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

FROM PEIRCE TO S KOLEM A NEGLECTED CHAPTER IN THE HISTORY OF LOGIC

STUDIES IN THE

HISTORY AND PHILOSOPHY OF MATHEMATICS

Volume 4

Cover photographs: Charles S. Peirce (upper left, official Coast Survey photograph c. 1875, courtesy of the Peirce Edition Project), Ernst Schr6der (upper right, from Generallandesarchiv Karlsruhe, signature A/Ac: S.106), Leopold L6wenheim (lower left, courtesy of Professor Dr. Christian Thiel), and Thoralf Skolem (lower right, courtesy of the family of Thoralf Skolem). Reproductions courtesy of Dr. Volker Peckhaus and John E. Muenning.

NORTH-HOLLAND AMSTERDAM- LONDON - NEWYORK- OXFORD- PARIS - SHANNON-TOKYO

FROM PEIRCE TO SKOLEM A

NEGLECTED

IN THE

HISTORY

CHAPTER OF

LOGIC

Geraldine Brady University r Chicago Chicago, USA

2000

ELSEVIER Amsterdam - London - New York - Oxford - Paris - Shannon - Tokyo

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For my mother

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Acknowledgments

This book began ten years ago as a master's thesis at the University of Chicago, u n d e r the direction of W. W. Tait and William A. Howard. During the time that I was beginning my research, I learned m u c h about m o d e r n 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 G r o n i n g e n i n t r o d u c e d 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. T h r o u g h 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 m u c h thanks fox their very helpful and t h o r o u g h 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 c o m m e n t s 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 c o m m e n t s on an earlier draft of this work. I e x t e n d my deepest thanks to Stuart A. Kurtz fox" lively a n d stimulating research sessions over the course of many years and for his patient help in d e c o d i n g the source materials for this book. I am also grateful to the C o m p u t e r Science D e p a r t m e n 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 N e r o d e for directing my study of logic and its history, fox" his incisive c o m m e n t s on my manuscripts in all their various stages, and for his expert advice on the texts. vii

viii

ACKNOWI.EDGMENTS

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 enc o u r a g e m e n t 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 J o h n Muenning for his help in typesetting this book. Finally, I would most especially like to thank my m o t h e r for her patience, support, and love.

Geraldine Brady Chicago, June 2000

Contents

1. 1.1.

Introduction The Early Work of Charles S. Peirce Overview of the Mathematical Systems of Charles S. Peirce Peirce’s Influence o n the Development of Logic Peirce’s Early Approaches to Logic

1.2. 1.3. Peirce’s Calculus of Relatives: 1870 2. Peirce’s Algebra of Relations 2.1. 2.1.1. Inclusion and Equality 2.1.2. Addition 2.1.3. Multiplication 2.1.4. Peirce’s First Quantifiers 2.1.5. Involution 2.1.6. Involution and Mixed-Quantifier Forms 2.1.7. Elementaiy Relatives Quantification in the C ~ ~ C L of I ~ Relatives LIS in 1870 2.2. 2.3. Summary 3. Peirce on the Algebra of Logic: 1880 3.1. Overvicw of Peirce’s “On the algebra of logic” 3.2. Discussion 3.2.1. The Origins of Logic 3.2.2. Syllogism and Illation 3.2.3. Forms of Propositions The Algebra of the Copula 3.2.4. 3.2.5. T h c Logic of Nonrelativc Terms 3.3. Conclusion 4. Mitchell on a New Algebra of Logic: 1883 Mitchell’s Rule of Inference 4.1. 4.2. Single-Variable Monadic I.ogic ix

1 9

9 11 14 23 24 27 29 30 38 39

42 44 46 48 51 51 54 54 56 60 64 70 73 75 75 78

4.2.1. Single-Variable Monadic Propositions 4.2.2. Disjunctive Normal Form Rules of Inference for Single-Variable Logic 4.2.3. 4.3. Two-Variable Monadic Logic 4.3.1. Mitchell’s Dimension Theory 4.3.2. Contrast to Peirce 4.4. Three-Variable Monadic Logic 4.5. Peirce on Mitchell Peirce on the Algebra of Relatives: 1883 5. Background in Linear Associative Algebras 5.1. The Algebra of Relatives 5.2. 5.2.1. Types of Relatives 5.2.2. Operations on Relatives 5.3. Syllogistic in the Relative Calculus 5.4. Prenex Predicate Calculus Summary of Peirce’s Accomplishments in 1883 5.5. 5.5.1. Syntax and Semantics 5.5.2. Quantifiers Peirce’s Appraisal of His Algebra of Binary 5.6. Relatives Peirce’s Logic of Quantifiers: 1885 6. On the Derivation of Logic from Algebra 6.1. 6.2. Nonrelative Logic Embedding Boolean Algebra in Ordinary 6.2.1. Algebra Five Peirce Icons 6.2.2. 6.2.3. Truth-functional Interpretations of Propositions 6.3. First-Order Logic 6.3.1. Infinite Sums and Products 6.3.2. Mitchell 6.3.3. Formulas and Kules 6.4. Second-Order Logic 7. Schroder’s Calculus of Relatives 7.1. Die Algebra dt7 Logik: Volume 1 Die Algebra dtrr Logik: Volume 2 7.2. 7.3. Die Algebra der Logzk: Volume 3 Peirce’s Attack o n the General Solutions of 7.3.1. Schroder Lectures VI-X and Dedekind Chain Theory 7.3.2. 7.3.3. Lectures XI-XI1 and Higher Order Logic 7.4. Norbert Wiener’s Ph.D. Thesis 8. Lowenheim’s Contribution 8.1. Overview of Lowenheim’s 1915 Paper X

78 80 82 86 86 88 89 90 95 95 98 98 100 104 106 109 109 110 111 113 113 116 116 121 125 127 127 128 129 132 143 144 147 149 153 155 160 165 169 171

8.2. 8.3. 8.4. 9.

Lowenheim’s Theorem Conclusions Impact of Lowenheim’s Paper

172 191 195 197

Skolem’s Recasting Appendices

1. 2. 3. 4. 5. 6.

7. 8.

Schroder’s Lecture I Schroder’s Lecture I1 Schroder’s Lecture 111 Schroder’s Lecture V Schroder’s Lecture IX Schroder’s Lecture XI Schroder’s Lecture XI1 Norbert Wiener’s Thesis

207 223 25 1 257 295 339 379 429

Bibliography Index

445 461

xi

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Introduction

T h i s b o o k is an a c c o u n t o f t h e i m p o r t a n t i n f l u e n c e o n t h e 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 C h a r l e s S. P e i r c e a n d his s t u d e n t O. H. M i t c h e l l , through the work of Ernst Schr6der, Leopold L6wenheim, and Thoralf S k o l e m . As far as we know, this b o o k is t h e first w o r k d e l i n e a t i n g this line o f i n f l u e n c e o n 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 t h e s e m i n a l p a p e r s o f L 6 w e n h e i m (1915) " O n possibilities in t h e c a l c u l u s o f relatives" a n d S k o l e m (1923) " S o m e r e m a r k s o n a x i o m a t i z e d set theory." T h e y s h o w e d o r d e r logic, if a s t a t e m e n t has an i n f i n i t e m o d e l , it also 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 t h a t s e c o n d - o r d e r have this p r o p e r t y ; witness t h e a x i o m s for t h e real n u m b e r

t h a t in firsthas a m o d e l logic fails to field. T h e i r

papers focused the attention of a growing n u m b e r of logicians, starting with K u r t G 6 d e l a n d J a c q u e s H e r b r a n d , o n m o d e l s o f t i r s t - o r d e r theories. ~ T h i s b e c a m e t h e 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 l a r g e 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 o v e r t h e rest o f t h e t w e n t i e t h c e n t u r y . In a d d i t i o n , t h e w o r k o f H e r b r a n d , b a s e d o n t h e n o t i o n o f S k o l e m f u n c t i o n , b e c a m e , t h r o u g h J. A l a n R o b i n s o n , t h e m a i n basis o f systems o f a u t o m a t e d r e a s o n i n g . A careful examination of the contributions

of Peirce,

Mitchell,

S c h r 6 d e r , a n d L 6 w e n h e i m s h e d s l i g h t o n several q u e s t i o n s : H o w d i d f i r s t - o r d e r logic as we k n o w it d e v e l o p ? W h a t a r e t h e 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 Heijenoort'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

INTRODUCTION

of Peirce, Mitchell, and Schr6der, over and above the better known c o n t r i b u t i o n s of Gottlob Frege, B e r t r a n d Russell, a n d David Hilbert? As a result of this investigation we c o n c l u d e that, absent new historical evidence, L 6 w e n h e i m ' s and Skolem's work on what is now known as the d o w n w a r d L6wenheim-Skolem t h e o r e m d e v e l o p e d directly from S c h r 6 d e r ' s Algebra der Logik, which was itself an avowed e l a b o r a t i o n of the work of the American logician Charles S. Peirce a n d his s t u d e n t O. H. Mitchell. We have b e e n unable to detect any direct influence of Frege, Russell, or Hilbert on the d e v e l o p m e n t of L 6 w e n h e i m a n d Skolem's seminal work, contrary to the c o m m o n l y held p e r c e p t i o n . This, in spite of the fact that Frege has u n d i s p u t e d priority for the discovery a n d f o r m u l a t i o n of first-order logic. This raises yet o t h e r intriguing questions. Why were the c o n t r i b u t i o n s of Peirce a n d S c h r 6 d e r neglected by later authors? Was it because Peirce p u b l i s h e d in American j o u r n a l s that were not easily available to Europeans? Was it because S c h r 6 d e r h a d a verbose and s o m e t i m e s obscure style as a writer? Was it because the logical notations used by Peirce and S c h r 6 d e r were simply less readable than those of Frege? After r e a d i n g this book, the r e a d e r should be able to form his or h e r own opinions. T h e r e is clear evidence that G6del, at the time he wrote his thesis in 1929, in which he prove d the c o m p l e t e n e s s t h e o r e m for the first-order predicate calculus, was directly a c q u a i n t e d with.at least the special term i n o l o g y used by L6wenheim. In the o p e n i n g p a r a g r a p h of his thesis, G6del uses the term "Ziihlaussage," in defining completeness, which he t h o u g h 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 papel, L 6 w e n h e i m 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, n o n e ranges over the relatives)," and which, of course, recurs in the statement 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 p a p e r 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 L6wenheim'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 H e i j e n o o r t and Hao Wang noticed the similarity of Skolem's 1923 K6nig's lemma-style p r o o f and G6del's 1929 completeness t h e o r e m proof and asked G6del about it. 3 Van Heijenoort 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 m u c h 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 t h e o r e m , the statem e n t "Every consistent statement has a countable model" replaces the earlier Skolem-L6wenheim formulation "Every statement with an infinite model has a countable model." The tree constructions of SkolemL6wenheim were justified semantically; those of G6del are identical, but are justified syntactically. Skolem p r o d u c e d several proofs of the L6wenheim t h e o r e m , the second not requiring the axiom of choice and close in spirit to L6wenheim's original proof. Skolem studied at G6ttingen in the winter of 1915-1916. We do not know whether he first learned of L6wenheim's "~This is r e c o r d e d in D r e b e n a n d van H e i j e n o o r 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 Annalen. Skolem's first paper re-proving L6wenheim's t h e o r e m (1920) introduced the notion of first-order proposition explicitly as a r e p l a c e m e n t for L6wenheim's first-order equations and d r o p p e d the relative sum and product notation that L6wenheim had adopted from Schr6der, and which originated with Peirce. Skolem's 1920 p r o o f 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 p r o o f 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 arg u m e n t to give a second proof of L6wenheim's theorem. This p r o o f avoided the axiom of choice. Skolem's second proof has the same root as H e r b r a n d ' 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 n e e d e d and is absent. Yet a n o t h e r proof Skolem gave of the L6wenheim t h e o r e m 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 p r o o f 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 m e t h o d of solution are natural extensions of material in volume 3 of Ernst Schr6der's Vorlesungen iiber die Algebra der Logik (1895). The L6wenheim language of relatives, infinite sequences with subscripts, c o n d e n s e d relatives, and fleeing subscripts can hardly be d e c i p h e r e d without a careful reading of Schr6der's volume 3, and L6wenheim's p r o o f 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.,

FROM PEIRCE TO SKOLEM

ematics. S c h r 6 d e r gave one of the first expositions of abstract algebraic structures in the form of a very extensive axiomatic d e v e l o p m e n t of lattices, based on both o r d e r and algebraic operations, in the first two volumes of his Algebra der Logik. He m a d e substantial investigations into a second-order theory of relatives, which 1.6wenheim, even in 1940, proposed as an alternative to set theory as a foundation 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 equation as a generalization of elimination theory in commutative algebra. This a m o u n t s to introducing a relation symbol that acts like a Skolem function and symbolically solves the equation as a function of its parameters. S c h r 6 d e r then used sequences of p r e n e x universal and existential quantifiers written as algebraic sums and products. Thus, the Skolem function technique itself can be seen as a direct d e s c e n d e n t of Schr6der's m e t h o d for alternating quantifiers. His h u n d r e d s 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 d e v e l o p m e n t of mathematics in the h i g h e r o r d e r theory of relations is thus the intellectual predecessor of Tarski's logic without variables, which does give an alternate f o u n d a t i o n for mathematics, as L 6 w e n h e i m had h o p e d it would. S c h r 6 d e r himself was not a disciple of or seriously influenced by Frege, while his work precedes that of both Russell and Hilbert. His research p r o g r a m was explicitly an extension of the calculus of relatives and the theory of quantifiers proposed by the American m a t h e m a t i c i a n and logician Charles S. Peirce. Peirce came from an algebraic tradition, t h r o u g h the intluence of his father, the great American algebraist, Benj a m i n Peirce, the a u t h o r of the p i o n e e r i n g work Linear Associative Algebras (1870). T h e 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 operations and the relative operations of relative product, relative sum, and converse, to which he a d d e d a theory of relations of all arities. In particular, he p r o p o s e d to develop a calculus of relatives that was an extension of Boole's calculus that would a c c o m m o d a t e quantification. In Peirce's earliest version of the calculus of relations (1870), existential and universal quantification are expressed by relational operations 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

INTRODUCTION

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 m o d e r n 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 stemmed 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, although 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 mention 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, j u d g e d Peirce to have had a tremendous 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

FROM PEIRCE TO SKOLEM

c a l c u l u s o f r e l a t i o n s , w h i c h w e r e n o t r e d u c e d to a s m a l l set o f f u n d a m e n t a l o n e s , as Tarski d i d later. However, notational complexity alone does not necessarily explain his n e g l e c 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 a r e o f t e n e q u a l l y u n r e a d a b l e , as is W h i t e h e a d a n d Russell's Principia Mathematica, e s p e c i a l l y 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 r d e r logic c o m e s f r o m n o n e o f t h e m ; it arrives full b l o w n in H i l b e r t ' s 1917 l e c t u r e s , w i t h o u t any r e f e r e n c e to a n y o n e . P e i r c e h i m s e l f was e x t r e m e l y critical o f S c h r 6 d e r ' s i d e a o f

Aufl6sung.

H e said t h a t if S c h r 6 d e r ' s n o t i o n o f a ( S k o l e m f u n c t i o n ) s o l u t i o n 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 l g e b r a i c e q u a t i o n t h a t o n e h a d s o l v e d it by i n t r o d u c i n g a f o r m a l f u n c t i o n o f t h e coefficients a n d saying its values w e r e t h e roots. H e d i d n o t g r a s p t h a t t h e i d e a o f a f o r m a l s o l u t i o n , o r S k o l e m f u n c t i o n , c o u l d b e useful. Russell, o n t h e 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 i d e a s freely, w h i l e r e j e c t i n g 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 . ~ B e c a u s e o f t h e w e i g h 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 v a l u e t h e o p i n i o n s e x p r e s s e d by Russell a b o u t d e f e c t s in P e i r c e ' s a n d S c h r 6 d e r ' s t r e a t m e n t o f relatives. T h i s d i d n o t go u n n o t iced. N o r b e r t W i e n e r , in his d o c t o r a l thesis ( 1 9 1 3 ) , c r i t i c i z e d t h e lack o f c r e d i t t h a t Russell gave to S c h r 6 d e r . H e t o o t h o u g h t t h a t Russell o w e d m u c h m o r e to S c h r 6 d e r , a n d h e n c e to P e i r c e , t h a n Russell was willing to a d m i t . W i e n e r , in t h e p a r t o f his thesis r e p r o d u c e d h e r e , g o e s so far as to s u g g e s t t h a t t h e a l g e b r a o f r e l a t i o n s as c a r r i e d o u t in

Principia

Mathematica

is t a k e n d i r e c t l y f r o m S c h r 6 d e r w i t h o u t c r e d i t . P e r h a p s t h e u l t i m a t e c a u s e for t h e n e g l e c t o f S c h r 6 d e r a n d P e i r c e c a n be t r a c e d to t h e i n f l u e n c e o f H i l b e r t . At t h e t u r n o f t h e t w e n t i e t h c e n t u r y , t h e Russell p a r a d o x a n d t h e Burali p a r a d o x c a u s e d H i l b e r t , w h o h a d a l r e a d y r e w o r k e d t h e 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 h o w to set u p logic a n d set t h e o r y as a f o u n d a t i o n for t h e m a t h e m a t i c a l In his discussion of the calculus of relations in

Principlesof 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 J o h n Venn and De Morgan's student W. Stanley Jevons, and by Peirce, who

10

PEIRCE'S EARLY WORK

gave a systematic lattice-theoretic treatment 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 manipulating 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 e x p e n d e d 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 n u m b e r 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 b o u n d 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 m e a n t 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 T h e noted category theorist E William Lawvere read an early version of the present manuscript and c o m m e n t e d 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 enriched categories (Eilenberg and Kelly 1965), where the internal or enriched hornobject represents the actual maps." The role of adjoints in giving c o m m o n form to all ttle rules of logic is summarized in Lawvere's 1994 paper "Adjoints in and a m o n g bicategories."

FROM PEIRCE TO SKOLEM

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 H e r m a n 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 formalizations 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 computational during the period of Peirce's a p p o i n t m e n t at the Johns Hopkins 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 develo p m e n t of the calculus of relatives on this later, highly algebraic presentation of the calculus of relatives by Peirce.

12

PEIRCE'S EARI.Y WORK

S c h r 6 d e r e x p a n d e d 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 foundation 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 D e d e k i n d ' s work on induction. Schr6der also gave an axiomatic t r e a t m e n t of Peirce's lattice-theoretic d e v e l o p m e n t of Boolean algebra in the first two volumes of Algebra der Logik. In his work on the calculus of relatives, Schr6der i n t r o d u c e d 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. L 6 w e n h e i m , however, did. L6wenheim took the essential idea of a Skolem function from S c h r 6 d e r and made it the first step of the p r o o f of his celebrated t h e o r e m , proving that if a first-order statement has an infinite model, thien it has a countable model. T h e second part of L 6 w e n h e i m ' s proof, giving his model-theoretic construction, has antecedents in Schr6der's " m e t h o d of elimination," which forces branchings in a tree of roots by e x p a n d i n g his f u n d a m e n t a l representation equation, 0 o' + 1 ' =o1 ' for various values of i and j. (In Schr6der's and L6wenheim's notation, 1' denotes the identity relation and 0' is its c o m p l e m e n t . ) L 6 w e n h e i m explicitly says t h a t h e got this idea from Schr6der. L 6 w e n h e i m ' s t h e o r e m was the basis for Skolem's work in the subs e q u e n t d e v e l o p m e n t of model theory. Conceptually, the nearest analogues to L6wenheim's p r o o f are the early topological compactness-style theorems, such as the Bolzano-Weierstrass theorem, at the point at which logic dissolves into topology. T h e closest analogue of all is the arithmetic u l t r a p r o d u c t 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 admitted his d e p e n d e n c e on Peirce or Schr6der. But, as Norbert Wiener claims in his doctoral dissertation and presents convincing evidence to show, Russell lifted his t r e a t m e n t of binary relations in Principia Mattaematica almost entirely from Schr6der's Algebra der Logik, with a simple change of notation and without attribution. T h e calculus of relatives (unions, intersections, relative products, etc.) remains in the mathematical curriculum without credit to Peirce in the introductory parts of mathematics books to this day. L 6 w e n h e i m claimed t h r o u g h o u t 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, i n t r o d u c e d a set theory without variables, which shows that a relational calculus basis for set theory can be fully realized. T h e abstract theory of allegories of Peter Freyd and Andre Scedrov is a n o t h e r intellectual d e s c e n d e n t of Peirce's

FROM

PEIRCE

TO SKOLEM

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 p r o g r a m m i n g , called relational programming, of which the work of James Lipton and Paul B r o o m e 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 develo p m e n t of first-order and higher order predicate logic, but the papers of Peirce do not seem to have influenced Hilbert, Skolem, H e r b r a n d , or G6del directly. The actual historical connection between Peirce and the later d e v e l o p m e n t of first-order logic runs t h r o u g h 1.6wenheim, via the calculus of relatives, and the link to Peirce results from Skolem extracting 1.6wenheim's t h e o r e m flom the calculus of relatives and stating it as a t h e o r e m 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 develo p m e n t of it, and the line flom Peirce to Hilbert and A c k e r m a n n 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 introduced 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 deduction systems looks sequential, but it is not clear that there is an actually historical d e p e n d e n c e . Although Peirce never published any of his writings on existential graphs (with one m i n o r exception), he presented the complete system in his I.owell Lectures, delivered to the philosophy d e p a r t m e n t at Harvard University in 1903-1904. To what extent Peirce's ideas on modal logic c o m m u n i c a t e d 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 unpublished 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 d e v e l o p m e n t on axiomatic arithmetic also must remain conjectural. In 1881, in his paper on "On the logic ot number," Peirce introduced discrete orderings with a first e l e m e n t satisfying the principle of induction. He further defined addition and multiplication by inductive definitions. Peirce used his recursion equations

PEIRCE'S EARLY WORK

14

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 H e r m a n Grassmann (1861).2 Peirce did not realize, however, as Dedekind did in his later work (1888), that these definitions of addition and multiplication n e e d e d to be justified, viz., that definition by induction is different from proof by induction. It was left to Dedekind (1888) to prove by the m e t h o d 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 i n d e p e n d e n t of Dedekind, Peirce also set out to capture finite sets, but he did it by characterizing these sets as images u n d e r one-to-one maps of initial segments of a discrete o r d e r with a first e l e m e n t that satisfies the induction axiom. This is equivalent to Dedekind's notion of finite set, that is, a set that cannot be m a p p e d one-to-one into a p r o p e r 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 other major work was with formal algorithms for solving ordinary differential and difference equations; it is therefore no surprise that his a p p r o a c h to Boolean logic was algorithmic rather than axiomatic. Peirce and Schr6der similarly first became attached to first and higher o r d e r relational algebra and merely used whatever algebraic identities they could discover as they went along to simplify reasoning. They made no early attempt 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. P e i r c e ' s Early Approaches to Logic Peirce's first publication in mathematical logic was a paper on Boolean algebra, "On an i m p r o v e m e n t 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

" a n d , " "or," a n d " n o t . " H o w e v e r , B o o l e ' s "or" was a p a 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 o n l y w h e n t h e a l t e r n a t i v e s w e r e e x c l u s i v e . :~I n his 1867 paper, Peirce introduced the operation of logical addition (inclusive " o r " ) , w h i c h is always d e f i n e d , in w h a t h e t h o u g h t to b e a n e x t e n d e d v e r s i o n o f B o o l e ' s system. P e i r c e d e f i n e s l o g i c a l a d d i t i o n , w h i c h h e den o t e s by "+," ( a n d / o r ) , as follows: Let the letters of the a l p h a b e t d e n o t e classes w h e t h e r of things or of occurrences. It is obvious that an event may either be singular, as "this sunrise," or general, as "all sunrises." Let the sign of equality with a c o m m a b e n e a t h it express numerical identity. Thus a--, b is to m e a n that a and b d e n o t e the same class--the same collection of individuals. Let a +, b d e n o t e all the individuals c o n t a i n e d u n d e r a and b together. T h e o p e r a t i o n here p e r f o r m e d will differ from arithmetical addition 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) P e i r c e r e t a i n s in his s y s t e m 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 with a " + " sign. P e i r c e ' s l o g i c a 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 h e s a m e as B o o l e a n " a n d " : Let a, b d e n o t e the individuals c o n t a i n e d at once u n d e r the classes a and b. (Peirce 1867, p. 4) P e i r c e also i n t r o d u c e d l o g i c a 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 a n i n v e r s e f o r l o g i c a l a d d i t i o n . L o g i c a l s u b t r a c t i o n , P e i r c e b e l i e v e s , will p r o v i d e f o r n e g a t i o n in his s y s t e m , 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 l o g i c a l s u b t r a c t i o n as a p a r t i a l i n v e r s e to l o g i c a l addition: Let - , be the sign of logical subtraction; so defined that If b +, x =, a

x =, a - , b.

