6der's Lecture III (Appen- dix 3), 251-256: Schr6der's Lec- ture V (Appendix 4), 257-294; Schr6der's Lecture IX (Appen- dix 5), 295-338; Schr6der's Lec- ture XI (Appendix 6), 339-377; Schr6der's Lecture XII (Appen- dix 7), 379-427

Second intensional logic, 133-142 Second-order domains, 339 Sequent calculus, 6, 10, 58 Set theory: of Cantor, 157; relativity of,

198-199, 204-205 Sets: finite, 14, 133-138, 140-141,157;

infinite, 138; in Schr6der, 207-208 Shelah, Saharon, 171 Skolem, Thoralf, 1-9; axiomatization

of set theory, 157; "Logico-combinatorial investigations in the satisfiability or provability of mathematical propositions," 197-199; L6wenheim's influence on, 3-4, 12, 171, 173-174; normal form theorem, 197,200; paradox of, 198-199; proofs of the L6wenheim-Skolem theorem, 3-4, 169, 173-174, 192, 198-203; Schr6der's influence on, 2, 163, 169, 197; "Some remarks on axiomatized set theory," 1, 198; source for Hilbert and Acker- mann's statement of the L6wen- heim-Skolem theorem, 195-196

Skolem function, 1, 2, 5, 7, 12, 153-155, 161-164, 168, 192-194; 195, 258

Souslin, Mikhail, 174 Steinitz, Ernst, 194-195 Stone, Marshall, 77, 120 Subsumption, 207 Subtraction: logical, 15-16, 29 Summation symbol (E) 992, 106, 128,

147, 150, 160-165, 223, 239, 251 Syllogism of transposed quantity,

138-140 Syllogisms: Aristotle's, 63-70; Mitch-

ell's representation of, 78-85;

467

Peirce's algebraic theory of, 51, 66-70, 104, 117

Sylvester, J. j., 11, 23, 95 Symmetry, 27-28 Syntax and semantics: 25-26; of pred-

icate logic, 109-110, 163

Tarski, Alfred, 5, 7, 37-38, 48, 95, 121, 163, 174, 251, 258; and Linden- baum algebras, 121; and relation algebras, 142; Schr6der's influence on, 5, 12-13, 159; semantics ofquan- tifiers for first-order logic, 111

Theory of quantification: Mitchell's, in 1883, 79-94; Peirce's, in 1870, 46-48; in 1880, 52, 73-74; in 1883, 101-102, 103, 110; Schr6der's, 147, 150, 160-165, 223, 251

Token, 133 Transitive closure of a binary relation,

145, 295 Transitivity, 27-28, 122, 145 Transposed quantity, syllogism of, De

Morgan's, 138-140 Tree argument, 180-191, 192-193,

203 Triple relatives, 87, 89n, 108-109 Tripod symbol (-<), 24; as implica-

tion, 51-73; as inclusion, 24, 27-28; as partial order, 101

Truth functions, I 16, 125; Boolean algebra of, 147-148; Truth value analysis, 125-127

Truth table, 125 Truth value analysis, 125-127 Type, 429

Universe of discourse, 60, 104, 212 Universe of relation, 78

van Heijenoort, Jean, l n, 3 Vaught, Robert, 172 Variables, 93, in first-order logic, 111 Venn, John, 9, 147 yon Neumann, John, 158

Wajsberg, Mordchaj, 123 Wang, Hao, In, 3, 172, 193

468

Weierstrass. Karl, 4, 157 Whitehead, Alfred North, 13 --Principia Mathematica, 2, 13,

165-168 Wiener, Norbert, 7, 429, comparison

of Schr6der's and Russell's treatments of relations, 7, 12, 165-168

INDEX

--Ph.D. thesis (Appendix 8), 430-444 Wigner, Eugene, 113 Wittgenstein, Ludwig, 58

Zariski open set, 257 Zermelo, Ernst, 133, 157 Zero, 30, 34, 46-47, 157 Zilber, B. I., 171

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From Peirce to Skolem - A Neglected Chapter in the History of Logic