H e r e it will be observed that x is not completely d e t e r m i n a t e . It may vary from a to a with b taken away. This m i n i m u m may be d e n o t e d by a - b. It is also to be observed that if the s p h e r e of b reaches at all beyond a, the expression a - , b is u n i n t e r p r e t a b l e . (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.

16

PEIRCE'S EARLY WORK

Logical subtraction is only d e t e r m i n a t e if x a n d b are disjoint, a n d thus is a partial inverse for exclusive "or," since the latter is only d e f i n e d in the disjoint case. Peirce uses a bar over a class term, d, r a t h e r t h a n logical subtraction, to d e n o t e the n e g a t i o n of a literal (Peirce 1867, p. 5) a n d r e p r e s e n t s the n e g a t i o n of an arbitrary class x by 1 - x, the c o m p l e m e n t of x, where the m i n u s 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 t h r e e advantages for his system over Boole's, all d u e to his new operations: 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) T h e first advantage is aesthetic: f r o m the p o i n t of view of m a t h e m a t i c a l duality, Peirce's system is s u p e r i o r to Boole's since Peirce's o p e r a t i o n of inclusive "or" is the natural dual of logical multiplication, a n d allows for the expression of i m p o r t a n t algebraic identities, such as De M o r g a n ' s law, w h e r e a s Boole's o p e r a t i o n s are n o t m a t h e m a t i c a l l y dual. This aesthetic gain is offset s o m e w h a t by Peirce's loss of additive inverses, however. T h e s e c o n d advantage is a p r a g m a t i c one: by a d d i n g the inclusive "or," Peirce p r o d u c e s a system that is not only m u c h m o r e c o n v e n i e n t to work in computationally, but also allows for an easy translation of n o n m a t h e m a t i c a l logical a r g u m e n t s into m a t h e m a t i c a l r e p r e s e n t a t i o n s o f logical a r g u m e n t s . Peirce's third claim, viz., that his system is m o r e expressive t h a n Boole's, is not correct. Peirce's i n t r o d u c t i o n of inclusive "or" is an ext r e m e l y useful c o n t r i b u t i o n , but his logical "or" can be e x p r e s s e d using the o r d i n a r y "and" ( c o m m a ) , Boole's restricted "or" ( " + " ) , a n d negation, simply as a +, b = a,/~ + a , b + ,4, b. At this level, Peirce's system does not have any m o r e expressiveness than Boole's. Peirce hints that t h e r e are Aristotelian notions that are expressible in his system but not in Boole's calculus. To substantiate 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 n o t clear what the expression a - , i, a means. O n the o n e h a n d , if a - , i, a m e a n s a minus the quantity i intersect a, that class will be a m i n u s a singleton, which may be m u c h larger than o n e individual. O n

FROM PEIRCE TO SKOLEM

x7

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, r a t h e r than a restricted to a singleton, as Peirce wants to claim. It is possible that there was a transcription error, and Peirce i n t e n d e d for the expression to read a-, i,d. 4 In any case, Peirce's idea fails because his notation does not completely capture how we use the word "some" in a m a t h e m a t i c a 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 properties of that individual. T h e p r o b l e m of existentially selecting an e l e m e n t from a has been replaced by the p r o b l e m of existentially selecting a class that contains only a single e l e m e n t of a. But it is not obvious that the second p r o b l e m is any easier than the first. T h e second p r o b l e m is simply one type level higher. Peirce is struggling toward the solution of the p r o b l e m of expressing quantification, but the real p r o b l e m is that the solution is not to be found within the confines of Boolean algebra. All Peirce's attempts were d o o m e d to fail until it occurred to him how to step out of the framework of purely Boolean operations. We know today that the decision p r o b l e m for the validity of a statement in propositional logic is decidable; a statement is valid if and only if the last c o l u m n of its truth table has only "T" values. We also know that the decision p r o b l e m for the validity of statements in predicate logic is undecidable; this was first proved by C h u r c h (1936), after he had given an exact definition of recursive or decidable. It can also be proved by the m e t h o d of p r o o f used by G6del for his incompleteness theorem. O n e c o n s e q u e n c e 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 p a p e r Peixce is e n g a g e d in presenting a cleaned-up version of Boolean algebra in which the "or" operation is not restricted to disjoint classes, and is dual to the Boolean "and" that is also not so restricted. But his i m p r o v e m e n t is algebraic. At the same time, Peirce has m u d d i e d the waters somewhat, although he did not realize it, by introducing a minus operation that is not, properly speaking, a function. In o r d e r to make clear that his efforts are not simply an improved, pedagogically m o r e useful presentation 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 o n g o i n g t h e m e in Peirce, his a t t e m p t i n g to reconcile Boole with Aristotle a n d solve the p r o b l e m of expressing quantification. 1 See l-Iailperin (1976) for a full analysis.

18

PEIRCE'S EARLY WORK

In an u n p u b l i s h e d m a n u s c r i p t written a r o u n d 1896, Peirce describes his early u n d e r s t a n d i n g o f 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 established that Boole arrived at B o o l e a n a l g e b r a while seeking to give a precise calculus o f probabilities. H e starts o u t with a p r o p ositional calculus a n d m a k e s probability a s s i g n m e n t s to p r o p o s i t i o n a l letters. H e t h e n discusses the rules for assigning probabilities to comp o u n d p r o p o s i t i o n s as a f u n c t i o n o f their parts. O f course, probabilities are b e t w e e n 0 a n d 1, a n d the rule for "or" is P(A o r B ) - P ( A ) + P(B) - P(A a n d B). This rule reflects his implicit class i n t e r p r e t a t i o n o f p r o p o s i t i o n a l letters, in which each letter d e n o t e s a subset o f a fixed (for simplicity of s t a t e m e n t a s s u m e d finite) set. In a d d i t i o n , o n e is adding u p the ( i n d e p e n d e n t ) probabilities o f the e l e m e n t s within the set d e n o t e d by the p r o p o s i t i o n a l letter. We now express this as a finite probability space i n t e r p r e t a t i o n o f p r o p o s i t i o n a l calculus, with e a c h letter d e n o t i n g an event. Boole o b s e r v e d that events with probability 1 a n d 0 (in m o d e r n terms, in finite spaces certain or impossible events, in infinite spaces almost certain or almost impossible events) have 0, 1 a s s i g n m e n t s associated with t h e m , these b e i n g o u r m o d e r n t r u t h assignments. H a v i n g d o n e this, w h e n e x t e n d i n g f r o m letters (atomic p r o p o s i t i o n s ) to c o m p o u n d propositions, instead o f i n v e n t i n g the B o o l e a n a l g e b r a o f 0 a n d 1 as values with o p e r a t i o n s restricted to t h e m , Boole instead copies the f o r m a t f r o m probability, w h e r e P(A o r B) is P(A) + P ( B ) P(A a n d B), a n d regards the B o o l e a n o p e r a t i o n s as restrictions o f the a r i t h m e t i c a l o n e s o n the real n u m b e r s . His successors e l i m i n a t e d this step, i n t r o d u c i n g the o p e r a t i o n s o n 0, 1 directly, n o t as restrictions o f o p e r a t i o n s o n the real n u m b e r s . This is what Peirce is r e f e r r i n g to in

FROM

PEIRCE

TO SKOLEM

19

the q u o t a t i o n above. W h e n the a r i t h m e t i c a l i n t e r p r e t a t i o n is a p p l i e d to the law of the e x c l u d e d middle, it gives the o r d i n a r y a l g e b r a e q u a t i o n x (1 - x) = 0. Peirce's "characters," i.e., p r o p e r t i e s , are thus e q u i v a l e n t to Boole's sets: each p r o p o s i t i o n a l letter d e n o t e s a p r o p e r t y of individuals, a n d the p r o p o s i t i o n a l connectives lead f r o m simple to c o m p o u n d p r o p e r t i e s . A s s u m i n g that these are all p r o p e r t i e s of e l e m e n t s of a fixed set, this is e q u i v a l e n t to Boole. If the p r o p e r t i e s are n o t p r o p e r t i e s of e l e m e n t s within a fixed set (e.g., x is a cardinal n u m b e r ) , t h e n Peirce's l a n g u a g e of p r o p e r t i e s is b e t t e r because it is m o r e g e n e r a l . T h e idea of equality of p r o p o s i t i o n s u n d e r a probability i n t e r p r e t a t i o n is clear: b o t h p r o p o s i t i o n s are assigned the same probability based on the probability a s s i g n m e n t to p r o p o s i t i o n a l letters used. W h e n a p p l i e d to 0, 1-valued assignments, this m e a n s the propositions are b o t h true or b o t h false u n d e r the a s s i g n m e n t , which is o u r usual s e m a n t i c equivalence. It is i n t e r e s t i n g that, like Boole, Peirce saw a p o t e n t i a l c o n n e c t i o n b e t w e e n probability t h e o r y a n d 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 "balancing 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) T h e s e r e m a r k s are e x t r a n e o u s to the m a i n lines of the work. Peirce is m e r e l y p o i n t i n g o u t that the f r e q u e n c y i n t e r p r e t a t i o n of the probability of an event as the ratio of cases in which the e v e n t holds to all cases w h e t h e r it holds or not is n o t the only perfectly n e a t way of int e r p r e t i n g probability. O n e could start by using, instead of probability, the ratio of favorable to u n f a v o r a b l e cases, the ratio of u n f a v o r a b l e to favorable cases, or the log of either. Each gives a d i f f e r e n t n o t i o n of f r e q u e n c y i n t e r p r e t a t i o n , each has s o m e intuitive c o n t e n t , a n d the calculus of probabilities of c o m p o u n d s t a t e m e n t s could be a l t e r e d 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 likelihood estimates, especially for o p t i m i z a t i o n .

20

P E I R C E ' S EARLY WORK

Peirce's motivation was similar to that of the m o d e r n statistician. In a way, it a m o u n t s to saying that o u r "ruler" of probabilities is to an e x t e n t quite arbitrary. In the n e x t a n d final p a r a g r a p h of tile 1896 m a n u s c r i p t , Peirce claims to have used index n o t a t i o n to d e n o t e individuals in or b e f o r e 1880, a l t h o u g h w h e t h e r he actually did so is q u e s t i o n a b l e . T h e r e is a f o r m of i n d e x n o t a t i o n in his 1867 paper, but it is not used to d e n o t e individuals; Peirce employs it r a t h e r to define a function 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. ab,,=a,b.

(Peirce 1867, p. 9) T h e index a in b,, d e n o t e s a class, not an individual. Peirce i n t r o d u c e d his calculus of relatives in 1870. Like Boole, he was g u i d e d by an analogy b e t w e e n the laws of p r o p o s i t i o n a l logic, the laws of classes, and the laws of arithmetic. H e was also g u i d e d by o t h e r m a t h e m a t i c a l analogies. Peirce's f a t h e r was the f o u n d e r of abstract line a r associative algebra; Peirce e d i t e d his father's work a n d m e n t i o n e d in his own that m a n y laws for the relative calculus o p e r a t i o n s were a n a l o g u e s of the laws of linear associative algebra. For instance, he n o t e d that an a p p r o p r i a t e multiplication of matrices with entries 0, 1 a l o n e a n d B o o l e a n o p e r a t i o n s on 0, 1 c o r r e s p o n d s to relative p r o d u c t . ~ Peirce saw an analogy b e t w e e n the laws of e x p o n e n t i a t i o n in arithmetic a n d universal quantification in logic. For instance, he i n t e r p r e t e d x y+'~ = x y , x ~ in the calculus of relatives as e x p r e s s i n g that to be in the relation x to every l n e m b e r of y or z is the same as to be in the relation x to every m e m b e r of y a n d in the relation x to every m e m b e r of z v v v . This can be r e g a r d e d as a primitive insight into the n a t u r e of quantification, notationally along the lines of the t r e a t m e n t of quantifiers in topos t h e o r y (Lawvere 1970; see also J o h n s t o n e 1977; Mac L a n e a n d Moerdijk 1992). Peirce was also g u i d e d by analogies to a r i t h m e t i c that were s o m e t i m e s conflicting. H e tried to establish an a n a l o g u e of the b i n o m i a l 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 a n d their identities, a l t h o u g h these a t t e m p t s are difficult to decipher. In p r o c e e d i n g by formal m a n i p u l a t i o n s , Peirce was following the tradition 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 PEIRCE TO SKOLEM

21

of Boole, the g r e a t e s t e x p e r t of his time on inverse differential o p e r a t o r f o r m a l solutions of differential equations. Like Boole, P e i r c e experim e n t e d with m a t h e m a t i c a l analogies to gain insight into a new subject. For Boole, the new subject was differential e q u a t i o n f o r m a l m e t h o d s a n d t h e n p r o p o s i t i o n a l logic. For Peirce, it was the calculus o f relatives. A l t h o u g h De M o r g a n i n t r o d u c e d the notions of relative p r o d u c t , converse, involution (forward a n d backward), a n d n e g a t i o n in his p a p e r "On the syllogism. IV" (1860), Peirce's discovery of the calculus of relatives 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 l e c t u r e 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 b e g i n n i n g , then, Peirce's work in logic first set o u t to e x t e n d Boole's t r e a t m e n t so that it covered the syllogisms of Aristotle in a n a t u r a l a n d satisfactory way. Peirce self-avowedly a c q u i r e d s o m e of his insights, as well as m u c h of his terminology, such as his "first-intentional logic of relatives," f r o m the scholastic logicians, themselves k e e n r e a d e r s of Aristotle. However, r e a s o n i n g in g e o m e t r y was in a highly stylized a n d finished state, a n d h a d b e e n for 2000 years. W h e n Peirce tried to use Boole's p r o p o s i t i o n a l logic with a s m a t t e r i n g of syllogistic to repr e s e n t g e o m e t r i c reasoning, he realized that the Aristotle plus Boole, or roughly what we now call m o n a d i c p r e d i c a t e logic, was n o t expressive

22

PEIRCE'S EARLY WORK

e n o u g h 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, m u c h if not all reasoning in Euclid can approached. We surmise that this was what Peirce m e a n t by the "various facts" that led him to desire a m o r e perfect logic, a n d 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

Introduction Peirce published his "Description of a notation for the logic of relatives" in 1870, eight years before the founding of the American Journal of Mathematics 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 Harvard 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 a p p e a r e d 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 c o n t e m p o r a r y 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 Algebras, 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 i m p o r t a n t 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," m e a n i n g 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 computations to deduce complex logical theorems from simple ones. Second, there is his codification of rules about relations to reveal the mathematical structure of formal logic itself. This was a precursor of metamathematics and p r o o f 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 p r o o f theory and syntax than to model theory as we know it. Peirce's belief that this was the most i m p o r t a n 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. T h r e e kinds of logical terms, called "absolute," "simple relative," and "conjugative." 2. A fundamental binary relation, d e n o t e d by -- 1 is identified with 1. T h a t is, 0x= 1 for x = 0 a n d 0x= 0 for x #: 0, so 0~ 0 for x = 0 a n d 0~ 1 for x ~ 0. Thus, the Boolean "or" of a + b, taking the s t a n d a r d repres e n t a t i o n of truth as 1 a n d falsity as 0, can be r e p r e s e n t e d as 0 ~ using the usual a r i t h m e t i c a l definitions of a d d i t i o n a n d e x p o n e n t i a t i o n . W h e n Peirce says that 0 ~176 is m o r e a c c u r a t e t h a n (l)o + (b)o, he robably m e a n s that, since 0 x normalizes every value of x to e i t h e r 0 or , 0 ~(z)'' +'(') " can r e p r e s e n t B o o l e a n "or" without a s s u m i n g the n o n a r i t h metical rule 1 + 1 = 1. Peirce's definition of multiplication is as a pointwise, scalar multiplication, the i n n e r p r o d u c t . H e calls this p r o d u c t "internal multiplication," a t e r m that he first used technically in c o m m e n t i n g o n G r a s s m a n n ' s two multiplications, i n t e r n a l a n d external, in 1877, in a b r i e f p a p e r on G r a s s m a n n ' s vector calculus (Peirce 1877, p. 102). G r a s s m a n n ' s i n t e r n a l a n d e x t e r n a l multiplication are o u r familiar i n n e r (scalar) a n d o u t e r (exterior) p r o d u c t in today's e x t e r i o r calculus. In his 1877 paper, Peirce viewed t h e m as two distinct multiplications of q u a t e r n i o n s a n d wrote G r a s s m a n n ' s a l g e b r a in his own system of linear associative relative alg e b r a (Peirce 1877, p. 103). Peirce lists t h r e e principal formulas for n o n r e l a t i v e a d d i t i o n a n d multiplication a n d their converses: Ifl--<sand

b-<s, then l+b--<s.

If s--< l a n d s - < b, t h e n s - < l, b. Ifl+b--<s,

then l-<sand

b-<s.

If s - < l, b, t h e n s--< l a n d s--< b.

(l+ b ) , s - - < l , s

+ b,s.

( l + s ) , (b + s) --< l, b + s. (p. 196) If we view --< as b e i n g a partial order, these f o r m u l a s are the axioms for a distributive lattice. In m o d e r n l a n g u a g e , Peirce is saying that relations with these two o p e r a t i o n s f o r m a distributive lattice. Peirce defines the relative p r o d u c t lb a n d 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.

P E I R C E ' S ALGEBRA OF RELATIVES

102

(lb)o = F,,, (l),,,(b),, j, (l t b) 0 = II~{(/),~ + (b)~#}. As in the case of nonrelative multiplication a n d a d d i t i o n , the logical o p e r a t i o n s on the left-hand side o f the e q u a t i o n s are d e f i n e d in terms o f arithmetical a d d i t i o n a n d multiplication on the right, w h e r e (l)~x a n d (b)xi are e i t h e r 0 or 1. Peirce recognizes explicitly that relative s u m a n d p r o d u c t i n c o r p o r a t e a species o f universal a n d existential quantification: 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) B o t h are o r d e r - p r e s e r v i n g o p e r a t i o n s , if 1--< s, t h e n l b - - < s b a n d 1 t b--< s t b; a n d if b ---< s, t h e n lb --< Is a n d l t b --< s. (p. 197) T h e definition of relative p r o d u c t is very familiar to a m o d e r n reader:

(lb)o = E,, (l),,,(b)~). This is the definition of matrix multiplication. In his 1870 p a p e r Peirce viewed relative p r o d u c t as the natural multiplication in his system a n d d i s t a n c e d h i m s e l f s o m e w h a t from the B o o l e a n p r o d u c t . F r o m a matrixt h e o r e t i c p o i n t of view, this makes sense: the natural o p e r a t i o n s for the calculus of relatives, viewed as matrix theory, are (1) relative p r o d u c t b e c a u s e it is matrix multiplication, (2) B o o l e a n a d d i t i o n b e c a u s e it is pointwise a d d i t i o n o f c o o r d i n a t e values, a n d (3) converse b e c a u s e it is m a t r i x transposition. It a p p e a r s that we are seeing B o o l e a n m a t r i x t h e o r y in Peirce's 1883 paper. Peirce would have r e c o g n i z e d (lb) O = I2,,(l)ix(b), 0 as matrix multiplication that will be e i t h e r 0 or 1. Relative p r o d u c t a n d sum obey the associative law:

lo3

FROM PEIRCE TO SKOLEM

I t (bt s ) = ( l t b) t s, l(bs) = (lb)s. Relative p r o d u c t 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 r e p r e s e n t e d by drawing a straight line over the sign for the relative itself' (p. 195). T h e negative includes every pair that the relative itself excludes. In addition, every relative has a converse, "produced by reversing the order of the m e m b e r s of the pair" (p. 195). Peirce's notation 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,

lb=[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 connecting relative and logical p r o d u c t (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 h e m as t h e o r e m s a n d uses t h e m in d e v e l o p i n g s e c o n d - o r d e r quantification r e d u c t i o n rules ( S c h r 6 d e r 1895, pp. 497ff). T h e first versions of these formulas a p p e a r e d in the last c h a p t e r of Peirce (1880).

5.3. Syllogistic in the Relative Calculus T h e n e x t part of Peirce's 1883 p a p e r (pp. 202-205) is d e v o t e d to d e m o n s t r a t i n g that syllogisms can be dealt with entirely in relative calculus equations. Peirce provides four n o r m a l forms (in the a l g e b r a of relatives) for universal propositions, four n o r m a l forms for p a r t i c u l a r propositions, forms for six forms of affirmative p r e m i s e s o c c u r r i n g in the syllogism, a n d r e d u c t i o n s c h e m e s for 21 forms of the Aristotelian syllogism. T h e six forms of affirmative premises, he says, "are the p r o p o sitions of the first o r d e r r e f e r r e d to in Note A." N o t e A itself is an a r g u m e n t for having quantifiers r a n g e over specific d o m a i n s , that is, for i n t r o d u c i n g the n o t i o n of a structure or a m o d e l . We h e r e list the principal points of Note A, with a brief c o m m e n t a r y . First, citing Boole and De Morgan, Peirce outlines 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) T h e u n l i m i t e d universe was that on which Frege based the semantics of his Begriffsschrift (1879). In addition, Peirce points o u t 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) T h e passage that follows is difficult to m a k e 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 PEIRCE TO SKOLEM

10 5

This assertion is r e m i n i s c e n t of the power set axiom (the universe is u n l i m i t e d if there exist always larger objects c o n t a i n i n g all subsets, which m i g h t be called a c o m b i n a t i o n of characters if we c o n s i d e r syntactical formulas describing the subset). Peirce's exclusion of the whole universe of characters seems to p o i n t to some awareness of trouble in allowing arbitrarily large sets of which o n e takes subsets. H e says "short of the whole universe of characters," m e a n i n g "not too large to get into trouble." It would be very r e m a r k a b l e if Peirce would have had a first glimpse of the troubles of set theory back in 1882. O n the o t h e r hand, if we i n t e r p r e t "character" as an atomic predicate of individuals of the d o m a i n a n d "combination" as finite c o n j u n c t i o n , Peirce's s t a t e m e n t would say that every finite c o n j u n c t i o n of atomic predicates is satisfiable in the domain. "Unlimited" would have the conn o t a t i o n that whatever finite set of atomic predicates o n e gives, there is s o m e t h i n g in the d o m a i n that possesses that set of properties. This is a constraint on the d o m a i n . Only n o n e m p t y atomic predicates would be allowed as characters, a n d every finite set of t h e m would be satisfied by s o m e t h i n g in the d o m a i n . (If infinite c o n j u n c t i o n s were allowed, we would get a primitive form of saturation.) In the above i n t e r p r e t a t i o n , we c o n c e d e that we have no idea what the phrase "short of the universe" m e a n s 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 e q u i r e m e n t on the d o m a i n of predicates (characters). H e r e is a possible i n t e r p r e t a t i o n for finite domains. For any set of individuals from the d o m a i n , there is a predicate (character) that applies to the e l e m e n t s of that finite set and to no o t h e r e l e m e n t s of the d o m a i n . For infinite domains, o u r i n t e r p r e t a t i o n would r e q u i r e that there be u n c o u n t a b l y many characters. Peirce just does n o t say what he means. In summary, in "Note A" Peirce observes that w h e n we start applying logic, we are usually not applying it to everything. Instead, we begin with a d o m a i n of individuals in which we are interested, a specific universe. For each such d o m a i n , he argues that we use a specific set of predicates (characters). H e goes on to argue that the predicates we use are built up from a class of basic predicates by a s t a n d a r d collection of operations, Boolean and otherwise. This is a hint of the m o d e r n notion of a signature of a model; the signature tells what d o m a i n of predicates is allowed, a n d the m o d e l tells

lo6

PEIRCE'S ALGEBRA OF RELATIVES

w h a t d o m a i n o f individuals is allowed. Peirce can thus be seen as positing a weak p r e c u r s o r of the n o t i o n of m o d e l .

5.4. Prenex Predicate Calculus At the close o f his 1883 p a p e r (pp. 2 0 5 - 2 0 6 ) , Peirce shows that Mitchell's g e n e r a l i z a t i o n , which allows two-dimensional p r o p o s i t i o n s o f a special sort (that is, with two quantifiers, b u t written in Mitchell's f o r m ) , can be written a n d derived in the calculus of relatives. Peirce m a k e s n o such claim o f representability for m o r e g e n e r a l p r o p o s i t i o n s (since L6wenh e i m later shows such claims n o t to be c o r r e c t in 1915, this is j u s t as well), b u t he allows, in a d d i t i o n to relative o p e r a t i o n s within expressions, quantifiers in front, o f 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 o c c u r explicitly in the subscripts; these are real m o d e r n quantifiers a p p l i e d to l o, that is, he is writing (3i)(3j)l(i,j) a n d (Vi)(3j)l(i,j). But he makes it clear that l 0 is r e g a r d e d as a p r o p o s i t i o n a l f u n c t i o n r a t h e r t h a n a f o r m u l a , a n d that I~ a n d II are r e g a r d e d as sums a n d products, respectively, of p r o p o s i t i o n a l f u n c t i o n s that at any a r g u m e n t are s u m m e d (or m u l t i p l i e d ) in the twoe l e m e n t set. T h a t is, II~Ejl 0 is 1 (i.e., > 0) if a n d only if the two-ary f u n c t i o n l o, s u m m e d first over j to get a one-ary f u n c t i o n in i, t h e n with a p r o d u c t o f these over all i, is 1. W h a t we see h e r e is the i n t r o d u c t i o n of quantifiers as o p e r a t i o n s o n p r o p o s i t i o n a l functions of i, j f r o m a d o m a i n , n o t formal l a n g u a g e quantifiers o n formal expressions. T h e n , leaving o u t the = 1 (or equivalently, > 0), Peirce writes I]~I]i/o a n d I I ~ j / o as propositions, which m a k e s t h e m look a n d act like formulas, b u t in fact they are abbreviations for an algebraic equality equal to 1:

FROM

PEIRCE

xo7

TO SKOLEM

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) Peirce

gives

several

examples

of

propositions

in

this

system:

IIiF., i (1)o(b) q m e a n s that e v e r y t h i n g is at o n c e a lover a n d a b e n e f a c t o r o f s o m e t h i n g ; IIiE i (l)q(b)j~ m e a n s that e v e r y t h i n g is a lover o f a b e n e factor o f itself; r, iF.,kIIi(l q + bik ) m e a n s that t h e r e is s o m e t h i n g which stands to s o m e t h i n g in the relation o f loving e v e r y t h i n g e x c e p t b e n e factors o f it. This last expression, (lq + bik), is an i n d i r e c t way o f introd u c i n g a new, triple relative, tii k = lit + bik. His r e m a i n i n g e x a m p l e s all deal explicitly with triple relatives. Thus, by 1883, Peirce u n d e r s t o o d the g e n e r a l n o t i o n o f a p r e n e x formula. H e u n d e r s t o o d the n o t i o n of a q u a n t i f i e r r a n g i n g over the universe a n d the n o t i o n of the d o m a i n , a n d he u n d e r s t o o d how to rewrite a p r e n e x f o r m u l a as a p r o p o s i t i o n a l f o r m u l a w h e n a c o n s t a n t is i n t r o d u c e d as a n a m e for each a n d every e l e m e n t a q u a n t i f i e r ranges over in a specific i n t e r p r e t a t i o n . Existential q u a n t i f i c a t i o n over i corr e s p o n d s to calling the e l e m e n t s over which it r a n g e s by individual n a m e s i l, i 2, ..., e l i m i n a t i n g the quantifier, substituting in succession e a c h o f i l, iz . . . . for i everywhere in the quantifier-free part, a n d taking the disjunction o f these formulas, each of which is p r e n e x b u t has o n e less q u a n t i f i e r at the front. If the l e a d i n g q u a n t i f i e r is H i , t h e n , n a m i n g the e l e m e n t s of the d o m a i n over which j ranges as jl,jz, ..., we o m i t the universal q u a n t i f i e r in each term, substitute e a c h j~ for all the j's in the quantifier-free part, a n d replace by the c o n j u n c t i o n . Since t h e r e are a finite n u m b e r o f quantifiers in the prefix, we finally e n d u p with a n e s t e d set of i n f i m u m s a n d s u p r e m u m s over the d o m a i n , a p p l i e d in the e n d to propositional logic statements, since all variables have b e e n r e p l a c e d by constants i k, jr, a n d so on. Thus, the validity o f this p r o p ositional logic s t a t e m e n t in this d o m a i n can be q u e s t i o n e d , b u t the s t a t e m e n t is infinitary. At this p o i n t Peirce uses the rules o f Boole's algebra, e x t e n d e d to infinitary disjunctions a n d c o n j u n c t i o n s : 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 d o m a i n only a n d i n t e r p r e t every quantifier as either a least u p p e r b o u n d or a greatest lower b o u n d (infinite disjunction of instances over the d o m a i n or infinite c o n j u n c t i o n over the d o m a i n ) , t h e n every first-order s t a t e m e n t b e c o m e s an infinitary propositional logic s t a t e m e n t (an infinitary Boolean algebra s t a t e m e n t ) . T h e n all logic ded u c t i o n s b e c o m e calculations using o r d i n a r y Boolean algebra rules plus the infinite distributive law. T h e r e is also the infinite De M o r g a n ' s law; t h e n every d e d u c t i o n can be d o n e by Boolean calculations, but infinitary ones. In fact, as Mitchell said, the calculations can be d o n e by c a n c e l i n g conjunctive hypotheses, a d d i n g terms to disjunctive hypotheses, and t h e n c a n c e l i n g terms after distributing. Peirce concludes his p a p e r with an e x a m p l e of a d e d u c t i o n in this system. T h e p r o b l e m he sets for himself is to eliminate the relative "servant" from 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 statements as

w h e r e ot d e n o t e s the triple relative "accuser t o - o f - , " E d e n o t e s the triple relative "excuser to - of," a n d l, b, and s d e n o t e , as before, the binary relatives "lover o f - , .... b e n e f a c t o r o f - , " a n d "servant o f - . " First, Peirce claims that "the s e c o n d yields the i m m e d i a t e inference, l'Ix~],,I'Iy~],,(e.y x + ~,~b~)" (p. 208). This follows from the s e c o n d premiss by three applications of Peirce's first rule, E i I I j - < IIjE;. Peirce then says, " c o m b i n i n g this with the first, we have ExE,,EyE~(e,,y~ + g~b,,x)(Ctx,~ + sy~ly,)" (p. 208). This follows by applying Peirce's s e c o n d rule to the p r o d u c t of the two premises, identifying h with x, u with i, y with j, and k with v. Lastly, a p p e a l i n g to the Boolean calculus, Peirce states:

lo9

FROM PEIRCE TO SKOLEM

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) T h a t is, d r o p p i n g the subscripts a n d i g n o r i n g 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" distribute over o n e another, we can multiply this expression out to obtain ec~ + e sl + gbc~ + gbsl. T h e t e r m gbsl drops out, a n d we can replace esl by el a n d gba by bc~ to get ec~ + el + bc~, which is the result that Peirce obtains. In summary, the c o n c l u d i n g material of Peirce's p a p e r is a p r e c u r s o r of p r e n e x first-order predicate calculus, but without f u n c t i o n symbols, constants, or equality. Peirce interprets logical o p e r a t i o n s in propositions as arithmetical o p e r a t i o n s in his semantics of propositions. Thus, a r i t h m e t i c is the basis for i n t e r p r e t i n g propositions. H e also sees linear algebra in the semantics of logic t h r o u g h Boolean matrices. This can be viewed as a r e d u c t i o n i s m , an algebratization a n d a r i t h m e t i z a t i o n of the semantics of logic. This same kind of r e d u c t i o n i s m was p r e s e n t in Peirce's father's work. Peirce senior identified c o m p l e x n u m b e r s with matrices using an e m b e d d i n g in 2 x 2 real matrices. Following in those footsteps, the y o u n g e r Peirce tried to build an i n t e r p r e t a t i o n of logic based on linear algebra and arithmetic.

5.5. Summary of Peirce's Accomplishments in 1883 5.5.1. S y n t a x a n d Semantics In his 1883 p a p e r Peirce discovered the syntax of p r e n e x formulas of p r e d i c a t e logic. However, the m o d e r n general inductive definition of ( n o n p r e n e x ) formulas is absent. Instead, he gives an i n f o r m a l semantics of p r e n e x formulas, based on an arithmetical i n t e r p r e t a t i o n of the logical operations.

110

P E I R C E ' S ALGEBRA OF RELATIVES

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 operation symbols, but in 1883 he makes a c o m m i t m e n t to spell out what the calculus of relatives is and how one puts things together in the system.

5.5. 2. Quantifiers Quantifier theory begins with Aristotle's syllogistic, and m o d e r n set theory and propositional theory begin with Boole. Boole's theory of quantifiers was practically nonexistent. Peirce's original theory used only relative p r o d u c t to capture (some) instances of quantification; we believe he t h o u g h t that relative p r o d u c t was m o r e 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 p a p e r Peirce interprets E and II arithmetically. He introduces the quantifiers as operations on propositional functions of i, j from a d o m a i n , and not as formal language quantifiers on formal expressions. It is somewhat difficult to describe the difference between the two. O m i t t i n g 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. O f course, we do the same thing, typically when we write 4~, which is s h o r t h a n d 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 operations 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 interpretation of plus as inclusive "or" or as the least u p p e r b o u n d in the two-element distributive lattice of truth values. This can also be r e g a r d e d as an operation on truth functions of i,j. This is quite a m o d e r n flavor, considering that Peirce has already laid out distributive lattices m o r e or less axiomatically in the paper. Similarly,

~i~jlii > 0

FROM PEIRCE TO SKOLEM

111

says that the Boolean sum is nonzero, and hence is equal to 1 in the two-valued interpretation. At several points, including here, Peirce waffles on w h e t h e r the operations are arithmetic or Boolean. T h e formulas on the very last pages of the paper, for example,

II, Ej(l),i(b),j,

II,~ i (l),j(b)j,,

F,,F,,IIj (lq + bjk) ,

also look m o d e r n 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 r e n e x formulas over the d o m a i n over which i a n d j range. T h e i and j act like individual variables, being indices ranging over a domain. T h e E and H m i r r o r existential and universal quantifiers, being the c o r r e s p o n d i n g least u p p e r b o u n d and greatest lower b o u n d over indices over a domain. Since this is d o n e 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 definition of truth value for a formula are not there. Perhaps we should regard Peirce's 1883 p a p e r as addressing not syntactical p r e n e x predicate logic, but rather quantifiers as infinitary least u p p e r b o u n d and greatest lower b o u n d operations on propositional 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 c o m p a r e d 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 c o m p l e x and many identities, the representation 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 a n d products are present, but where both the n e e d e d 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 quantifier calculus. H e r e 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

112

P E I R C E ' S ALGEBRA OF RELATIVES

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 intervening 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. T h e n (l t b)ik means Ilil Ov 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 formalities. Fourth, it requires the constant introduction of 0, 00, l, i, in complicated ways, the meaning of which is far from evident, and loading 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 t h a t relatives, i n t r o d u c e d first by De M o r g a n ( n o t B o o l e ) , were u n d e r s t o o d by Peirce, with relative p r o d u c t as t h e basic a l g e b r a i c o p e r a t i o n . H e u n d e r s t o o d t h a t t h e r e was a l i n e a r associative a l g e b r a with o r d e r e d pairs as a basis, a n d t h a t r e l a t i o n s h a d r e p r e s e n t a t i o n s as s u m s of these pairs, with coefficients 0, 1. H e p u t syllogisms into relative calculus a n d , following Mitchell, a d o p t e d q u a n tifiers with s u m a n d p r o d u c t n o t a t i o n s , b o t h as an a d d i t i o n to relative calculus a n d as s o m e t h i n g to investigate o n 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 Mitchell's n o t a t i o n by p u t t i n g the s u m s a n d p r o d u c t s all in f r o n t in t h e style o f calculus a n d infinite series, r a t h e r t h a n as subscripts. In o t h e r words, P e i r c e is n o w p o i s e d to d e v e l o p w h a t we w o u l d call a calculus o f p r e n e x formulas.

6. Peirce's Logic of Quantifiers" 1885

Introduction Peirce's p a p e r " O n the a l g e b r a of logic: A c o n t r i b u t i o n to the p h i l o s o p h y of n o t a t i o n " was written in the s u m m e r of 1884, while Peirce still h e l d an a c a d e m i c position at the J o h n s H o p k i n s University. It was p u b l i s h e d in the AmericanJournal of Mathematics a n d was to be P e i r c e ' s last technical p a p e r on logic to a p p e a r in a m a j o r scientific j o u r n a l . T h e final two sections of this paper, "First-intentional logic of relatives" a n d "Secondi n t e n t i o n a l logic," p r e s e n t a p r e n e x f o r m of first- a n d s e c o n d - o r d e r p r e d i c a t e logic, which is, in retrospect, o n e of his m o s t i m p o r t a n t scientific a c h i e v e m e n t s .

6.1. On the Derivation of Logic from Algebra In the first section of his 1885 paper, Peirce m a k e s a c u r i o u s a n d int e r e s t i n g o b s e r v a t i o n a b o u t the n a t u r e of m a t h e m a t i c s , which is an ant e c e d e n t to s o m e very influential p a p e r s of the m i d - 2 0 t h century, in particular, E u g e n e W i g n e r ' s "The u n r e a s o n a b l e effectiveness o f m a t h e matics." Peirce's a r g u m e n t does n o t deal with exactly the s a m e q u e s t i o n as Wigner's, b u t 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. O n e might naively assume, then, that the apparently m u c h larger experimental repertoire of the physicist would be reflected in a much m o r e 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 c o m p u t e r 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 deduction of the motions of the planets was based on laws obtained by generalizing Galileo's observations of falling bodies and cannonball trajectories, and on Kepler'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 m e t h o d 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 further 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).

FROM

PEIRCE

TO SKOLEM

115

m a n y e x p e r i m e n t s we m a k e in the way of c o n s t r u c t i n g e x a m p l e s a n d c o u n t e r e x a m p l e s . As for physics, will it ever be d o n e either? Peirce t h e n 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) T h e first objective is to e x t e n d logical algebra over the whole of its realm (logic). To Peirce, "logical algebra" m e a n t Boole's algebra of logic. Since he had already e x t e n d e d logic to include the realm of relatives, a n d Mitchell had e x t e n d e d it to contain p r e n e x quantifiers, we can safely assume that Peirce h e r e intends to e x t e n d logical algebra to the realm of relative algebra plus quantifiers. T h e s e c o n d objective is to illustrate "the principles which u n d e r l i e all algebraic notation." Since in his 1870 p a p e r Peirce h a d already stated the laws that he felt should hold before an addition or multiplication symbol could be used for an operation, this s e c o n d objective can be viewed, in m o d e r n terms, as explaining the abstractness of axioms and what a m o d e l of abstract o p e r a t i o n symbols with axioms is (i.e., what an algebra is). This follows Peirce's father's abstract a p p r o a c h to linear associative algebra. T h e third objective is to m a k e clear what rules of i n f e r e n c e are used for d e d u c i n g c o n s e q u e n c e s from premises, in algebraic terms if possible. It is h e r e that we would e x p e c t the r e a p p e a r a n c e of the notion of a prepartial o r d e r from Peirce's 1880 paper, in which the d e d u c t i o n rules allowed him to have a transitive closure such that b d e d u c i b l e f r o m a is a prepartial o r d e r a_< b. T h e r e is h e r e an unstated t h e m e that, if premises are written algebraically a n d there is a desired conclusion, then algebraic transformations of the premises using algebra laws plus the i n f e r e n c e partial o r d e r (i.e., a _< b if a n d only if a implies b) will lead to the desired conclusion if the latter is a c o n s e q u e n c e of the premises. This idea surely goes back to Leibniz, who m a d e an explicit proposal for a five-year t e a m project to c o m p l e t e l y formalize all logical deductive r e a s o n i n g for use as a p r o o f tool.

116

PEIRCE'S LOGIC OF QUANTIFIERS

In this p a p e r Peirce tackles I.eibniz's p r o b l e m , c o n s t r u c t i n g an algebra of logic e x t e n d i n g Boole, using additional o p e r a t o r s and algebraic laws a n d covering relative algebra a n d quantifiers. His p o i n t of d e p a r t u r e is algebra, a n d his a p p r o a c h will be to show that logical ideas can be e n c o d e d within algebra, a l t h o u g h on those occasions w h e r e this does not work out in any simple or natural way, he will a u g m e n t the algebra in o r d e r to be able to a c c o m m o d a t e his u n d e r s t a n d i n g of s o m e logical construct. T h e result of this process will be, he hopes, a generalization of algebra and logic that strongly resembles ordinary algebra. 6.2. N o n r e l a t i v e L o g i c

6.2.1. Embedding Boolean Algebra in Ordinary Algebra Algebra to Peirce was primarily c o m p u t a t i o n a l , a n d the idea of m a k i n g logic algebraic m e a n t to him converting logical a r g u m e n t s into a comp u t a t i o n a l system. H e thus begins with propositions, a n d tries to e m b e d the Boolean algebra of propositions into o r d i n a r y algebra, e x t e n d e d to include two constants, r e p r e s e n t e d by v a n d f. H e 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 truth values will be allowed for truth functions, even t h o u g h arithmetical o p e r a t i o n 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 considering the roots of equations; in this system, 0 is true a n d n o n z e r o is false; p r o d u c t is inclusive "or," a n d sum is "and." T h u s ( x - f ) ( v - x) - 0 is the s t a t e m e n t that x is e i t h e r true or false, i.e., the law of the e x c l u d e d middle. Peirce is h e r e describing a function f r o m logical formulas to algebraic formulas, d e f i n e d by [x] ~ ( v - x ) , [ :~] ~ ( x - f), and [x A y] = [x] + [y] and [x v y] = [x][y].

ax7

FROM PEIRCE TO SKOLEM

Peirce translates the logical statement "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 e n o u g h 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 manipulation he makes is to take (x - f)(v - y) = 0

(1)

( y - f ) ( v - z) = 0,

(2)

( x - f ) ( v - y ) ( v - z) = 0,

(3)

( x - f ) ( y - f ) ( v - z) = 0,

(4)

and

multiplying (1) by ( v - z),

and (2) by (x - f),

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 ordering properties. To explicate this, we give a translation of Peirce's a r g u m e n t into modern equational language. For this purpose, we need to r e p r o d u c e 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 m o d e r n 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 a x i o m s for a B o o l e a n a l g e b r a (B, v , A ,--,, T , F ) , b a s e d o n t h e s e o p e r a t i o n s r a t h e r t h a n o r d e r , r e a d n o w a d a y s as follows: I d e m p o t e n c e : a A a = a, a V a = a; C o m m u t a t i v i t y : 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 i g h 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); Left Distributivity: a A ( b V c ) = ( a A b ) V ( a A c ) , aV(bAc) =(aVb) A

(av c); Truth: avT=T, aAT=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 o b t a i n s t h e 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 associative laws allow us to i g n o r e t h e o r d e r o f f a c t o r s a n d d r o p p a r e n t h e s e s , t h e i d e m p o t e n c e laws allow us to d r o p r e p e a t e d factors, a n d t h e De M o r g a n laws allow us to drive n e g a t i o n s over the other operations. T h e d e r i v a t i o n rules for e q u a t i o n s a r e as follows: D e r i v e a = a for all a; F r o m a = b a n d b - c, d e r i v e a = c; F r o m a = b, d e r i v e b = a; From a=b, derive cVa=cvbforany F r o m a = b , derive c A a = c A b f o r a n y F r o m a = b, d e r i v e "-,a = --,b.

c; c;

T h e s e easily i m p l y t h a t Ifa=band

c=d, then aAc=bAd,

aVc=bvd,--,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 a n e q u a t i o n by any e l e m e n t . W i t h this r e c o n s t r u c t i o n in m i n d , P e i r c e ' s p r o o f r e a d s as follows: T h e p r e m i s e s are --,xV y = T ,

(1)

--,y v z = T,

(2)

f r o m w h i c h we m u s t o b t a i n t h e c o n c l u s i o n --,xV z = T .

FROM

PEIRCE

TO SKOLEM

1

19

Take the s u p r e m u m of (1) on the left with z (i.e., p r e m u l t i p l i c a t i o n ) to obtain zV (--,xV y) = zV T.

(3)

z v T = T.

(4)

By the truth axioms,

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 obtain -',x V (--,yV z) = --,x V T.

(6)

--,xV T = T.

(7)

By the truth axioms,

By the transitivity of equality, (6) and (7) give -',x V (-',y V z) = T.

(8)

Take the i n f i m u m 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 c o m m u t a t i v e 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 p r o p e r t y of equality to obtain (--,x v z) v (y A --,y) = T.

(11)

Apply the law of contradiction ( c o m p l e m e n t ) a n d p r o p e r t i e s of equality to obtain (--,x v z) v F = T.

(12)

Apply the falsity axioms and properties of equality to obtain --,xV z = T. This is an exact m i r r o r of Peirce's a r g u m e n t . T h e alternative partial-order style of p r o o f for this p r o p o s i t i o n is quite different. For propositional functions x and y, x < y is d e f i n e d to m e a n that w h e n e v e r x is true, then y is true. This is the partial o r d e r of propositional functions. Peirce's two premises are equivalent to x < y a n d y_< z. By transitivity of the prepartial truth function o r d e r i n g , x_< z, which is equivalent to the conclusion desired. This is the form the

120

PEIRCE'S LOGIC OF QUANTIFIERS

a r g u m e n t 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 l e m e n t 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 u p p e r 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 c o m m e n t 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.

FROM

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TO SKOLEM

121

S c h r 6 d e r ' s v o l u m e 1 defines B o o l e a n algebras as distributive 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 f o r m u l a t i o n , the law of the e x c l u d e d m i d d l e exactly says that every e l e m e n t has a c o m p l e m e n t , a n d thus c o m p l e t e s S c h r 6 d e r ' s axioms for an abstract B o o l e a n algebra. But this is n o t what Peirce is d o i n g here; he is simply s h o w i n g how o n e writes the law o f the e x c l u d e d m i d d l e in a l g e b r a notation. T h e r e is no special significance to this. Tarski's t h e o r y o f L i n d e n b a u m algebras shows that every B o o l e a n a l g e b r a is i s o m o r p h i c to a L i n d e n b a u m algebra o f a p r o p o s i t i o n a l calculus; that is, is a h o m o m o r p h i c image o f a free B o o l e a n algebra. Lind e n b a u m algebras are exactly the algebras of propositions. Any distinction b e t w e e n g e n e r a l B o o l e a n algebras a n d algebras o f p r o p o s i t i o n s is thus a distinction w i t h o u t a difference.

6. 2.2. Five Peirce Icons In his t r e a t m e n t o f propositional logic, Peirce restates the rules f r o m his "algebra o f the copula" in his 1880 paper, which he now calls "icons of algebra." T h e first of these icons is his principle of identity f r o m 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 p r i n c i p l e of illation, which is n o t h i n g o t h e r t h a n the i n t r o d u c tion a n d e l i m i n a t i o n rules for implication, says "from x we can infer y" if a n d only if we can infer "x implies y." Applying this to the first icon, we get that f r o m x we can infer x, which is r o u g h l y his own e x p l a n a t i o n . T h e s e c o n d icon restates the s e c o n d rule o f 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-.~..... >.,, is satisfied. Thus we obtain o n e or m o r e solutions of the first step, that is, assignments d e t e r m i n i n g the classes and relations in such a way that Ul. 1..... ~.>.~..... y. is satisfied. (Skolem 1923, p. 294) W e n e x t a s s o c i a t e w i t h e a c h level k a s i n g l e f i n i t e 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 h e d i s j u n c t i o n o f t h e f i n i t e l y m a n y g r o u n d instances

of

X

of

the

X1 . . . . . Xm is a s e q u e n c e sponding

sequence

x ( X 1, . . . , X m,

Y1. . . . .

Y,)

f r o m {1 . . . . . Ck}" a n d

form

Y1. . . . .

11, is t h e c o r r e -

of successive integers introduced

n a m e s to w i t n e s s x ( X ~ . . . . .

X .... Y1 . . . . .

such

that

at s t a g e k + 1 as

Y,,). S k o l e m e x p r e s s e s this as:

T h e second step consists in choosing, for x~ . . . . . x,,,, every p e r m u t a t i o n with repetitions of the n + 1 n u m b e r s 1,2 . . . . . n + 1 taken m at a time, with the exception of the p e r m u t a t i o n 1,1 . . . . . 1, already c o n s i d e r e d in the first step. For at least one of the solutions o b t a i n e d in the first step, it must then be possible, for each of these (n + 1) .... 1 p e r m u tations, to choose y~ . . . . . y,, a m o n g the n u m b e r s 1,2 . . . . . n + 1 + n ( ( n + 1 ) ' - 1) in such a way that, for each p e r m u t a t i o n x I. . . . . x,,, taken within the s e g m e n t 1,2 . . . . . n + 1 of the n u m b e r s e q u e n c e , the proposition U,,~......... y~..... y, holds for a c o r r e s p o n d i n g choice of Yl . . . . . y,, taken within the s e g m e n t 1,2 . . . . . n + 1 + n ( ( n + 1)"' - 1). T h u s from certain solutions gained in the first step we now obtain certain continuations, which constitute solutions of the s e c o n d step. It must be possible to c o n t i n u e the process in this way indefinitely if the given first-order proposition 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 c a n b e s h o w n , u s i n g t h e 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 h e 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 b e 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 w a y t h a t e a c h

i n s t a n c e o f ~ is t r u e . B u 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 . R a t h e 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 r d e r now to obtain a uniquely d e t e r m i n e d solution for the entire n u m b e r s e q u e n c e , we must be able to choose a single solution from a m o n g all those o b t a i n e d in a given step. To achieve this, we can always take the first from all of the solutions o b t a i n e d in an arbitrary step, o n c e they have b e e n o r d e r e d in a s e q u e n c e in the following way. (Skolem 1923, p. 294)

SKOLEM'S RECASTING

202

We n o w have, in p r o p o s i t i o n a l logic, an infinite s e q u e n c e o f finite d i s j u n c t i o n s ~,, e a c h satisfiable by at least o n e t r u t h v a l u a t i o n Tk o n its a t o m i c s e n t e n c e s . In a d d i t i o n , e a c h d i s j u n c t i o n ~k+l i m p l i e s ~bk, in t h a t every v a l u a t i o n m a k i n g t h e first t r u e m a k e s t h e 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 n u m b e r 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 p o i n t , we m u s t i n v o k e t h e 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 f o r m in o r d e r to c o n c l u d e : since every finite s u b s e t o f this c o u n t a b l e set o f p r o 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 r e satisfiable at o n c e by a single t r u t h v a l u a t i o n T. This T d e f i n e s t h e r e q u i r e d m o d e l , a n d the d o m a i n is t h e set o f all i n t e g e r s o c c u r r i n g in at least o n e ~bk. T h e a t o m i c r e l a t i o n s d e c l a r e d t r u e in that d o m a i n a r e t h o s e v a l u e d t r u e by T. O f c o u r s e , the c o m p a c t n e s s t h e o r e m h a d n o t b e e n f o r m u l a t e d in 1922. ( S k o l e m d e l i v e r e d this p a p e r as an a d d r e s s in J u l y o f 1922.) W h a t d i d S k o l e m offer as proof?. This is t h e o n e p o i n t o n w h i c h t h e 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 continuation of L,,,L, , ( n > v). Then, if n'> n, a~ _< a,. But, since the n u m b e r 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 n u m b e r sequence, q.e.d. (Skolem 1923, p. 294) n

ta t

An i n c o m p l e t e a p p e a l s e e m s to be m a d e at this p o i n t to i n f i n i t a r y p r o p o s i t i o n a l logic, b e c a u s e we are l o o k i n g at t h e c o n j u n c t i o n o f an infinite n u m b e r o f finite d i s j u n c t i o n s . T h e intuitive b u t n o t p r e c i s e a r g u m e n t w o u l d be to write this u s i n g a distributive law as a d i s j u n c t i o n o f infinite c o n j u n c t i o n s . E a c h o f t h e c o n t i n u u m - m a n y infinite c o n j u n c tions is a c o n j u n c t i o n o f o n e c h o i c e o f an a t o m i c s t a t e m e n t f r o m e a c h ~bk. Since t h e d i s j u n c t i o n is n o n z e r o , a single infinite c o n j u n c t i o n is

FROM

PEIRCE

TO SKOLEM

203

nonzero, and this defines the required truth valuation. Notation 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 c o n t i n u u m in number. This is the best we can do to conjecture as to what Skolem and L 6 w e n h e i m 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 argum e n t here because the distributive law used (which is correct for sets) does not hold in these infinitary propositional logics. The compactness t h e o r e m 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 r g u m e n t 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 u n d e r 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 statement true form, u n d e r 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 t h e o r e m for predicate logic. Divorced from inessential details of notation, he takes the phrase "inconsistent" used in Skolem's (1920) formulation of the L6wenheim t h e o r e m in a semantic sense of having no models and checks that, if a formal definition of consistency relative to p r o o f rules is substituted, exactly the a r g u m e n t just given still produces a model. That is, if a statement does not lead to a formal contradiction, then its Skolem normal form does not either, and the step-by-step construction 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 s u p p l e m e n t e d by the Skolem (1929) a r g u m e n t can be carried out by simply observing what formal deduction rules are n e e d e d to replace the use of elements of M. What was an obscure tree a r g u m e n t in L6wenheim (1915) becomes a very simple closure a r g u m e n t in Skolem (1920), but in constructing it, Skolem uses the axiom of choice. It is transformed by Skolem (1923) into a choiceless argument, by transforming to a world of constants. He

204

SKOLEM'S RECASTING

justifies that the n e e d e d disjunctions are true by semantic i n t e r p r e t a t i o n in the m o d e l M. In 1929 G6del t r a n s f o r m e d this p r o o f to observe that the hypothesis of formal consistency is sufficient to replace the hypothesis of truth in a m o d e l M. In this way the G6del c o m p l e t e n e s s t h e o r e m was born. H e r b r a n d (1930) took the p r o o f and replaced all existential quantifiers by function symbols and thus replaced the r e q u i r e m e n t of satisfying the original s t a t e m e n t with satisfying an equivalently satisfiable universal statement. This he does using all the terms built up from a c o n s t a n t symbol using the function symbols as his d o m a i n , or a Herb r a n d universe. He proceeds after that to find, for a consistent universal statement, a similar family of disjunctions as above. In o t h e r 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. T h e existence of the single truth valuation a n d the m o d e l in which the universal s e n t e n c e is satisfied if consistent are f o r m u l a t e d in terms of instantiations in the H e r b r a n d universe, not in the integers. H e r b r a n d was a constructivist who did not accept the K6nig l e m m a as constructive. H e did not accept the Bolzano-Weierstrass t h e o r e m either. Thus, his f o r m u l a t i o n and interests went in a different direction. We m i g h t say that, d u e to his constructivist prejudice, he refused to c o n c l u d e the c o m p l e t e n e s s t h e o r e m , a n d the credit went to G6del. In the last p a r a g r a p h 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. (Skolem 1929, p. 273) T h e astute r e a d e r may note that there is an unstated a s s u m p t i o n b e h i n d this paragraph. Skolem a n d Fraenkel had freed set theory of the s e c o n d - o r d e r quantifier "for all properties P of sets," as in Zermelo,

FROM

PEIRCE

TO SKOLEM

205

in favor of first-order expressible properties (classes), and it was becoming a p p a r e n t that c o n t e m p o r a r y mathematics could be d o n e in this purely first-order system. Skolem then reverses g r o u n d and assumes that every mathematical notion should be stated in first-order logic. Then, with the Skolem-L6wenheim theorem, he finds that set theory c a n n o t specify the notion of cardinal, in that sets in a one-to-one correspondence in one model (by a c o r r e s p o n d e n c e in the model as a set) may not be in one-to-one c o r r e s p o n d e n c e in a n o t h e r model. Skolem then, having assumed that all formal theories really should be expressible in first-order logic without saying so explicitly, asserts a rather absolute noncharacterizability of mathematical notions such as cardinality. Von N e u m a n n and Bernays saw this gap, and when they i n t r o d u c e d 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 m o r e or less i n d e p e n d e n t l y 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.

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Appendix 1" Schr6der's Lecture I

Introduction T h e r e are s o m e i m p o r t a n t m e t h o d o l o g i c a l points o f S c h r 6 d e r ' s work that can be e x p l a i n e d by a careful r e a d i n g o f this l e c t u r e as a selfc o n t a i n e d whole, w i t h o u t i m p o s i n g the p r e j u d i c e s or l a n g u a g e o f later g e n e r a t i o n s . Peirce p r e f e r r e d implication i n t e r p r e t e d as i n f e r e n c e as the basis o f logic, m u c h as in natural d e d u c t i o n systems today. S c h r 6 d e r explicitly a c k n o w l e d g e s this, a n d claims to be d e v e l o p i n g the a l g e b r a c o r r e s p o n d i n g to logic, n o t logic itself. His a l g e b r a is b a s e d o n e x t e n d i n g the f o r m a t of the identity calculus to a calculus o f relations. T h e identity calculus is formal B o o l e a n a l g e b r a based o n the B o o l e a n o p e r a t i o n s a n d identities alone. Thus, De M o r g a n ' s law a n d the distributive law are stipulations (axioms) for him. T h e rules o f i n f e r e n c e are the infere n c e rules for equality, a n d the reilexive, symmetric, transitive, a n d substitution o f equals for equals rules. In his algebraic setup, every p r o p osition is stated as an identity, a n d proofs are essentially strings of identities, e a c h o b t a i n e d f r o m previous ones by these rules. This is in c o n t r a s t to Peirce's p r e f e r e n c e for logic systems based o n m o d u s p o n e n s a n d 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 rules r a t h e r than a l g e b r a systems based solely o n identities. T h e s e two f o r m u l a t i o n s are, however, fully equivalent. W h e t h e r o n e does cylindric or polyadic algebras b a s e d o n identities, or p r e d i c a t e logic based on Hilbert-style axioms a n d m o d u s ponens, is a m a t t e r o f taste a n d historical p r e c e d e n t . S c h r 6 d e r h a d an algebraist's taste, Peirce a logician's. ~ S c h r 6 d e r has often b e e n accused o f n o t b e i n g able to distinguish a f r o m 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

SCHRODER'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 s u p r e m u m s 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 u p p e r 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 r e a t m e n t of least u p p e r bounds a n d greatest lower bounds. This is the way he thought. If we start with a prespecified collection of atoms and build in m o d e r n terms the complete completely distributive Boolean algebra it generates, we get what S c h r 6 d e r specified as the unary relatives of a d o m a i n with superscript 1. He does use the infinite distributive laws in later lectures, assuming t h e m 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 g e n e r a t e d structures, the question of unit sets versus their elements simply does not arise. Given a d o m a i n of atoms, Schr6der introduces the set of o r d e r e d pairs of atoms, without worrying about what an o r d e r e d pair is. Individual binary relatives are simply individual o r d e r e d pairs. Taking the o r d e r e d pairs of the d o m a i n as atoms, he then builds the complete, completely distributive Boolean algebra they generate, now having least u p p e r 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 i n t r o d u c e d 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 elements and their unit sets simply does not come up. T h e r e is absolutely no analysis of the notion of o r d e r e d pair. O n e simply takes two atoms a n d puts a semicolon in between the symbols that d e n o t e them. This is a very algebraic way to proceed. In Schr6der's original text, Schr6der typographically distinguishes his c o m m e n t s from his main discussion by setting them in smaller type; we have d o n e the same here and in the appendices that follow.

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First Lecture

Introduction w 1. Outline. The Operational Sphere of the Algebra of Binary Relatives Page

1

Page 2

c~) It is a grandiose discipline, rich in expressions and powerful m e t h o d s of inference, almost too rich in propositions (although they are of matchless p r o p o r t i o n ) , into which I will try to introduce the reader. Although the first beginnings of this discipline--with Augustus De M o r g a n - - d a t e back no further than to the middle of this century, the literature in this field is already of considerable bulk; the u n d e r s t a n d i n g 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 c h a n g e d 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 " r o u n d 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 n u m b e r of directions into which it is capable of being developed, the multitude of applications in highly diverse fields--to which the notions "finiteness" [Endlichkeit], "number" [Anzahl], "function," and "substitution" belong as m u c h as, for example, the term " h u m a n relations h i p s " m a n d 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 n u m b e r of accurately formulated, f u n d a m e n t a 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 m u c h later lectures to the foundation of the discipline, in o r d e r 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 interpretations of expressions or formulas of the calculus appear, at most, at the same time in the form of side glances, with the intention 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 10

SCHRODER'S

LECTURE I

additional details and summarize only later in a separate section what o u g h t to be said to acknowledge the contributions of o t h e r 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 n u m e r o u s 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. T h e r e will be cases where both are used d i f f e r e n t l y m a l t h o u g h not in propositions [Aussagen]myet for binary relatives (as well for those of a still h i g h e r o r d e r ) m a point to which I will return later. After having said all this, I immediately p r o c e e d to try: /3) First, to give a brief overview of the operational sphere [Operationskreis] of relative l o g i c ~ b y c o m p a r i n g it to the operational sphere of Page 3 arithmetical algebra. I thereby focus exclusively on the most i m p o r t a n t part of the former, the Algebra of Binary Relatives (called "dual relatives" by Peirce), which constitutes the natural point of d e p a r t u r e of the whole theory. It alone has so far e x p e r i e n c e d some upgrading, and perhaps the science may limit itself to it in o r d e r to deal with its most i m p o r t a n t problems. In the identity (system [Gebiete] or class) calculus, we had to get acq u a i n t e d with three modesm"species"---of calculation [Rechnungsarten]: identity multiplication, identity addition, and negation. O f these, the first two are "tying' [kniipfende] operations which presuppose at least two o p e r a n d s (terms) as given for their execution; the latter, a "nontyin~' operation, can be carried out on one o p e r a n d (term). T h e 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 Hauptstufe] of e l e m e n t a r y operations. But three m o r e species have to be a d d e d as a second main stage [zweite Hauptstufe]: the three "relative" elementary operations, that is relative multiplication (or composition), relative addition, and conversion; the first two are tying and associative, but (in general) not commutative operations; the latter is a nontying operation 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,

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211

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 t h o r o u g h g o i n g 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, c o u p l e d as a tetrad, a quadruple, a "group"[Gespann] of propositions (or formulas); by means of contraposition, 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 conditions and guarantees. 7) T h e r e f o r e , if the identity calculus appears as a m e r e part of relative l o g i c B t h e most e l e m e n t a r y u t h e latter being thus an extension [Erweiterung] (special m o d e of application and continuation) of the former, there are obviously two possibilities for founding the algebra of relatives. T h e first: as a continuation of the present course [Lehrgang] in which we started from the notion of subsumption in o r d e r to reach at the e n d a scientific definition of the individual. T h e 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 d e p a r t u r e from the consideration of "elements" (or individuals), and c o m p a r i n g the thereby created, totally distinctive foundation for the entire logic with o t h e r foundations can only be instructive. We therefore follow this course, especially since the "continuation" we m e n t i o n e d will be very easy and quickly accessible.

w 2. The Domains of Various Orders [Ordnung] and Their Individuals As given, s o m e h o w conceptually d e t e r m i n e d , we consider the following "elements" or individuals A,B,

C,D,E,...

1)

of a " c o m m o n " 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 12

SCHR(~DER'S LECTURE I

a n d they have to e x c l u d e each o t h e r (to be mutually disjoint), so that n o n e of the e l e m e n t s could be i n t e r p r e t e d 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 e p r e s e n t the totality of the e l e m e n t s c o n s i d e r e d by linking their n a m e s with a plus sign, as an "identity sum" (logical a g g r e g a t e ) a n d call it the original or universe of discourse of the first order [Denkbereich der ersten Ordnung]: 11 (read: o n e to the power o n e ) , so that II=A+B+C+D+

... "

2)

T h e universe of discourse 11 is to c o n t a i n more than one e l e m e n t . This a s s u m p t i o n is necessary for the validity of almost all the p r o p o s i t i o n s of this theory. We will call the case w h e r e the universe c o n t a i n s but one e l e m e n t , the exceptional case. For s o m e formulas it will even be necessary to assume that the universe c o n t a i n s more than two e l e m e n t s in o r d e r to claim validity. Such f o r m u l a s are to be m a r k e d by an asterisk ( . ) a d d e d to the n u m b e r of the formula. In addition, the set [Menge] of e l e m e n t s which o u r universe c o m p r i s e s can be "finite" [endlich] (or b o u n d e d [begrenzte]), w h e r e b y the universe consists of an arbitrarily c h o s e n " n u m b e r " [Anzahl] of e l e m e n t s . O r the system of e l e m e n t s is "infinite" [unendlich] ( u n b o u n d e d [unbegrenzte]); h e r e o n e c a n n o t talk of its "number." In the latter case, the e l e m e n t s may be "discrete," such as when f o r m i n g 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 e x a m p l e , a " c o n t i n u u m " as the points of a line, a surface, a solid, in p a r t i c u l a r of a straight 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"numPage 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 on a straight line (viz., a is generally known, the of a part of it, such as its

"prefer' a universe which consists of all the poznts straight line o p e n at both ends) (to which, as real n u m b e r s c o r r e s p o n d ) = - o r consisting only half: that ray to which the positive real n u m b e r s

FROM PEIRCE TO SKOLEM

2x3

c o r r e s p o n d - - o r p e r h a p s simply of the integers consisting of e q u i d i s t a n t points on o u r straight line, or their positive half. However, this will always have to r e m a i n u n i m p o r t a n t . In o u r theory, we may n o t p r e s u p p o s e that the e l e m e n t s are in a definite 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 immediately 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. T h e " e l e m e n t s " n e e d not exist simultaneously; they may r e p r e s e n t "events" f r o m the past, present, a n d future (coexistent or s i m u l t a n e o u s , as well as successive). It is sufficient that they can be thought together. For the p u r p o s e s of exemplifying a logical system, it is r e c o m m e n d e d to i n t e r p r e t the e l e m e n t s as "persons" of the h u m a n society or m a n k i n d in g e n e r a l .

Page 7

In a d d i t i o n to u p p e r c a s e r o m a n letters which have to stand for definite e l e m e n t s , in case we wish to call special a t t e n t i o n to s o m e of t h e m , we also n e e d a c a t e g o r y of signs to r e p r e s e n t or as n a m e s for i n d e f i n i t e or general e l e m e n t s . This n e e d a p p e a r s already at the first a n d m o s t simple a c t m w h i c h we are n o t g o i n g to deal w i t h m t h e act of p u t t i n g any two e l e m e n t s in relation [Beziehung] to e a c h o t h e r or to c o n s i d e r t h e m f r o m such a perspective. T h e e l e m e n t s b e t w e e n which, let us say, "a relationship" [Verhiiltnis] is said to be established, can be e i t h e r different, or they can also be the same or "equal." T h e perspective, the "fundamentum relationis," can, for e x a m p l e , be the love of o n e p e r s o n for a n o t h e r . W h e n p e r s o n A loves p e r s o n B, the pair of e l e m e n t s is c o n s i d e r e d as "A : B" f r o m this perspective. If, at the s a m e time, p e r s o n B does n o t love p e r s o n A, the pair of e l e m e n t s "B: A" (which t h e r e f o r e is to be d i f f e r e n t i a t e d f r o m the previous pair) c a n n o t be c o n s i d e r e d . If p e r s o n A loves himself, the "pair of e l e m e n t s .... A : A " has to be considered. T h e s e two cases c a n n o t be dealt with t o g e t h e r D w i t h i n the limitations of the p r e s e n t stock of s i g n s D b e c a u s e the a s s u m p t i o n B - A , which would s u b s u m e the s e c o n d case to the first, is in c o n t r a d i c t i o n with the c o n d i t i o n A 4: B, which was i n t r o d u c e d a n d has to be firmly m a i n t a i n e d . With this p r o c e d u r e , o n e will not go b e y o n d sticking to e x a m p l e s .

214

SCHRt~DER'S LECTURE I

We n e e d new s i g n s m a n d these we select, for the time being, exclusively f r o m the following s e q u e n c e :

i j, h , k ,

l, m, n , p ,

q

3)

r a i n o r d e r to r e p r e s e n t any one of the e l e m e n t s A, B, C, D . . . . of o u r universe 1 l. If we now take the pair of e l e m e n t s i : j again to state that p e r s o n i loves p e r s o n j, both the a s s u m p t i o n j = i a n d the a s s u m p t i o n j g: i rem a i n valid; all advantages of generality for o u r e x a m i n a t i o n are assured, in a d d i t i o n to the advantages which the i n t r o d u c t i o n of a m i n i m a l sign l a n g u a g e [ Zeichenschrift] g u a r a n t e e s . Thus, while two of the letters A, B, C . . . . always have to r e p r e s e n t two d i f f e r e n t e l e m e n t s , two d i f f e r e n t letters f r o m the s e q u e n c e i, j, h . . . . are n o t subject to this restriction. T h e r e a s o n for this d i f f e r e n c e lies in the fact that while A, B, C . . . . r e p r e s e n t to us definite, that is to say, "specifiear' e l e m e n t s , the symbols i, j . . . . are used as r e p r e s e n t a t i v e s of Page 8 s o m e , of any, unspecified or "generalelements." In a r i t h m e t i c , the f o r m e r c o r r e s p o n d to n u m b e r s , the latter to literals or variables. As the latter, they are essential a n d allow us to realize a n a l o g o u s advantages. T h e g e n e r a 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 c o n s i d e r t h e m as such for the time being. I n d e e d , now we can write the e q u a t i o n , 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 c o r r e c t i n t e r p r e t a t i o n , for an "analysis" or "evaluation" of the righth 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 m e a n i n g , o r E a s o n e may s a y E t o "run t h r o u g h " the totality of individuals of the universe 11. This process, which in our theory is to be thought through with each expression 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. T h e r e f o r e , every s u m m a t i o n sign related to a symbol of s e q u e n c e 3) has to be u n d e r s t o o d with the "extension" [Erstreckung] d e s c r i b e d b e f o r e (unless otherwise stipulated). T h e universe of discourse 11 forms a m a n i f o l d [Mannigfaltigkeit] for which the e n t i r e "identity calculus" could be used without p r o b l e m s . But we shall not use this fact until we get to a new f o u n d a t i o n of the latter.

FROM P E I R C E T O SKOLEM

2 15

E a c h (identity) sum of elements o f this u n i v e r s e 1' will simply be called an "absolute term," later on, a "system" (Gebiet) o r a "class" (class t e r m ) . N o w we can f o r m u l a t e in g e n e r a l w h a t was s h o w n with t h e a b o v e example: If, in o u r u n i v e r s e 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 w h a t t h e r e a s o n , n o Page 9 m a t t e r w h a t t h e p e r s p e c t i v e - - t h e result of t h e j u x t a p o s i t i o n m a y be r e p r e s e n t e d by m e a n s o f a c o l o n

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 c o m m o n language when the words "relation" and "ratio" are almost synonymous. Furthermore, the arithmetical 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," t h a t is to say, i is t h e a n t e c e d e n t (the first t e r m ) o r t h e relate, j is the c o n s e q u e n t (following t e r m ) o r t h e c o r r e l a t e o f the pair. After w h a t has b e e n said, j : i a p p e a r s as another p a i r o f e l e m e n t s t h a n i: j, as s o o n as j is d i f f e r e n t f r o m i. O r we m a y s t i p u l a t e

(i =j) = (i : j = j : i),

(i e: j) - (i : j e: j : i)

6)

as valid for all i a n d 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.

216 Page 10

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 r e m a r k , too, serves only for the p r e s e n t i n t r o d u c t i o n ; 6) will be proved as a proposition later f r o m the fundam e n t a l stipulations of the next section. T h e same is true for the s t a t e m e n t

i:j , 0

7)

by which we express that each pair of e l e m e n t s is different from nothing. T h e pair of e l e m e n t s j : i is called "conversg' of the pair of e l e m e n t s i:j. Because of the general validity of this stipulation, it will be permissible to e x c h a n g e the terms i a n d j; t h e r e f o r e also the pair of elem e n t s i : j has to be converse to j : i. T h e relationship b e t w e e n converse pairs of e l e m e n t s is reciprocative [gegenseitig]. All conceivable pairs of elements, for the f o r m a t i o n of which the universe of discourse 1 ~ provides the elements, may be arrayed or arr a n g e d in a "block" or tab/e:

A:A, A'B, A'C, B : A , B B, B C, C : A , C B, C C, D : A , D B, D C,

A'D .... B D, C D, D D,

8)

We want to r e m a r k that these "specified" or special pairs of e l e m e n t s will have to be set forth or d e n o t e d as "individual binary relatives" of which any one can be generally r e p r e s e n t e d by i:j. T h e totality of these individual binary relatives or pairs of e l e m e n t s forms a new, a u n i q u e universe of discourse which we designate as the

"universe of discourse of the second ordd' [Denkenbereich der zweiten Ordnung], r e p r e s e n t e d 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 p a r e n t h e s e s a r o u n d the pairs of e l e m e n t s can also be left out---or the abbreviation, which is possible because of the s u m m a t i o n Page 11 sign, 12 - E~F~,(i'j) = E, Ei(i'j) = ~o(i'j).

10)

217

FROM PEIRCE TO SKOLEM

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 [Verbindungen] (combinations) and arrangements (permutations). It a p p e a r s as self-evident (but we will not m a k e essential use of it) that this universe of discourse, too, p r e s e n t s a m a n i f o l d on which the identity calculus can be applied. T h e investigation of the t h e o r y in this v o l u m e will deal with this universe almost exclusively. T h e r e f o r e , we will d e n o t e it for short with a I only---dropping m o s t of the time the e x p o n e n t 2 in all f o r m u l a s (and m o r e rarely in the text). To s u m m a r i z e , we r e p e a t e q u a t i o n s 9) a n d 10) in their m o s t simple m o d e of e x p r e s s i o n l = Eoi'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 a n d in the c o l u m n marked j---if the e l e m e n t s i, j were natural n u m b e r s , we could say in an a b b r e v i a t e d way: in the ith row a n d in the jth column. A l t h o u g h , as already e x p l a i n e d , the s u p p o s i t i o n of a definite o r d e r or a r r a n g e m e n t of the e l e m e n t 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 e m e n t s of the s e c o n d universe; we shall nevertheless a c c e p t the previous m o d e s of e x p r e s s i o n for the sake of c o n v e n i e n c e a n d for overall clarity: If we speak of such individual relatives i ' j , i ' h , i" k . . . . . which all a g r e e in the relate, we will state f r e q u e n t l y that they c o m e f r o m the s a m e horizontal row or line, a n d "the row b e l o n g i n g to i" of the table 12 will simply m e a n the totality (identity sum) of all pairs of e l e m e n t s Page 12 f r o m o u r s e c o n d universe which have i for a relate. Also all pairs of e l e m e n t s i ' j , h ' j , k ' j . . . . . which a g r e e in the c o r r e l a t e , will be attribu t e d to the s a m e vertical row or c o l u m n . And we t h e r e f o r e distinguish in o u r table 12 "rows" of pairs of e l e m e n t 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 l e m e n t s of o u r universe 12, we see two kinds, d e p e n d i n g on w h e t h e r i = j or i ~ j is in i ' j . In the first case, we have an pair of e l e m e n t s of the f o r m i" i. It s h o u l d be called an individual (binary) "'self-relative." In the case o p p o s e d to it of i g= j, we call i ' j an individual (binary)

"'aliorelative."

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 c o m p o u n d 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 a c c o m m o d a t e any of these suggestions. 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. T h e word "other-relative" gives rise to awkward associations in case we have to talk of other relatives or even of other other-relatives. T h e other expressions seem less apt, and cover the idea less accurately. Although "self-relative" does not quite do justice to the international considerations m e n t i o n e d above, I will keep it and leave it to the r o m a n c e 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 t a b l e 12, t h e i n d i v i d u a l self-relatives A : A , B: B, etc., all s t a n d in a s t r a i g h t line w h i c h cuts across this table f r o m t h e u p p e r left to t h e l o w e r r i g h t side, a n d is c a l l e d t h e "main diagonal" o f t h e t a b l e - - a c c o r d i n g to a c o m m o n e x e r c i s e in t h e t h e o r y o f d e t e r m i n a n t s a n d m a t r i c e s . U n d e r t h e 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 l u s t r a t i o n s w h i c h s e e m to p r e s u p p o s e a d e f i n i t e Page 13 a r r a n g e m e n t o f pairs o f e l e m e n t s o n a s u r f a c e - - w e t h u s u n d e r s t a n d n o t h i n g else t h a n t h e totality ( i d e n t i t y s u m ) o f all i n d i v i d u a l self-relatives from our second universe. T h e i n d i v i d u a l alio-relatives lie o u t s i d e , s t a n d at t h e side of, a b o v e and below the main diagonal. Each individual self-relative is the converse of itself. O n t h e o t h e r h a n d , i n d i v i d u a l alio-relatives, c o n v e r s e to e a c h o t h e r , s t a n d "symmetrically" to t h e m a i n d i a g o n a l so that, if o n e looks at t h e l a t t e r as a m i r r o r line, a n y o n e o f t h e two m u s t be t h e m i r r o r i m a g e o f t h e o t h e r . S i n c e we n o w k n o w a b o u t t h e "individual b i n a r y relatives, " t h e quest i o n arises: w h a t d o we a c t u a l l y m e a n by a "binary relativg'?

2x9

FROM P E I R C E TO SKOLEM

A l t h o u g h this question is not to be treated systematically until the n e x t section, we would like to anticipate the answer: it is an identity sum (an aggregate) of any individual binary relatives. We can pick any pairs of e l e m e n t s from o u r universe 12 a n d c o m b i n e t h e m by m e a n s of identity a d d i t i o n m b e it collectively to a "system of pairs of elements," be it generally to a "class of pairs of elements." T h e result is to be called simply a binary relative. T h e perspective from which we p e r f o r m such selections of pairs of e l e m e n t s is primarily that we j o i n all those individual relatives i : j in a class or identity sum of pairs of e l e m e n t s in which the relate i stands to the correlate j in a "relation" of a definite kind, a relation c h a r a c t e r i z e d by a certain "fundamentum relationis," on which o u r interest is mainly concentrated. Just as we did not set any limits to the (wider) e x t e n s i o n a l logic of class f o r m a t i o n , a n d the individuals of a class were not held t o g e t h e r to constitute a regular "concept," c o r r e s p o n d i n g to the r e q u i r e m e n t s of the (narrower) intensional logic, likewise the possibilities for selecting pairs of e l e m e n t s which are to be j o i n e d in a binary relative s h o u l d not Page 14 be limited in any way; this perspective, as the o n e m e n t i o n e d above, may be decisive as a reason for their selection f r o m the universe 12; however, its existence is not an essential c o n d i t i o n for the selections which can be carried out arbitrarily (from the start) (and be m a i n t a i n e d from there on). A c o m b i n a t i o n of any three e l e m e n t s i, j, a n d h f r o m o u r original universe 1~, if they are written in this definite order, may be called an "triple of elements" or an "individual ternary relative" a n d 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 l e m e n t s , constitutes a new universe of discourse which we call the "the universe of discourse of third order" a n d d e n o t e it by 13, so that we can say 1~= E,,EiE,(i'j" h) =Eohi" j" h.

13)

Specifically, the triples of e l e m e n t s can only clearly be r e p r e s e n t e d in the form of a block--filling, for e x a m p l e , a book--on its first page would stand the pairs of e l e m e n t s of 12 in 11) with " : A" a d d e d ; on the s e c o n d page (or better, front page of the s e c o n d sheet), the s a m e pairs of e l e m e n t s with ": B" added; on the third page (or third s h e e t of its front page), the same pairs of e l e m e n t s with " : C" added; a n d so on. But we abstain h e r e from their specific illustration. Mathematically speaking, the universe 1~ consists of the "third class

220

SCHRODER'S LECTURE I

of variations (or p e r m u t e d c o m b i n a t i o n s ) with repetitions" of e l e m e n t s f r o m the original universe 11. A c c o r d i n g to w h e t h e r all t h r e e e l e m e n t s 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 t h r e e ways---only two, or, eventually, n o n e of t h e m c o r r e s p o n d s to o n e of the other, i.e., all t h r e e b e i n g different, we would have to distinguish b e t w e e n five types of individual ternary relatives, for which the e x a m p l e s

A:A:A, Page 15

B:A:A,

A:B:A,

A:A:B,

A:B: C

14)

are illustrative. An identity sum of triples of e l e m e n t s , s o m e h o w selected f r o m the universe 13, is now to be called a "ternary relative." Because all individual t e r n a r y relatives have to be c o n s i d e r e d as differ.ent f r o m each o t h e r a n d f r o m n o t h i n g , the identity calculus (as system, as well as class calculus) is applicable, at least for the time being, to the universe 13 a n d the t e r n a r y relatives possibly c o n t a i n e d in it. A n d so on. It is clear how o n e can c o n t i n u e in this way a n d c o m b i n e all possible q u a d r u p l e s , quintuples, sextuples . . . . of e l e m e n t s of the universe 11 to the universe 14 , 15 , 16 . . . . . of the fourth, fifth, sixth . . . . order, a n d state with it the n o t i o n of quaternary, quinary, senary, ... relative. To c o n c l u d e , we may r e m a r k 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 h o u g h t to be built f r o m e l e m e n t s of the original universe 11, can also be c o n s i d e r e d , r e p r e s e n t e d , or d e n o t e d as "uninary* relatives" (called "simple relatives" by P e i r c e ) - - a s i n d e e d t h e r e is no p r o b l e m to call the e l e m e n t s i of the universe 11 "individual uninary relatives." We t h e n have the u p p e r m o s t reason for the classification of all imaginable relatives: the "order" [Ordnung] of them. We have to distinguish b e t w e e n relatives of the first, second, third, etc., order. A relative of a definite o r d e r is n o t h i n g m o r e than the (identity) sum of any "individual" relatives of the same o r d e r ; by "individual relatives" of a definite o r d e r we have to u n d e r s t a n d the "variations (with repetitions) in the s a m e class" f r o m e l e m e n t s of the first universe of discourse. T h e latter are r e p r e s e n t e d symbolically--for the h i g h e r o r d e r s (the "variations in the first class" are, as is well known, the e l e m e n t s t h e m s e l v e s ) - - b y conn e c t i n g with a colon the e l e m e n t s which e n t e r into t h e m in a definite 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.

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Finally, we want to draw attention in advance to the fact that the theory of relatives makes it possible and offers the p r o c e d u r e to reinterpret Page 16 expressions, as well as relations, formulas, or propositions of relatives of a definite o r d e r [Ordnung] from their c o m m o n universe of discourse into a universe of another order. Namely, either to "preinterpret" t h e m into a universe of higher order; then all the relatives (of a given order) constituting the expression, or entering 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 interpret t h e m back into a universe of lower order. Certain m o m e n t s (elements) of our knowledge about the constitution of the relatives in question are lost when interpreting back, that is to say, they are ignored, one abstracts from them, or, in o t h e r words, certain parts of our knowledge are d r o p p e d , relinquished, and will not be regained in the course of possible following reinterpretations, are not restituted, so that the loss of knowledge caused by this interpretation is p e r m a n e n t - - o f course, without d e t r i m e n t to the validity and justification of the whole process. In the correct grasp of these processes, in the a d e q u a t e interpretation and use of the formulas set up for one of the universes in a n o t h e r of o u r universes, lies the main difficulties that the proper understanding of our theory may encounter; they have to be overcome by aiming at m a k i n g it understandable. The new interpretation from the second into the first u n i v e r s e u a n d converselyuwill illustrate this point. T h e r e f o r e , we want to deal with this question again only after we have a t h o r o u g h knowledge of these two universes. We now p r o c e e d to an in-depth e x a m i n a t i o n of the second universe, 12; w e will focus on this p r o b l e m almost exclusively for a long time, i.e., we limit o u r e x a m i n a t i o n to the algebra and logic, the theory of binary relatives.

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Appendix 2: Schr6der's Lecture II

Introduction In this lecture, which presents the basic identities of the algebra of binary relatives, S c h r 6 d e r begins by introducing the two-element Boolean algebra of 0, 1. He notes that if a 0 is a 0, 1-valued function of i, j r u n n i n g t h r o u g h a domain, then a binary relative is d e t e r m i n e d as a =

~oao(i'j), and every relative is so d e t e r m i n e d . If we think of 1 and 0 as the largest and smallest elements of the Boolean algebra, and think of sums as least u p p e r bounds and products as greatest lower bounds, this makes perfect sense. Thus Schr6der is here i n t r o d u c i n g the characteristic function of a relative on a domain. He points out (his page 24) that the operations on relatives can be carried out by simple formulas of combinations 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 c o m p l e m e n t 0' is the characteristic function of unequal. O n his page 27 he says that these characteristic functions are to be r e g a r d e d as proposition functions of pairs. He goes on to introduce p r o d u c t and sum over the d o m a i n of the proposition functions, which he knows to c o r r e s p o n d to greatest lower b o u n d s and least u p p e r b o u n d s in the Boolean algebra of relations. This introduction of p r o d u c t and sum over a fixed d o m a i n is his way of i n t r o d u c i n g 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 u p p e r b o u n d as sum, greatest lower b o u n d as product, not based on ordinary algebra. S c h r 6 d e r prefaces this lecture by stating that less than or equal (subsumption) instead of equality can be taken as a basis for all o t h e r relations. This is in the sense of the first volume of the Algebra der Logik, 223

SCHRO.DER'S LECTURE II

224

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. Second Lecture

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 m a n n e r 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"0=

O" 1 = 1 " 0 = 0 , 1 91 = 1 ,

i = O,

0 :(: 1, 1~0, 1 +1=

1 =(= 1, (2) 1+0=0+1=1, 0+0=0,

b = 1.

(3)

(4)

After having already postulated more than half of the formal foundations of

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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 (identically zero) a n d 1 (identically o n e ) , we have fully s t a t e d - - a l t h o u g h only in nuce--the laws of inclusion a n d n o n i n c l u s i o n , o f equality a n d inequality, in a d d i t i o n to a calculus which has as basic m o d e s o f calculation the " t h r e e identity species": multiplication, addition, a n d negation. T h e f o u r c o n v e n t i o n s (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, a n d which are not. T h r e e have validity, b u t n o t the fourth. 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 i g h t c o n v e n t i o n s (3) r e p r e s e n t the "abacus," the multiplication table a n d the addition table, for the d o m a i n of values restricted to the symbols 0 a n d 1. For this d o m a i n of values, they c o m p l e t e l y d e f i n e the product a" b o r ab a n d the sum a + b of two values a a n d b - - h o w e v e r these latter "general" values are d e t e r m i n e d , a s s u m e d , or t h o u g h t of within that d o m a i n of values. T h e m u l t i p l i c a t i o n table c o r r e s p o n d s exactly to the n u m e r i c a l table, as it w o u l d be e x p r e s s e d for the dyadic n u m b e r system or as p a r t o f the global d e c i m a l m u l t i p l i c a t i o n table k n o w n to e v e r y o n e . T h e a d d i t i o n table shows only o n e deviation f r o m the n u m e r i c a l add i t i o n table which is h e r e stated as 1 + 1 = 1. To m o t i v a t e this deviation, the following r e m a r k may n o t be wasted: b e c a u s e in o u r discipline only what is identical, same, is admissible as "equal," a r e p e a t e d use o f " e q u a l s " is n o t o t h e r w i s e possible t h a n in the f o r m of a tautological a n d t h e r e f o r e m e a n i n g l e s s repetition---comparable to the action o f the child w h o gives his f r i e n d "the same" object repeatedly. It is exactly to this d e v i a t i o n that o u r discipline mainly owes its w o n d e r f u l symmetry. T h e two c o n v e n t i o n s (4) d e f i n e in g e n e r a l the negation d ("a bar" or " n o t - a ' ) for every value a of this d o m a i n . 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 previous principle I - - a n d 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

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SCHR(~DER'S LECTURE II

one wants a further reduction--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 s u m - - s t a t e d 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 n u m b e r of indep e n d e n t stipulations for what has been said. But one could also produce a larger n u m b e r (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 subsumptions by virtue of (1). Therefore, I do not want to argue about the exact n u m b e r of stipulations for i n d e p e n d e n 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 departure and prepare a clear overview. Indeed, the 15 data m e n t i o n e d 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 whatever way it would be formulated. This core appears here as mere artless enumeration 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 b e c o n f i r m e d , t h a t o u r system o f c o n v e n t i o n s is without contradictions', t h e y a p p e a r f r o m t h e b e g i n n i n g as independent of each other. B o t h insights s t e m f r o m t h e o b s e r v a t i o n t h a t Page 20 e a c h o f t h e s t i p u l a t i o n s (1) to (4) d e f i n e s "a new s y m b o l , " so to s p e a k , w h i c h was n o t b e e n m e n t i o n e d in any o f t h e p r e v i o u s s t i p u l a t i o n s , so t h a t t h e s e c o u l d n o t affect t h e m e a n i n g o f t h e n e w s y m b o l . S o - - t o begin with the end: even if we first agree on what we understand 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 b o u n d by anything), the a g r e e m e n t will not be contained in it or any previous agreeme,~-ts, nor be able to contradict them. With the abacus ( 3 ) - - t h e 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 operation 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 numerical 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.

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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. As already m e n t i o n e d , the previous 15 stipulations f o r m the basis a n d sufficient f o u n d a t i o n for a calculation with letters, a "calculus," in which we have to s u p p o s e of each " g e n e r a l value symbol" or l e t t e r n a s a, b, c, . . . m t h a t it r e p r e s e n t s the two values 0 a n d 1. T h e formal laws, propositions, a n d f o r m u l a s of this l e t t e r calculus are no o t h e r t h a n those of the "propositional calculus" (which is even r i c h e r in f o r m u l a s ) . T h e p r o p o s i t i o n a l calculus is a subcase of the "identity calculus," which we m e t in volumes 1 a n d 2. We now d e m a n d f r o m the r e a d e r that he convinces h i m s e l f thoro u g h l y of that, that is, at least that he checks that the definitions a n d Page 21 principles taken as the f o u n d a t i o n for the calculus in the place cited result f r o m o u r 15 stipulations w h e r e b y their collective c o n s e q u e n c e s are also g u a r a n t e e d . This can be d o n e in an u n s o p h i s t i c a t e d way by s h e e r verification of the f o u n d a t i o n s f r o m the abacus, f r o m o u r stipulations. For e a c h letter we only have to distinguish two cases: w h e t h e r it m e a n s 0 or w h e t h e r 1; in a f o r m u l a in which only 1, 2, or 3 letters appear, only 2, 2 2= 4, or 2 3= 8, respectively, substitutions (of values 0 or 1 for the letters) are to be carried 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 o n e stroke we may p r e s u p p o s e that the r e a d e r is fully a c q u a i n t e d with the c o m p l e t e f o r m a l i s m of the p r o p o s i t i o n a l calculus (implicitly a l o n g with it also the identity calculus). T h e i m p o r t a n t fact is thus assured: that we are at any p o i n t free to i n t e r p r e t 1 as a "true" proposition, 0 as a "false" propositionmwhere all true p r o p o s i t i o n s are equal to a n o t h e r ("equivalent"), are valid, a n d likewise

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SCHR()DER'S LECTURE II

all false p r o p o s i t i o n s - - i f we i n t e r p r e t at the s a m e time the s u b s u m p t i o n b e t w e e n propositions, the p r o p o s i t i o n a l n e g a t i o n , the p r o p o s i t i o n a l p r o d u c t , a n d the p r o p o s i t i o n a l sum in the usual way. T h e s e r e m a r k s do not at all anticipate the applications which we i n t e n d to show with the same value symbols 0 a n d 1 within the d o m a i n o f values of the relatives. T h e s t u d e n t has to r e m e m b e r , a n d always pay close a t t e n t i o n to this fact that if we now draw conclusions f r o m additional stipulations, that these i n f e r e n c e s always has to follow the laws of this p r o p o s i t i o n a l calPage 22 culus, whose f o u n d a t i o n s are f o r m e d by the previous stipulations a n d which are no o t h e r than those of general logic--also of traditional logic, in its m o s t succinct a n d strictest version, however. T h e f o u r t h c o n v e n t i o n (2) f o r m u l a t e s the o p p o s i t i o n "true" a n d "false" for propositions, a n d expresses it in a concise way. For relatives, it will only play an essential role w h e n p a r t i c u l a r j u d g m e n t s are taken into c o n s i d e r a t i o n , a n d we will t h e r e f o r e do without it for the time being.

A f u r t h e r g r o u p o f - - s e v e n - - f u n d a m e n t a l stipulations serves to d e f i n e the general "binary relativg' a n d certain special relatives of this (the seco n d ) 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 r o m the universe 12--and, that is to say, of none, of some, or also of all. T h e g e n e r a l f o r m 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 a n d j in the sum E 0 have to run t h r o u g h all the e l e m e n 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

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TO SKOLEM

of e a c h o t h e r m 0 n condition that o n e restricts the "coefficients" a 0 (said: a sub ij), with which the pairs of e l e m e n t s i : j (as the a c c o m p a n y i n g "constituents") are related or "multiplied," to the d o m a i n of the two values 1 a n d 0, which the following f o r m u l a would express:

(a 0 =1) + (a 0=0) =1.

(5c~)

Its first coefficient value would g u a r a n t e e , by m e a n s of the stipulation l ( i : j ) = 1 9(i: j) = i : j ,

(5~)

the existence of i : j as a t e r m of the sum; the latter coefficient value would g u a r a n t e e , by m e a n s of the stipulation 0 ( i : j ) = 0" (i: j) = 0,

(5./)

the absence of the pair of e l e m e n t s i : j as a t e r m of the sum. T h e "relative coefficients," known from the suffix which is always a t t a c h e d to t h e m (as a rule in the f o r m of a double i n d e x ) , are t h e r e f o r e to be subject to the laws of the pure propositional calculus. T h e o p e r a t i o n s which will have to be p e r f o r m e d on, with, or between t h e m are fully e x p l a i n e d by o u r first 15 stipulations a n d are r e g u l a t e d a c c o r d i n g to their laws. O t h e r o p e r a t i o n s than the three identity species ( c a n n o t and) will n o t be c o n s i d e r e d . From now on, we will always express binary relatives with simple letters f r o m the lowercase latin a l p h a b e t , that is, those without suffix, i n c l u d i n g those which we have already reserved for the r e p r e s e n t a t i o n of indices, irrespective of this use; this calls for an e x p l a n a t i o n to be given later ( a n d gradually). E q u a t i o n (5) by itself would not yet be u n d e r s t a n d a b l e b e c a u s e the original obscurity of the coefficients a o a n d their workings has n o t yet b e e n e x p l a i n e d , unless o n e c o m b i n e s t h e m with their verbal corollaries or a d d i t i o n a l formulas which---cf. (5ol)massign each of the coefficients to o n e of the two values 0 a n d 1 a n d explain by m e a n s of (5/3) a n d (53') the effect of factors 1 or 0 on a constituent. In a rational way o n e can only set down the four s t a t e m e n t s (5), (5c~), (5/3), a n d (53') as "one stipulation," a n d to c o n s i d e r this (5) n o t only as a f o r m u l a , b u t also to look at it in c o n n e c t i o n 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 verbalform 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,

23 ~

S C H R O D E R ' 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 multiple 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 u n d e r s t o o d as representative of i, j a n d not as the "product" of i a n d j (which, of course, would have to be r e p r e s e n t e d correctly as i j). Since the notation for the s u m m a t i o n variables is i n d i f f e r e n t to begin with, n a m e l y because we have any o t h e r term which has not b e e n already allotted at o u r disposal, we can, of course, write (5) as

a = Ehk ahk (h: k), a n d we would have to make such a c h a n g e in the d e s i g n a t i o n of the indices by using s c h e m e (5) in cases where o n e of the terms i a n d j ( p e r h a p s to r e p r e s e n t a definite e l e m e n t ) has b e e n allocated otherwise a n d is thus no longer at o u r disposal. It is possible to substitute b or c in (5) for a, a n d 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 r e p r e s e n t a binary relative or which we can consider as binary relative. T h e c o n v e n t i o n (5) o u g h t to be set forth as general a n d provide the "scheme" for all binary relatives. Insofar as the left-hand symbols in the following e q u a t i o n s refer to binary relatives, (5) implicitly states the following: Corollary to (5) 1 = Eolo(i:J)'

0 = ZoOo(i:j),

i = [;hk ihk(h: k),

ab = Zij (ab)o(i : j), a ; b = Eo(a; b)ii(i : j),

1'= E 01'0(i 9j),

0 ' = E000' ( i ' j ) ,

i : j = Enk (i : j)hk(h : k),

a + b = ~0 (a + b)o(i : j), a j- b = ~o (a j- b)o(i : j),

d = ~0 ( a)0(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 following.Page 25 If o n e knows for a definite universe of discourse for every possible suffix 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 particular suffix ij), one also knows which pairs of e l e m e n t s e n t e r the sum a exclusively, o n e knows how the binary relative a is c o m p o s e d of individual binary relatives of the universe 12. In o t h e r words, one knows the binary relative a itself. By stating all its coefficients, namely, all their relevant values, a relative can be " d e t e r m i n e d , " sufficiently described, be m a d e known. For the

231

FROM PEIRCE TO SKOLEM

determination, complete specification of a binary relative, in o t h e r words, for the "definition" of a special binary relative, it suffices, and is even necessary in view of (5), to d e t e r m i n e 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 t h r o u g h the following six stipulations 1;j = 1,

0o= O,

(6)

0 0 : (i r j ) ,

(7)

0 0' = 0 , 0q=l

(7)

!

1'ij = ( i = j ) - - o r , better distinguished

{1~= 1,, 10=0

for i C j ,

for i c j ,

and

ihk = l'i;, , (i "j)hk

=

(8)

1'i;,1'~,

(9)

--which are to be thought as appointed for every suffix i, j, respectively--for (8) and ( 9 ) - - h , k - - t h e symbols 1, 0, 1', 0' and i, as well as i : j (also)

Page26

have found their definition as "binary relatives." These stipulations form, together with (5), the "second" group of f u n d a m e n t a l 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 m o d u l e 1 (one) is marked as binary relative which coincides with the universe of discourse 1 "~. It is the universum, the full 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 m o d u l e 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, although 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 o r d e r to avoid confusion with an incomplete equation of which the right side still has to be produced, the symbol 0 has to a p p e a r from now on in such cases where all terms on the one side fall away.

232

SCHRODER'S LECTURE lI

T h e identity m o d u l e s r e p r e s e n t the limiting 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 e m e n t s t h a n m o d u l e 1, n o n e can c o n t a i n less than m o d u l e 0; t h e r e f o r e o n e c o u l d postulate 1 as "the m a x i m a l relative," 0 as "the m i n i m a l relative." T h e admissibility o f these limiting cases has b e e n discussed u n d e r the g e n e r a l d e f i n i t i o n o f a binary relative. In a d d i t i o n to these two "identity" m o d u l e s , two special (binary) relatives a p p e a r in the t h e o r y which are d e f i n e d by the two c o n v e n t i o n s (7), namely, the two "relative modules" 1' a n d 0 ' - - - p r o n o u n c e d as "one-ap" a n d "zero-ap" (as abbreviation of "one-apostrophe" etc.). F r o m (5), we have the r e p r e s e n t a t i o n :

1' - ~o (i =j)(i "j) = ~ ( i " i),

O' = Eq (i :1: j)(i " j).

Namelymleft of c e n t e r m i 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. Page 27 This m e a n s 1' is the sum, "the universum, the d o m a i n " o f all individual self-relatives o f the universe o f discourse 12, 0' is the sum o f all individual aliorelatives of this universe a n d forms "the universe o f the aliorelatives"

(cf. w 9).

O u r t h e o r y o f binary relatives thus accepts four "modules," 1, 0, 1', 0 ~. T h e i r d e s i g n a t i o n can soon be m o t i v a t e d f u r t h e r . ~ Before e n t e r i n g into the discussion of c o n v e n t i o n (8), we w o u l d like to anticipate a n d m e n t i o n the following, a l t h o u g h 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 o f relatives, for its i n t e r p r e t a t i o n a n d a p p l i c a t i o n ~ i t is thus mainly in the interest o f the applications that we i n t e n d e d to weave it into the topic o f a l g e b r a for the p u r p o s e of illustration. T h e relative coefficients which, as we m e n t i o n e d earlier, are subject to the p r o p o s i t i o n a l calculus, can be i n t e r p r e t e d any time as propositions. Thus, we can read: a/j = (i is an a o f j). T h e n a m e a o f the binary relative can be r e c o g n i z e d as a "relative n a m e " (cf. vol. 1, p. 76ff) e q u i v a l e n t to "an a o f - , " such as "a lover o f - , a p i c t u r e o f - , an effect o f - , a f a t h e r o f - , " etc.; as a t e r m which n e e d s an a d d i t i o n a l correlate for its c o m p l e t i o n . T h e same n a m e s can be used as "absolute terms," since o n e can speak o f "lovers, pictures, effects, fathers," e t c . ~ w i t h o u t the a d d i t i o n of correlates. T h e stipulation (8), which was essentially goes back to P e i r c e ~ o f

FROM PEIRCE TO SKOLEM

233

which I only presented the most succinct v e r s i o n m f o r m s the basis for the transition of using names as relatives to using t h e m 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 r e p r e s e n t e d as a binary relative. It will later follow naturally how an "absolute t e r m " - - n a m e l y , a "system" or a "class" as the identity sum of elements, the individuals i of the universe l~mis to be r e p r e s e n t e d at any time as a binary relative; or, conversely, how binary relatives are to be i n t e r p r e t e d in the original universe of discourse, in o t h e r words, how they can be i n t e r p r e t e d back from 1z into 1~. It does not b e c o m e me to declare the establishment of convention (8), insignificant as it may appear, as the highest and most consequential Page 28 a c h i e v e m e n t in the whole theory. T h e student will only gradually be able to u n d e r s t a n d the full range of its consequences, its p r o p e r application, and its usages. For the representation 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 r e p r e s e n t e d as the sum of all the pairs of elements which have i as their relate; the relative i comprises those terms that are m a r k e d with the e l e m e n t i in the table 12. T h e algebra of binary relatives can be developed to a high stage without ever using convention (8). It may therefore be advisable for the r e a d e r to ignore this convention during the first and general part of the theory; otherwise, he may e n c o u n t e r difficulties in understanding---confusions, objections may a p p e a r 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 e a s e m b e c a u s e we will be treating it systematically. However, for the sake of completing the e n u m e r a t i o n of the formal foundations, we t h o u g h t it necessary to m e n t i o n this convention (8). T h e convention (9) leads, according to the preceding: (7) left, (5), and (3) left, mainly to the acceptance of the equation

i:j=i:j. It lets us recognize that the general coefficient (i : j)hk = (i = h)(k =j) of a binary relative d e n o t e d 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.

234

SCHRODER'S LECTURE II

In o t h e r words, c o n v e n t i o n (9) g u a r a n t e e s the admissibility of the pair of e l e m e n t s i : j as a (0ne-term, m o n o m i a l ) s u m of pairs of elements; it arranges in a row the pairs of e l e m e n t s formally u n d e r the "binary relative" a n d gives us s u b s e q u e n t a n d expressive i n d e m n i t y for having previously taken the liberty to p r e s e n t or to call this pair of e l e m e n t s Page 29 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 r o u p o f - - s i x - - f u n d a m e n t a l stipulations defines those binary relatives which are deducible, by m e a n s of the six species or basic m o d e s of calculation of w 1; it explains the results of these six o p e r a t i o n s (on or with binary relatives) as again binary relatives. T h e s e are (ab)q - aob 0 ,

d0

(a + b)ii - a 0 + b 0 ,

or

(a~ b) o = IIh(a~h + bhj),

(a; b)o = F,haihbhj , di~

(d)aj= (aii),

or

(d)q=at,,

(10) (11) (12)

(13)

a n d s h o u l d be u n d e r s t o o d 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 c o n v e n t i o n s m s i m i l a 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 e c e d i n g the proposition, to be placed in braces {}, by which the c o n v e n t i o n a p p e a r s s t a t e d - - f o r the last, for example, by m e a n s of II0{~ 0 = aji}. In view of what has been said by (5), the first three (10) a n d (11) of the above stipulations define the identity product a " b or ab, further, the identity s u m a + b of two relatives a and b, as well as, finally, the negative ("the negation") d (read a bar) of a relative a. Because a c c o r d i n g to the known t h e o r e m s of the propositional calc u l u s - a l s o c o m p a r e the abacus ( 3 ) - - t h e (ab) O above can be equal to

FROM PEIRCE TO SKOLEM

235

1 only if a 0 and bO both have the value 1, whereas (a + b)o will already be equal to 1 if a 0 or b0 have the value 1; then one sees that the identity p r o d u c t ab will be that relative which contains exclusively the pairs of elements common to the factor relatives a and b, whereas the identity sum a + b contains all pairs of elements, and those alone, which either Page 30 b e l o n g to a or b, or to both summands. T h e 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. T h e 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 relat 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 n e e d to apply. In view of what has been said by (5), the last three (12) a n d (13) of the stipulations above define T h e Relative Product a; b (said: a of b)

T h e Relative S u m act b (I say: a piu b)

of two binary relatives a,b, and finally: T h e Converse ("the conversion") ~ - - s a i d : 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 d e n o t e 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 insinuations--whereas the English term " c o m p o u n d " 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 "benPage 31 efactor of-." Incidentally, we have to m e n t i o n t h a t - - b e c a u s e of the n o n c o m m u tativity of relative o p e r a t i o n s - - t h e two relative factors e n t e r the notion of the relative p r o d u c t in very different ways; they have thus to be distinguished as "first relative factor," "relative prefactor" or "multiplicant" and "second relative factor, .... relative postfactor" or "multiplicator." W h e n a ; b is formed, we will have to say that one can "relatively premultiply" b with a, or "relatively postmultiply" a with b. Likewise, when f o r m i n g a relative sum ac~ b, the "first (relative) suxnmand," the "first relative term"

236

SCHRODER'S LECTURE II

a is "preadded" to the "second' b, or the latter postadded to the former. W h e n we speak of the relative addition or "summation" (respectively multiplication) of given terms, one has always to consider the o r d e r in which they were given: one then joins the terms in the o r d e r 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. 1 0 ) - - t h e 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 m a n n e 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 j o i n e d 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 definition and u n d e r s t a n d i n g of the two relative operations, only the knowledge of the propositional calculus is necessary. Because of the abacus ( 3 ) - - i n addition to ( 4 ) - - t h e s e operations of the propositional calculus are always applicable, and in every case of their application they give a "single-valued' [eindeutig] or fully determined result. T h e identity p r o d u c t and identity sum of any two values from the d o m a i n of values 0, 1 is in each case again a fully determined value from exactly this d o m a i n of v a l u e s - - n e i t h e r m o r e nor less than the negation of one such. T h e 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 o u r six species for obtaining the coefficients, for which an easily accessible m e t h o d is prescribed to p r o d u c e them, namely, a definite p r o c e d u r e consisting of the types m e n t i o n e d before--with the exception (if you will) of the converse; there a mere exchange of the two indices has to be m a d e which causes the r e p l a c e m e n t of a coefficient of the o p e r a n d a with a n o t h e r of its known coefficients. In short, we can say so much: T h e six species of our discipline--the identity as well as the relative m o d e s of calculation--are "totally single-valued" operations. They are unconditionally applicable in our universe; if a, b m e a n given binary relatives, then the symbols * In a way that reminds the mathematician of the (row-by-column wise) "multiplication of determinants."

237

FROM PEIRCE TO SKOLEM

ab,

a + b,

d,

a;b,

aj-b,

5,

which m a r k the results o f these species as such, are never m e a n i n g l e s s o r undefined signs, also never multivalued n a m e s , t h a t is, o n e a n d only o n e value b e l o n g s to t h e m - - t o t a l l y d e t e r m i n e d - - i n t h e class o f b i n a r y 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. T h e last of o u r f u n d a m e n t a l stipulations, which we a d d to the t h i r d g r o u p , is the d e f i n i t i o n o f 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;) )

Page 33

(14)

a n d leads back to the k n o w n n o t i o n o f inclusion b e t w e e n c o r r e s p o n d i n g coefficients o f this relative, which has b e e n d e f i n e d by m e a n s o f stipu l a t i o n (2). For two b i n a r y relatives a a n d b, a is said to be included in b, a@b, if a n d 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 t h o s e o f b. We can t h e n also say: a is part ( e i t h e r a p r o p e r p a r t o r also the w h o l e ) o f b, is contained in b. B e c a u s e o f (1), as can easily be seen, it follows: C o r o l l a r y to (14) (a = b) = Hi)(a 0 = b0) - - a c c o r d i n g l y , t h e n two relatives can be called e q u a l to e a c h o t h e r if a n d only if they a g r e e o n t h e i r c o r r e s p o n d i n g coefficients, t h a t is, identically involve the s a m e pairs of e l e m e n t s exclusively. T h u s , o u r verbal c o n s i d e r a t i o n s above, t h a t a b i n a r y relative be det e r m i n e d by its coefficients, 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 , a n d the s t i p u l a t i o n s given in e q u a t i o n s (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 L E C T U R E 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 t h e d e v i a t i o n s o f m y s y s t e m o f n o t a t i o n f r o m P e i r c e ' s , r e s p e c t i v e l y t h e o n e w h i c h c o m e s c l o s e s t to ours"

Because of the noncommutative character of relative addition, I have formed the 'piu'-sign asymmetrically, whereas Peirce (1883) used the formal cross we c o m m o n l y find in obituary notices. For similar reasons, I chose for relative multiplication the semicolon as an asymmetrical connection sign; while I also express symmetrically the identity multiplication as a commutative connection, be it by means of the period as the sign for multiplication, be it--as in most cases--by simple juxtaposition of factors (without any particular sign of connection). In this respect, I deviate considerably from Peirce. Peirce denotes the identity p r o d u c t by "a, b." This comma, as a sign of multiplication, seems less suitable for a commutative c o n n e c t i o n because of the asymmetry to the left and right; furthermore, I must declare the hyphen, the often used dividing line in punctuation, 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 arguments have to be considered which are also separated by commas (cf. vol. 1, p. 193ff). Peirce expresses relative multiplication virtually symmetrically by means of the simple juxtaposition of factors. If for n o t h i n g else, I could not accept this proPage 34 c e d u r e because it is used elsewhere. However, two circumstances support Peirce's procedure. T h e first one is not significant: if a relative a is interpreted 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 repetition of "of." I diffuse any criticism by saying a can be i n t e r p r e t e d 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. T h e second weightier circumstance is the following: U n d e r the notion of binary relative falls also--as we will s e e - - t h e 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 t h e r than its actual multiplication which the theory of substitution has been expressing for a long time without c o n n e c t i n g signs, by mere juxtaposition of the factor symbols. I do not want to deprive the substitution theory of the advantages of such a simple type of n o t a t i o n - - a s long as the theory has (as up to now) to deal only with the one operation of c o m m o n (therefore "relative") multiplication. Simplifying deviations from systematic notation for use in a special field are always admissible, however such practices for a discipline so very general are not applicable. T h e following circumstance is against Peirce: the simple juxtaposition 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 operation ab

FROM

PEIRCE

239

TO SKOLEM

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 c o n s i d e r as f u n d a m e n t a l stipulations the rules (which we o m i t t e d initially) that define a n d r e g u l a t e the use of the product a n d 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 definite (to be c o n s i d e r e d as s o m e h o w given) d o m a i n of values shall be assigned. This d o m a i n of values is called the "extension" [Erstreckung] of the "product H," respectively the "sum ~," "taken with respect to u" a n d will, in the m o s t g e n e r a 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 c a u g h t sight of after this s i g n - - w e u n d e r s t a n d a "function of u," f(u), i.e., an expression which is c o n s t r u c t e d in s o m e given way by m e a n s of o p e r a t i o n s f r o m the g r o u p of six species in o u r discipline, f r o m u itself, a n d any o t h e r relatives a, b, c . . . . . x, y, .... of which the m e a n i n g (value) has to r e m a i n constant, even if the m e a n i n g of u (within that e x t e n s i o n ) changes. T h e s e latter relatives are c a l l e d m i n c o n t r a s t to the "argument" u n t h e "parameters" of the f u n c t i o n f(u), a n d can be u n d e r s t o o d b o t h as general relatives, as well as having special values; in particular, they, or s o m e of t h e m , can be r e p l a c e d by u

u

modules. F u r t h e r m o r e , the f u n c t i o n f(u) is itself a binary relative whose value for e a c h assigned value of u a n d the fixed values of the g e n e r a l letter p a r a m e t e r s has to be fully d e t e r m i n e d - - t h e r e a s o n b e i n g that the results of the o p e r a t i o n s or species, constituting the e x p r e s s i o n f(u) a n d t h e r e i n

SCHRODER'S LECTURE II

240

prescribed, are defined to be single-valued, by courtesy of o u r 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 c o m b i n e d application of our six schemes (10) to (13) as a proposition function [Aussagenfunktion] of the general coefficient of both the a r g u m e n t u a n d all of its parameters according to a totally d e t e r m i n e d , prescribed Page 36 p r o c e d u r e . 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 d o n e by means of a general statement 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 .

If we include these, we will have in total 29 + 2 = 31

stipulations.

(15)

fundamental

In fact, there can be no doubts about the m e a n i n g and value of the coefficient (i.e., propositional) products or sums on the right side by which o u r coefficient on the left is to be m a d e explicit. In view of the goals pursued in our ninth lecture (w167 23 and 24), it is i m p o r t a n t 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 propositional p r o d u c t 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 imp o r t a n t 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 m o r e 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-

241

FROM PEIRCE TO SKOLEM

sents any p r o p o s i t i o n r e f e r r i n g to an object of t h o u g h t u, i.e., a "proposition about u," t h e n o f the two symbols

HA,, u

Page 37

and

EA,, u

- - - e x t e n d e d over any given d o m a i n of "values" as the m e a n i n g s that symbol u s h o u l d take o n - - t h e first o f these has to r e p r e s e n t : the p r o p osition that A,, is true for every o n e o f these objects u (within the "extension"); the s e c o n d tells us: the p r o p o s i t i o n that A,, is true for certain u (within this e x t e n s i o n ) , that t h e r e is at least one u in the e x t e n s i o n [Erstreckungsbereiche] for which A,, is true. T h e r e f o r e , the truth value 1 is applicable to the p r o p o s i t i o n I I A , if a n d only i f f 0 r each of the aforesaid u, A,, = 1; o n the o t h e r h a n d , the t r u t h value 0 is applicable if t h e r e is at least o n e u for which A u is n o t true; t h e r e f o r e , w h e r e A,, = O. T h e truth value 1 will apply to the p r o p o s i t i o n E Au if t h e r e is any u at all in the e x t e n s i o n of u for which A u = 1; o n ~he o t h e r h a n d , the truth value 0 applies if a n d only if t h e r e is no such u, i.e., if for e a c h u in its e x t e n s i o n A,, is n o t true, A , = 0. If v r e p r e s e n t s a value, arbitrarily p r o d u c e d for u f r o m the e x t e n s i o n o f u, we m u s t have u

(HA.= u

1)=(= (A,, = 1) =(=(EA,. = 1), u

(EA,,=O)={c--(A,~= 0)

= ( = ( I I A . = 0)

u

u

or, shorter: m

HA,, a@ A,, =r r, A,, , u

E A,, =t-=A,, =C-HA,,,

u

u

u

o~)

w h e r e b y it is given:

HA,, = A,,IIA,,, u

u

EA,, = A,, + EA,,. u

u

13)

T h e last f o r m u l a shows that for e a c h value (v or u) f r o m the e x t e n s i o n , the so-called " g e n e r a l factor" A,, o f the propositional-II can also be c o n s i d e r e d a n d p r e s e n t e d as a real ("actual") "factor" o f the propositional " p r o d u c t " in the m o s t n a r r o w sense, already e x p l a i n e d as a "binary" (two-factor) p r o d u c t , w i t h o u t the H-sign; a n d also that the so-called " g e n e r a l term" o f a propositional-[; is an actual (or "real") summand o f a binomial p r o p o s i t i o n a l sum, i.e., o f a p r o p o s i t i o n a l s u m in the n a r r o w e s t sense, as which o u r propositional-E has to be always c a p a b l e o f b e i n g understood. Based o n what has b e e n said above, it m u s t f u r t h e r m o r e be clear that the negation o f o u r propositional-II a n d E has to be "carried out" acc o r d i n g to the following scheme:

242

SCHRODER'S LECTURE II u

H A , = EA,, , u

EA u = IIA,.

u

u

u

3/)

In the "dictum de omni et de nuUo," a p p l i e d here, by which we have g a i n e d all the p r e c e d i n g f o r m u l a s (of which we i n d e e d d e m a n d accepPage 38 t a n c e ) , a real "axiom" is n o t to be seen; the d i c t u m has only the c h a r a c t e r o f a "principle" (in the sense of v o l u m e 1); it acts as a substitute f o r m a n d is in this sense n o t h i n g m o r e t h a n - - a definition o f the n o t i o n s "every" u, respectively, "some" o r "certain"* u ("in g e n e r a l o n e " u, s h o r t e r " o n e u," a u) - - t h e definition of which c o u l d p r o b a b l y n o t be given "formally," as a " s t a n d a r d definition " Cf. pp. 67ff. If the e x t e n s i o n of u contains only one object v, it is easy to see that the m e a n i n g of b o t h HA a n d EA is the p r o p o s i t i o n A r e f e r r i n g to this o n e v, n a m e l y that

IIA,,=A v =EA,. u

u

T h e II a n d E consist h e r e of only one term; they are " m o n o m i a l . " If the e x t e n s i o n of u contains the two objects v a n d w, we r e c o g n i z e easily that the m e a n i n g of

IIA,,=AvA,,, u

E A , , = A v + A,o 11

is d e f i n e d by the binary p r o d u c t , respectively the binary (= b i n o m i a l ) s u m of individual p r o p o s i t i o n s r e f e r r i n g to v a n d w, established t h r o u g h abacus (3) (without the II- a n d E-signs). If the e x t e n s i o n of u contains exactly t h r e e o b j e c t s m t o express it now m e r e l y 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, t h e n we c o u l d see that the m e a n i n g o f HA is d e f i n e d by the t e r n a r y (threefactor) 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 o f the two binary p r o p o s i t i o n a l p r o d u c t s A,,(A,,A,,,) a n d (A,,A,,)A~,---established o n the basis o f the associative law for p r o p o s i t i o n a l multiplication. A n d so on. In o u r theory, we can omit e x p l a i n i n g that the p r o p o s i t i o n a l - H (with a b o u n d e d e x t e n s i o n restricted to an arbitrary " n u m b e r , " a "finite set" o f objects u) can be derived t h r o u g h successive, binary multiplication o f its factor propositions, we can o m i t stating this a n d m a k i n g essential use o f it. At least until the n i n t h lecture, in which the " i n f e r e n c e f r o m n to n + 1" will be rigorously proved. However, until this h a p p e n s , the m a t t e r at h a n d will have to be c o n s i d e r e d as settled by A p p e n d i x 3 o f v o l u m e 1. [We have to take even less advance notice of a d o m a i n o f values consisting o f 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").

FROM PEIRCE TO SKOLEM Page 39

objects u, o u r plication of

243

IIA

could be derived as a "limit" t h r o u g h binary multithe factor propositions continued 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 m e n t i o n of u in the proposition which figures here as the general term, then we may omit (also in the formulas) the suffix u in the proposition A,, as u n i m p o r t a n t , and represent it merely with A. T h e n the result certainly is HA=A

and

EA=A.

6)

Likewise, if B,, is a proposition with respect to u and B is a constant proposition with respect to u, we have further the scheme:

rI(A~-B,,)=(A=~c--HBu),

H(A,,~-B)=(EA,~--B),

e)

both of which can be c o m b i n e d into the general scheme: II or IIII(A,,~(=B,,)=(E,A,,~IIBv), u,1;

u

11

t')

1~

where the extension of v is arbitrary [posssibly different from 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 c o m b i n e d to the m o r e general ~2 or

u,~

F,r,(A,,=~-B~)=(H,A,,~-I2B~). u

IJ

1~

0)

If one specifies in e) and r/) A = 1 (by taking A hereafter for the r e m a i n i n g B) or B = 0, then the following schemes result: { II ( a , = 1) = (HA,, = 1), u

Z, (a,, = 0) = (II a , = 0),

Page 40

II (a,, = 0 ) = ( E A , = 0), u

I~, ( a , = 1) = (E au = 1),

t)

of which the first and last do not express anything [nichtssagend], in view of the "specific principle" of the propositional calculus: (A=I) =A; the o t h e r two, however, are very often used. Finally, we have to m e n t i o n the propositional scheme: I Text (other than G e r m a n terms) enclosed in brackets t h r o u g h o u t the appendices is Schr6der's unless otherwise noted.

244

SCHRODER'S LECTURE II

{ (H, A,,::~-.H B,,) (IA,,~-EB,,)

(A,,~B,,) ~

14

E (A,, :~--B,,)

K)

u

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,,, u

A + IIB,.= II (A + B,,),

u

tt

X)

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~)

u, v

u

Ip

u , 1~

lz

Ix) 9

The counterpart to X)"

AI/B,, = II AB,., u

Page 41

A + E B,. = ~ (A + B.),

u

u

u

1,)

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

[Erstreckung] .

245

FROM PEIRCE TO SKOLEM

We could f u r t h e r add schemes for multiple sums and products. T h e most r e m a r k a b l e new t h e o r e m a m o n g these is this one, based on Peirce: If A,,.~ represents a proposition referring to two objects u a n d v, which are to be t h o u g h t of as variable in their own respective extensions, then

E II A,,.v ~-- II E A,.,,

o)

-,---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 proposition A is valid, then there is also for every v at least one u (i.e., the one we just m e n t i o n e d ) such that the proposition A is true of both. This conclusion is obviously not convertible.

Page 42

We shall use the assumed schemes predominantly, if not exclusively, for such objects u, v, ... which are not only general relatives but also m e r e "elements" i,j, ..., or "individuals of the first universe of discourse." In such cases, we add the r u n n i n g indices to I2 and II (as d o n e so far) as suffixes instead of writing them under ~ or II. In o r d e r not to be r e d u n d a n t and not repeat ourselves too much, we want to consider the relevant or still u n e x a m i n e d 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 d o m a i n of values of o u r propositional-II and I2 may also always be a "continuum," as, for example, are the real n u m b e r s or the points forming a straight line. In such cases, the signs II and I] are definitely indispensable and it would never do to r e p r e s e n t the propositional p r o d u c t explicitly, as an "actual" p r o d u c t with all its factors, which is r e p r e s e n t e d "symbolically" by, for example, II. In our theory, it will always b e m i f not m e r e propositional products, respectively propositional sums, then at least "identity" products II and sums I2, which these signs help us represent. In o t h e r 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 n e e d e d 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 n e e d in-depth consideration, perhaps in relation to the d e t e r m i n a t i o n of general coefficients. T h e student has to think seriously about the above propositional schemes a)...0) [as we have carried out for illustration u n d e r 0)] and try to absorb them succum et sanguinem.

246

SCHRODER'S LECTURE II

The practice of the rest, which our theory permits, will further contribute to a better understanding .... 2

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, d e p e n d i n g on whether we take as a starting point: the f u n d a m e n t a l stipulations asserted in w 3; or, the geometric representation of relatives and the operations to be p e r f o r m e d with them, as we c o n n e c t e d it to the consideration of the matrix; or, finally, the verbal interpretation 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, although it is not completely exhaustive, I shall call the three different evidences: the analytic, the geometric, and the rhetorical evidence. The purity of the m e t h o d will require that we do not mix them, that we prefer one of t h e m - - f o 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 t h r o u g h 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 p r o o f is to be given by calculation [rechnerisch], in which at each step we realize by which scheme of the propositional calculus we p e r f o r m 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. A n o t h e r u n a m b i g u o u s n a m e 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 indirectly 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.

247

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

and operations between relatives from a "geometric" point of viewmfor example, by means of a comparison through mental superposition, making parts coincide, connecting 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 other for the entire universe of discourse 12, in other words, that they are negations of each other. O r m t o illustrate the "geometric evidence" with one m o r e e x a m p l e m i t will make immediate geometric sense that (a=0)=(a;l=0)

Page 66

and

(a~: 0 ) = ( a ; 1

g: 0)

after we have recognized that the relative a ; 1 is always obtained from a relative a when the rows of the latter occupied with (one or more) filled circles are c h a n g e d 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 propositions 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 construction of the theory, the geometric evidence o u g h t 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 attention to it in the theory; yes, the theory will even justify the tendency to gradually "replacg' the coefficient evidence by the geometric evidence in an analytically well grounded way. T h e third type of evidence, the rhetorical evidence, is effective for ordinary thinking. We notice it, we b e c o m e aware of it, as soon as we e x e m p l i f y ~ w i t h Peirce~relatives of a special nature, such as l = lover (of), b - benefactor (of), s -- servant (of), with those general relatives that a p p e a r as letters in our formulas. Anybody will find the following proposition immediately illuminating: 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 s o m e b o d y ) m a s the following formula expresses [somehow shorter and also m o r e generally]: l; (bs) ~ (/; b)(/; s). Nobody will refrain from extending the proposition of the given rel~ See page 50 of S c h r 6 d e r ' s second lecture.

24 8

S C H R O D E R ' S L E C T U R E 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 o u r 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 c a n n o t but recognize a high Page 67 degree of direct intuitiveness in the more simple formulas of o u r theory. The logic, if one wants to develop the methods and schemes of deductions and inferencesmfor relative as well as for absolute not i o n s - - c a n n o t avoid to strive for the most complete possible registration and schematization of: the a priori, self-evident, identical, analytic, or irreducible j u d g m e n 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 i n t r o d u c e d into a most perplexing variety of them" when entering the logic of relatives. In addition to the fundamental conventions compiled in w 3, there is no o t h e r "principle" in the theory. And if somebody raises questions about the axiomatic foundation of our discipline of the algebra and logic of relatives, I can only agree with Mr. Peirce who c o m m e n t s 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 n e e d no p r o p e r "axioms." In o r d e r 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, p e r h a p s 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 t h o u g h t to be

FROM PEIRCE TO SKOLEM

Page 68

249

at the basis of it. These axioms are not at all analytic or tautological [nichtssagenden] j u d g m e n t s ; even if we call them "psychologicaUy necessary" in view of the nature of our spatial intuition, we c a n n o t call t h e m necessary in the logical sense, and thus g e o m e t r y is more than a m e r e b r a n c h of logic; it is the most e l e m e n t a r y 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 n a m e "axiom" for the "principles" appearing in the theoretical analysis. Those "principles" are only h i d d e n definitionsm"are m e r e substitutes for definitions of the universal logical relations." In so far as the universal logical relations can be d e f i n e d m P e i r c e has a right to saymwe can do without any "principles" (all axioms may be dispensed with). This idea, I think, will find further 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 c a n n o t revert too m u c h 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 overtake, readily speed ahead of, the first evidence; for relatively complex propositions, the third evidence in particular remains far behind---often 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 t h e r two types of evidence requires for the u n t r a i n e d s t u d e n t a considerable a m o u n t of thinking and "headache."

This Page Intentionally Left Blank

Appendix 3" Schr6der's Lecture III

Introduction T h e first page of Schr6der's third lecture in volume 3 is r e p r o d u c e d in translation here, followed by a n o t h e r part of the third lecture, b e g i n n i n g on page 97 to end of section 6. This is Schr6der's algebraic t r e a t m e n t of quantifiers as arising from least u p p e r bounds (sums) and greatest lower bounds (products). S c h r 6 d e r 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 elements of the domain: (v i) (u For a single finite domain, of course, d e t e r m i n i n g w h e t h e r an identity holds reduces to propositional calculus. S c h r 6 d e r 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, S c h r 6 d e r tries to rewrite all firsto r d e r statements about binary relations as equations in the relational calculus. As was proved by Korselt (L6wenheim 1915), this c a n n o t be done. But in 1941 Tarski a n n o u n c e d that if there is a decision m e t h o d for telling w h e t h e r a relational equation is an identity, then there is a decision m e t h o d for telling w h e t h e r a first-order s t a t e m e n t about binary relations is true, contradicting C h u r c h ' s solution to the Entsheidungsproblem. It was later proved that no finite set of true identities implies all true identities (Monk 1964). T h a t 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

S C H R O D E R ' S LECTURE III

Third Lecture

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. T h e most i m p o r t a n t laws of the six basic types of calculation have been Page 76 discussed with considerable completeness by Peirce. T h e lowercase latin letters will always represent general binary relatives and in addition d e n o t e the "elements" m e n t i o n e d in 3) on page 7. O f course, the laws of operation [Kniipfunggesetze] which have already c o m e to light in the simplest operations will also play a role in the m o r e complicated operations; they will, in all expectation, be the basis of m o r e complex laws applying to the propositions with m o r e 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 n d e p e n d e n t and arbitrary relatives. Thus, we can roughly limit the field to the rules of inference [Folgesiitze] or the "laws" which can be called f u n d a m e n t a 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) t h r o u g h 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 [Einordnung] or equality) can be justified between these coefficients based on the t h e o r e m s of the propositional calculus [Aussagenkalkuls]. T h e n we will also have gained the conviction that our compilation of propositions is complete, or, at least, gained the knowledge that n o t h i n g i m p o r t a n t has been overlooked . . . .

w 6. The II and ~ of Relatives Finally, we have to stipulate the most i m p o r t a n t 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 i n d e p e n d e n t l y 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 w h e t h e r 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.

253

FROM PEIRCE TO SKOLEM

results of binary identity operation species between arbitrarily many terms which it would be c u m b e r s o m e 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.

Page 98

Most of the propositions are known and valid from the identity calculus. They form counterparts [Gegenstiick], pendants, possibly also heavily modified (that is to say, weakened or defective) analogues to the schemes of the identity calculus which we have collected u n d e 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 r e m a i n without a c o u n t e r p a r t (for relatives), and, from the point of view of our newly f o u n d e d basis, the identity calculus will prove to be the o n e which allows fewer conclusions c o m p a r e d to the propositional calculus which is richer in f o r m u l a s m w e find that the last reason for this lies in the circumstance that the f u n d a m e n t a l stipulation (14), constructed with YI0, lacks a "propositionally dual" c o u n t e r p a r t constructed with E0' a n d remains definitely inadmissible there (cf. volume 2, pp. 43ff). In o r d e r 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 used. If a is constant, we have the law of tautology IIa = a = E a

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 c h a n g i n g terms (a), over which the II and ~ have to extend consistently [iibereinstimmend]---cf. o~) of w 3. O n e such Ha, = II a, o u g h t to be replaced by the following expression which is m o r e general from a formal point of view or seemingly m o r e comprehensive: 1-I4>(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

S C H R O D E R ' S L E C T U R E III

However, instead of specifying the d o m a i n of values as the e x t e n s i o n o f H, which we have to a d d mentally to the a r g u m e n t a o f the g e n e r a l t e r m qS(a), in o t h e r words, which a itself has "to run t h r o u g h " a n d t h r o u g h which the c o r r r e s p o n d i n g values o f 4~(a) are u n i q u e l y [ eindeutig] d e t e r m i n e d , we can i m m e d i a t e l y c o n s i d e r the d o m a i n of the latter values as given. This d o m a i n forms a new e x t e n s i o n [Erstreckungsbereiche], which is in g e n e r a l d i f f e r e n t from that o f a, and, if we use it, instead o f the previous one, as a basis, o u r e x p r e s s i o n II 4~(a) will now be r e p l a c e d by a

II

~(a)

Page 99

~b(a).

However, the two terms may n o t be p o s e d as formally equal despite their identity b e c a u s e in such an equality, o n a c c o u n t of the d i f f e r e n c e o f the e x t e n s i o n s o n b o t h sides, the sign II would a p p e a r as o n e u s e d with a " d o u b l e m e a n i n g " [~b(a) does n o t have to take or r u n t h r o u g h the value of a ! ] D t h e r e f o r e : b e c a u s e o f the passage over the equal sign, we w o u l d have a c h a n g e in the d e n o t a t i o n principle [Bezeichnungsprinzipien] e n t e r i n g in the t e r m i n o l o g y or n o m e n c l a t u r e , which is n o t permissible! T h e r e are n o m o r e obstacles to i n t r o d u c i n g a s h o r t e r d e s i g n a t i o n , in the f o r m of the letter c, for the formal term 4~(a), a n d let 4~(a) = c; thus we get the e x p r e s s i o n

I-It, r

which has the same form as the previous II a in which only the e x t e n s i o n is to be c o n s i d e r e d as different, that is to say, consisting o f the values o f ~(a), instead of the values o f a. If we c o n d u c t a new or i n d e p e n d e n t e x a m i n a t i o n in c o m p l e t e generality, we may use from the b e g i n n i n g the t e r m a instead of the letter c, a n d we t h e n get back to o u r previous e x p r e s s i o n as o n e which is only apparently less general" a

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 simplification for the e x p r e s s i o n s E4~(a), ~b(b), ~ ~(b), which we can replace by the s i m p l e r ~ a, H b, ~ b, a~ t h e n the converse r e m a i n s valid. It is r e c o m m e n d e d to d o this b e c a u s e o f the gain in the clarity of the f o r m u l a s a n d the ease which we achieve. O n this a s s u m p t i o n , we have, as c o u n t e r p a r t ~:o -y) o f w 3:

IIa = Ed , n e x t to which we i m m e d i a t e l y posit:

Ea = IId ,

19)

255

FROM PEIRCE TO SKOLEM

~ a = ZS,

l i a - liS,

20)

I;a- IIa, f u r t h e r m o r e , as c o u n t e r p a r t to ~), ~) of w 3:

21)

~h (a :(= b) = (E a :(= H b) - - w h i c h can also be r e p r e s e n t e d by II(a ~ b) = (Ea =(= lib)

also sufficiently expressive and not equivocal. T h e schemes r/) and 0) of w 3 lack an exact analogy in o u r theory a n d bring no formulas of the same s c h e m e for our relatives, e x c e p t the weakened: E(a =(= b) =~: (Ha =~- Eb). Page 100

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 w e a k e n e d 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 part li(a =(= b) =~:

(Ha =(= lib), (Ea @ Eb),

25)

whereas the last part or the e n d of that s c h e m e r e m a i n s without counterpart. 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 t, ~ ' (Ea) E b= E ab a,b

b

a + li b = li (a + b) h t, ' I I a + li b= li (a + b)

'

a

b

a,b

26) '

a n d to y) and () of w 3, we have the propositions ali b = b

~

ab

a + E = Y (a + b),

~

(Ha) H b = li ab a,b

b

'

b

r a + ~ b = r (a + b) a

'J

a.b

27) '

- - f o r the latter, in case the extension of both li (respectively, E) is the same, the d o u b l e p r o d u c t (respectively, the d o u b l e sum) can also be c o n t r a c t e d into a simple one [cf. 5) of w 7]:

256 II H ck(a)~,(b) = II ~(a)~(a), a

~

SCHRODER'S

LECTURE

III

Ea EIJ {4)(a) + ~(b)} = Ea {4)(a) + ~(a)}

a

n f i n a l l y , we have the proposition, as c o u n t e r p a r t to o) of w 3: E H a :(= II Ea.

28)

For the relative operations, we only have to add the following e x t e n s i o n of propositions 5) and 6) of this section: a;~b=~a;b,

a# IIb=II(a#b),

"(~ a); b = E a ; b,

II a j- b = II (a c~ b),

b

b

b

a

a

(~ a)" ~ b : F, a" b '

a,b

'

a ~b=~---~a;b, I(~ a) ;b:~-' a; b, (IIa);IIb IIa'b Page 101

b

a,b

'

29)

a

II a~ O b = II (a# b)

'

a

J

\ -r

b

~

a,b

'

E(a#b)~- ~ b , E (a# b) ~= aE a ~ b, E (a#b)~.~a# Eb. b

'

a,b

a

30)

b

We can read the signs II = IIII = I I I I ,

a,b

arbitrarily.

a

b

b

a

I2 = I~I2 = E~;

a,b

a

b

b

a

31)

Appendix 4: Schr6der's Lecture V

Introduction T h e 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 c o m p l e x 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 elimination theory, a project for giving conditions for solvability of sets of many-variable algebraic equations over finitely g e n e r a t e d integral domains. In the context of the c o m p l e x n u m b e r s as g r o u n d 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 o p e n basic set, this t h e o r e m states that the projection of a Zariski o p e n set is Zariski open. These are also called constructible sets in algebraic g e o m e t r y (see H a r t s h o r n e ' s standard graduate textbook Algebraic Geometry), a n d this is the t h e o r e m that they are closed u n d e r projection. In Schr6der's fifth lecture, he replaces a finite set of algebraic equations a n d inequations by a finite set of relational equations a n d 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 u n k n o w n 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 T h i s is also the m a i n t h r u s t o f Tarski's decision m e t h o d for algebraically closed fields. His m a t h e m a t i c s is a special case of K r o n e c k e r ' s , which also applies to m a n y o t h e r fields.

257

258

S C H R O D E R ' 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 m e a n 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 u n d e r 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 i n d e p e n d e n 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. T h e 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.

259

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

Fifth Lecture

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

T h e most remarkable aspect i n h e r e n t in the results d e m o n s t r a t e d in the previous lectures may be the f a c t ~ s h o w n in formulas 5) in w 1 0 ~ t h a t in o u r algebra each inequality can be c h a n g e d 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 c o r r e s p o n d i n g schemes next to each other; we will begin with the s c h e m e 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 better to say, left) side 1: (a:/: 0) - ( 0 ~ d ~ 0 = 0 ) = (1 ; a ; 1 = 1),

(a:/: 1) = ( 0 0 ~ a : t 0 = 0 ) = (1 ; d ; 1 = 1).

1)

This fact is of great i m p o r t a n c e 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" a n d "universal" j u d g m e n t s is final. As we have proved in volume 2, w 40, the most general proposition, mathematically "secondary" in Boole's sense, is constructed out of "primary" propositions. A primary proposition has e/ther the form of a subsumption or an equation--which is really the same, because the one form can always be c h a n g e d into the o t h e r - - o r the negation thereof, that is, a nonsubsumption or a inequation--which, again, can be t r a n s f o r m e d into each other. Page 151

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 r o u g h t to 0 or 1--already in the identity c a l c u l u s - - a n d also every inequation (and nonsubs u m p t i o n ) , and since the latter propositions can be rewritten in equations, after preparations 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 proposition of the data of a p r o b l e m is built from only (primary) equations by rules from the propositional calculus. After "executing" the negations which may be prescribed in a prop-

260

SCHRODER'S LECTURE V

osition function (and converting inequations which may have been introduced 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 m e n t i o n e d 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, (0j-aj-0)(00~ba-0)=00~ab0~0,

1;a;l" 1;b;l=l;a;1;b;1, 1;a;l+l;//;1-1;(a+b);1,

3) 4)

which can be easily extended from two to any n u m b e r of terms, and should be t h o u g h t 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. Page 152 First, in o r d e r 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:

261

FROM PEIRCE TO SKOLEM

L = (1 ;a) ;1 + (1 ;b) ;1 = (1 ; a +

1;b);l={1;(a+b)};l=R.

By the propositions 1) t h r o u g h 4) above, we now r e a c h o u r goal m e n t i o n e d previously of easily r e d u c i n g the e q u a t i o n s (likewise the inequations) 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+/~+~?+"" =11 ; ( d + / ~ + = (abc"

k

(a~

0 = 1),

=(0#d/~?'"#0=0) ={1;(a+

(a ~ 1) + (b :/: 1) + (c:/: 1 ) ' "

~ + . . . ) ; 1 =01

9 =1)

= (O~abc'"~

0) + ( b ~ 0) + ( c ~ 0 ) " "

b + c + "") ;1 =11

= (O~abc"'~-O=O) :11 ; ( d + / ~ +

(a=0) + (b=0) + (c=0)""

6)

=0)

~ + " " ) ; 1 : 1},

=(1;a;1;b;1;c;l'"-0) = (Oct ~o~Oct/~o~Oct ~ t O "

= 1)

=(1;,~;1;/~;1;?;1""=0)

(a=l)+(b=l)+(c=l)'-"

7)

8)

= (Oj-a#O#bj-O#c#O'"=l),

(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)

= (Oj-a~Oj-beOeceO

=0)

9)

= (1 ; d ; 1 ;/~;1 ;~;1 " " = 1).

Page 153

To prove these schemes, for 6) and 7)--the latter coming from 6) by contrapositionmno 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 equations in the p r o p o s i t i o n f u n c t i o n which r e p r e s e n t s the total proposition of the data of a p r o b l e m , as p e r

262

SCHRODER'S LECTURE V

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 i m p o r t a n t 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 b r o u g h t 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 Page 154 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 :(=fuwill not be mentioned 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" relatives~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 u n t r u e (false). T h e n 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

FROM

PEIRCE

263

TO SKOLEM

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 coefficients, possibly its g e n e r a l c o e f f i c i e n t in o n e stroke). If we i n d e e d o b t a i n 0 as the value o f f (or o f each o f its c o e f f i c i e n t s ) , t h e n t h e e q u a t i o n f = 0 was correct b e c a u s e it will give 0=0. It is t h e n a d m i s s i b l e to a s s u m e or assert it, e v e n t h o u g h t h e a s s u m p t i o n can be called e m p t y o r w i t h o u t c o n t e n t [nichtssagende] (self-evident) (its self-evidence o r validity may, however, be q u i t e a r d u o u s to p r o v e ) . If, however, we find t h a t the value of f is d i f f e r e n t f r o m 0 ( w h e n at least one c o e f f i c i e n t o f f is - 1), t h e n t h e e q u a t i o n f = 0 was false a n d r e m a i n s i n a d m i s s i b l e as a s s u m p t i o n as well as an assertion. It can o n l y be a d m i t t e d provisionally as an a s s u m p t i o n o r c o n d i t i o n in o r d e r to prove its final rejection "apagogically" by d e d u c i n g absurd Page 155 c o n s e q u e n c e s . We m a y call t h e e q u a t i o n f = 0 i t s e l f " a b s u r d " o r n o n s e n s i c a l , a n d k e e p it for o u r discipline ( t h e t h e o r y of relatives) as t h e original, or t h e r e p r e s e n t a t i v e for all a b s u r d e q u a t i o n s o f t h e s c h e m e 1=0 k n o w n f r o m t h e i d e n t i t y 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 n w h i c h , 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 s o l u t i o n p r o b l e m o r 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 w h i c h we have e x a m i n e d so far. It is d i f f e r e n t in t h e " s e c o n d " case, w h e r e u n d e t e r m i n e d relatives in t h e e x p r e s s i o n o f t h e p o l y n o m i a l f in the e q u a t i o n f = 0 occur.

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Even by the sheer posing of the e q u a t i o n m b y p r o n o u n c i n g or stating this equation "f= 0," be it as a (conditional or unconditional) assertion, be it to make the equation an assumption for an investigationwwe impose on the reader, we c o m m i t ourselves to the following: to think of or imagine a system of values u n d e r 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 u n d e t e r m i n e d 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 unPage 156 d e t e r m i n e d relatives are also called "unknowns," are designated as "the unknowns." T h e d e t e r m i n a t i o n of a system of roots is called "a solution" of the equation, and the d e t e r m i n a t i o n (sometimes also simply the statement) 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 b e e n established to "solve" it for the u n d e t e r m i n e d relatives that occur in it as "unknowns"; the equation involves or states a problem. T h e r e b y two limiting cases can occur: First, the case in which there is no system of roots at all. In this case, the equation posits a d e m a n d which it is impossible to fulfill; the p r o b l e m remains unsolvable, and the equation is inadmissible (its " r o o t s " w i f we can still talk about roots here, where there are none, although they have names in the form of x, ...; they have obtained t h e m prematurely, so to s p e a k m w o u l d 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 equation 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 t h e r case, we call it "synthetic"); its roots remain undetermined, that is to say, arbitrary. We also say the "resultant" of the elimination of all u n k n o w n s (or any of them) from the equation is 1, or 0 = 0, or the equation 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 u n d e t e r m i n e d 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 equation

f= 0."

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Also b o t h these limiting cases, which are o f t e n n o t easily r e c o g n i z a b l e a n d p r o v a b l e as such (in spite of the n a t u r e of self-evidence o f o n e o f t h e m ) , offer n o real solution p r o b l e m [Aufl6sung~oblem]. O n the o t h e r h a n d , solution p r o b l e m s arise in every o t h e r case; a n d we now want to deal with t h e m in d e p t h . T h u s , o u r r e m a i n i n g o b s e r v a t i o n s are based o n t h e c o n d i t i o n t h a t in the d o m a i n of binary relatives t h e r e is at least o n e system ( a n d possibly m a n y systems) of values which m a k e the e q u a t i o n true, u n d e r s t o o d with r e s p e c t to the " u n k n o w n s " x, y, z . . . . , a, b. . . . o c c u r r i n g in the e q u a t i o n ; b u t the value o f these variables may n o t always be a s s u m e d arbitrarily if the e q u a t i o n is to be satisfied, in o t h e r words, if the e q u a t i o n is to be n e i t h e r a b s u r d , n o r formal, b u t is r e q u i r e d or s u p p o s e d to satisfy a real "relation" b e t w e e n all u n k n o w n s . "Relation b e t w e e n - " 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 unknowns which will enter into it. If only one unknown enters into such a "relation"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.

Page 158

If we e m p h a s i z e a m o n g the u n k n o w n s any p a r t i c u l a r o n e - - l e t us call it x - - t h e n only o n e of the two following cases can apply with r e s p e c t to the o t h e r u n k n o w n s y, z . . . . . a, b. . . . : E i t h e r the latter c o u l d be a s s u m e d arbitrarily, w h e r e b y t h e r e is a value or values of x for e a c h system of values that may be a d d e d to it, which t o g e t h e r f o r m a "system of roots" of the e q u a t i o n f = 0. O r this is n o t true. In the first (sub-)case, we say that the e l i m i n a t i o n of the u n k n o w n x f r o m the e q u a t i o n f---0 gives "no" resultant, or "the r e s u l t a n t o f this e l i m i n a t i o n " is the e q u a t i o n 0 = 0; f r o m 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, a n d t h e r e is no relation b e t w e e n the r e m a i n i n g u n k n o w n s , these b e i n g unrestricted g e n e r a l relatives ("parameters"). In fact, we may t h e n a t t r i b u t e to the latter an arbitrary value system, a n d it will only m a t t e r to o b t a i n the a p p r o p r i a t e values o f x which t o g e t h e r with it f o r m a system of roots, in o t h e r words, to e x p r e s s the u n k n o w n x in t e r m s of the o t h e r u n k n o w n s . I f this case can be proved to be correct, we have a "pure" solution p r o b l e m , the p r o b l e m of the s o l u t i o n of equa-

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tion f = 0 of the one u n k n o w n x; t h e n we have i d e n t i f i e d this e q u a t i o n as an "unconditionally" solvable o n e , a n d we can p r o c e e d to its solution. For the s e c o n d (sub-)case t h e r e are s o m e value systems (at least o n e ) to which may not be given to the u n k n o w n s y, z . . . . . a, b. . . . b e c a u s e t h e r e is n o value of x t o g e t h e r with which they c o u l d r e p r e s e n t a system of roots. Each p r o p o s i t i o n which truly states that a system of values y, z . . . . . a, b. . . . is inadmissible, "excludes it," can be c o n s i d e r e d a "relation" b e t w e e n these o t h e r u n k n o w n s , which, if we so desire, can again be e x p r e s s e d in the f o r m of an e q u a t i o n , a n d it may be called "a resultant o f the e l i m i n a t i o n of x f r o m the e q u a t i o n f = 0" b e c a u s e it d e p e n d s o n t h e e q u a t i o n f = 0 (necessary for its fulfillment), b u t the n a m e of the u n k n o w n x does n o t a p p e a r in it. T h e u n i t e d e q u a t i o n , total p r o p o s i t i o n of all resultants (of e l i m i n a t i o n o f x f r o m f = 0), m u s t n o t only be a necessary b u t also a sufficient condition for the solution of the e q u a t i o n f = 0 for the u n k n o w n x; we call it "the" complete or "full resultant" of the aforesaid e l i m i n a t i o n . It e x c e p t s all i n a d m i s s i b l e systems of values of the o t h e r u n k n o w n s y, z . . . . . a, b. . . . . Each system of values of this u n k n o w n that is sufficient is an "admissible" o n e , w h i c h yields, t o g e t h e r with s o m e values of x, a system of roots of the e q u a t i o n f = 0; it can also be c h a r a c t e r i z e d as a relation b e t w e e n t h e u n k n o w n s w i t h o u t x, which d e p e n d s o n the e q u a t i o n f = 0, the satisfaction of which g u a r a n t e e s the solubility "for x" of the e q u a t i o n f = 0, that is, it g u a r a n t e e s the e x i s t e n c e of at least o n e r o o t value x which satisfies this e q u a t i o n ; it states the "validity c o n d i t i o n " for x. Since we can only deal with the solution of the e q u a t i o n f = 0 for the Page 159 u n k n o w n x in the cases in which the e q u a t i o n is solvable, in which t h e r e are values of x that satisfy it, the solution for x has to be preceded by the d e t e r m i n a t i o n a n d the fu~llment of its (full) resultant (of the e l i m i n a t i o n o f 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 o u r resultants, is again a s o l u t i o n p r o b l e m which, however, has (at least) o n e u n k n o w n (x) less. O u r original solution p r o b l e m has c h a n g e d , a n d in its place we now have a s i m p l e r one. For the latter, the same principles apply as those e s t a b l i s h e d o r those that we will establish for the original p r o b l e m . 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. T h e result which we have o b t a i n e d in the s e c o n d 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

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first subcase, as well as to the two limiting cases previously discussed; thus we can call it entirely general:

In the algebra of relatives, every solution problem is inseparably connected with the elimination problem (just as in the identity calculus); the s o l u t i o n o f an e q u a t i o n for o n e (or for a system of) u n k n o w n s c a n n o t r e a s o n a b l y be b e g u n b e f o r e we have o b t a i n e d the full r e s u l t a n t o f t h e e l i m i n a t i o n o f this u n k n o w n , which, in turn, is solved in t e r m s o f all o t h e r (or t h e t h e r e i n o c c u r r i n g ) u n k n o w n s ; it requires, first, the c o m p l e t i o n of this e l i m i n a t i o n as a f o r e g o i n g o r preliminary task. T h e results of the o t h e r cases are s u b s u m e d u n d e r this o n e , insofar as the p r o o f of the absence of a r e s u l t a n t can be e s t a b l i s h e d a n d was established, to get a r e s u l t a n t 0 - 0 (that is, to achieve t h e p r o o f that "the resultant" leads only to 0 = 0), and, further, the p r o o f o f the impossibility of the solution, or the absurdity 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 r e s u l t a n t leads to 1 - 0 .

Page 160

The problems mentioned before, which we occasionally characterized as difficult problems, will have to be dealt with under elimination problems. If we assume, for the time being, that the set of u n d e t e r m i n e d relatives o r " u n k n o w n s " which o c c u r in the e q u a t i o n f = 0 is limited (its n u m b e r b e i n g "finite"), t h e n these two m a i n p r o b l e m s r e m a i n : First, to eliminate one u n k n o w n f r o m one e q u a t i o n . S e c o n d , "to calculate" an u n k n o w n f r o m the e q u a t i o n , t h a t is, to solve the g e n e r a l e q u a t i o n for it, if "the" r e s u l t a n t o f the e l i m i n a t i o n is fulfilled. Let us s u p p o s e that we can h a n d l e b o t h these p r o b l e m s , the p r o b l e m o f the e l i m i n a t i o n of one u n k n o w n a n d the s o l u t i o n for one u n k n o w n in every case; t h e n we can satisfy every r e q u i r e m e n t f = 0, which is n o t a b s u r d , in general: We e l i m i n a t e o n e u n k n o w n after the o t h e r in any s e q u e n c e , until they have all "fallen out" a n d we have r e a c h e d the r e s u l t a n t 0 = 0. This will h a p p e n , at the latest, with the e l i m i n a t i o n of the last u n k n o w n . 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 elimination of an x does certainly not "mention" or "contain" this eliminant as a term, operational term, or argument; but there can be other unknowns unrepresented 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 o n e , a n d p e r h a p s even m o r e , c o n t a i n s fewer u n k n o w n s t h a n its p r e d e c e s s o r . T h e original equation f = 0 itself may be c h a r a c t e r i z e d as "zer0th resultant" w h e r e a s the identity 0 = 0 is to be seen, as we have already said, as its "last." T h e satisfaction of any R' of these resultants is a n e c e s s a r y a n d suf-

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ficient c o n d i t i o n for the solvability o f any i m m e d i a t e l y p r e c e d i n g R for any o n e o f the u n k n o w n s which fell o u t with the e l i m i n a t i o n ; that is, which are n o l o n g e r in R', b u t c o u l d be in R. W h e r e a s the o t h e r "sup e r f l u o u s " u n k n o w n s can be a s s u m e d arbitrarily, we n e e d for the satisfaction of R, as s o o n as R' is satisfied, to solve R only for one o f t h e m , Page 161 in o t h e r words, to express o n e of these s u p e r f l u o u s u n k n o w n s in terms o f the o t h e r s (and already d e t e r m i n e d f r o m R'), the first o f which will then remain undetermined. We now have the rule: to satisfy the resultants in o r d e r [Reihenfolge], in the reverse o r d e r f r o m that in which they were o b t a i n e d t h r o u g h successive elimination. We thus in the first place satisfy the s e c o n d b u t last r e s u l t a n t by m e a n s o f its solution for any o n e 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, a n d o t h e r u n k n o w n s fallen o u t f r o m its predecessors; it has to be solved unconditionally, b e c a u s e the last r e s u l t a n t 0 - 0 is certainly satisfied. We t h e n substitute the system o f values o f roots, thus g a i n e d for the u n k n o w n s u n d e r c o n s i d e r a t i o n , into all previous resultants (to a n d i n c l u d i n g the e q u a t i o n f = 0 ) , a n d t h e n deal likewise with the previous resultant, a n d 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 eliminated. This o b s e r v a t i o n seems to justify the fact that we will o c c u p y ourselves f r o m now o n only With the a p p a r e n t l y very m u c h m o r e specialized problem o f e l i m i n a t i o n a n d solution which has only one relative as its elim i n a n t or u n k n o w n . 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 "unknowns" to begin with and have to be dealt with just as the x, y . . . . . .

The astute reader will easily understand why our discipline differs in this respect from arithmetical analysis.

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w 12. General and Rigorous Solutions Let

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, w h i c h is "solvable," t h a t is, has at least o n e r o o t x. Page 162

If other u n d e t e r m i n e d 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 " c o m p l e t e " s o l u t i o n of e q u a t i o n 1) for x (p. 156) t h e specification of all relatives which, w h e n s u b s t i t u t e d for x, satisfy t h e e q u a t i o n a c c o r d i n g to the laws of relative a l g e b r a , s e p a r a t e f r o m all relatives w h i c h d o n o t satisfy it. T h o s e r e l a t i v e s - - t h e "roots" of the e q u a t i o n n c a n be c o l l e c t e d in t h e o r y as well as in p r a c t i c e into o n e united expression w h i c h i n c l u d e s all o f t h e m , b u t only t h e m , a n d is thus called the "general root (or solution)" of the equation. We n o w aim to establish s o m e f u n d a m e n t a l p r o p o s i t i o n s a b o u t t h e m , w h i c h i m p r e s s u p o n o u r w h o l e discipline 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 w h i c h we call arbitrary if t h e u n k n o w n x has n o o t h e r d e t e r m i n a t i o n s t h a n to satisfy e q u a t i o n 1), w h e r e , f u r t h e r , f signifies a s o m e "function in the sense o f t h e a l g e b r a o f b i n a r y 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 r o o t - - o r "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

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SCHRODER'S LECTURE V

considered for all possible values of u; it will be the same in all cases, it will be 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 = 1l

f(u) },

3)

w h e r e the sum on the right has to e x t e n d over all possible relatives u within .12. And, third, I claim: that we can impose certain o t h e r r e q u i r e m e n t s , which I call "adventivg' because they are not already i n c l u d e d in the n o t i o n of a general solution, on a function f which, a c c o r d i n g to 3) (and also, f u r t h e r m o r e , a c c o r d i n g to 2)), is capable of r e p r e s e n t i n g "a g e n e r a l s o l u t i o n " m t h a t is, all roots exclusively--of e q u a t i o n 1), a n d 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 p r e f e r r e d , forces itself u p o n us as i m m i n e n t l y useful. We want to begin the p r o o f of o u r claims by showing that the equivalence 3) expresses the notion o f f ( u ) as the general root of e q u a t i o n 1). If an expression 2) is to r e p r e s e n t this general root, it m u s t possess two properties. First, it has to yield for every value of u a correct root x of o u r e q u a t i o n F(x) = 0, such that

Flf(u)} =0

5)

is an identity; in o t h e r words, this e q u a t i o n is valid as g e n e r a l f o r m u l a for any arbitrarily c o m p o s e d relative u. This m e a n s that o u r expression 2) may yield only roots from o u r e q u a t i o n 1). T h e p r o o f that this is i n d e e d true for a definite function f(u) will be called "Proof I"; this function represents the general solution of e q u a t i o n 1). More completely than by 5), this r e q u i r e m e n t will be e x p r e s s e d by II [{x =f(u)} =(= IF(x) = 0} ],

Page 164

u

6)

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

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FROM PEIRCE TO SKOLEM

condition, the hypothesis, the conditional statement of the propositional subsumpfion 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], u

7)

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 b e c a u s e the p r e d i c a t e of the p r o p o s i t i o n a 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, c o n s t a n t in view o f the p r o d u c t variable u, 6) can be rewritten e q u i p o l l e n t l y a c c o r d i n g to p r o p o s i t i o n s already k n o w n (namely, a c c o r d i n g to Th. 3+)) as E {x =f(u)] ~ IF(x) = 01. u

8)

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. S e c o n d , o u r e x p r e s s i o n f(u) also has to yield every r o o t x o f o u r equation 1); that is, if x r e p r e s e n t s any o n e given relative in such a way that it satisfies the e q u a t i o n F(x) = 0, t h e n t h e r e also has to be a relative u for which o u r f(u) will be equal to this x. This r e q u i r e m e n t is correctly e x p r e s s e d by IF(x) = 0} :(= E If(u) = x}.

9)

T h e proof, for a definite f u n c t i o n 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) r e p r e s e n t s the g e n e r a l solution o f e q u a t i o n 1). T h e two r e q u i r e m e n t s 8) a n d 9) tell us that the e x p r e s s i o n f(u) Page 165 "yields," or comprises, only roots a n d also every r o o t o f e q u a t i o n 1); they are now necessary a n d sufficient conditions, so that we can call f(u) the g e n e r a l solution of e q u a t i o n 1); they c h a r a c t e r i z e f(u) as "the g e n e r a l root" o f 1). T h e two 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 8) a n d 9) can be p u l l e d t o g e t h e r as forward- a n d b a c k w a r d - e q u i p o l l e n t to the p r o p o s i t i o n a l equation 3),

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which f o r m u l a t e s n o t h i n g m o r e than the notion of 2) as the " g e n e r a l solution" of 1), as was c l a i m e d ( u n d e r the "second" claim), q.e.d. To c o n t i n u e with the justification of o u r claims, the m a i n task is now the following: We a s s u m e e q u a t i o n 1) to be solvable; it has thus at least one root. Let the relative a be o n e such root; t h e n we h a v e m n o t j u s t as an e q u a t i o n which has to be fulfilled, b u t o n e which is t r u e m F(a) = 0

10)

- - w h e r e a s for an arbitrarily c h o s e n x, we have in g e n e r a l F(x) :/: O, a n d the e q u a t i o n F(x) = 0 has to be c o n s i d e r e d n o t as satisfied, b u t r a t h e r as "a f o r m u l a " which has to be satisfied by a p p r o p r i a t e d e t e r m i n a t i o n o f x. If we now f o r m the e x p r e s s i o n f ( u ) = a " 1; F(u) ;1 + u " {O ~ F(u) ~ O},

11)

then, indeed,

x =f(u) m u s t be a f o r m of the g e n e r a l solution of 1), satisfying r e q u i r e m e n t 3). Proof. C o n s i d e r i n g that, a c c o r d i n g to 1) of w 11, the relative 1;F(u);1-

1 if F(u) :/: O, 0 i f F ( u ) =0,

and, conversely, the relative O ~ F(u) o~ 0 -

0 if F(u) :/: O, 1 i f F ( u ) =0,

we see i m m e d i a t e l y that we have f ( u ) = a" 1 + u " 0 = a, w h e n u is not a r o o t of the e q u a t i o n F(x) = 0, but, on the o t h e r h a n d , we get f ( u ) = a 90 + u 91 = u, w h e n u satisfies the e q u a t i o n F(u) = 0, a n d can now be Page 166 a s s u m e d as a r o o t x of it. In particular, for the a s s u m p t i o n u = a, b o t h o f these results coincide. If we assume that the relative u is c h o s e n arbitrarily, t h e n it is e i t h e r n o t a r o o t of e q u a t i o n 1) or it is one: u--- x. In b o t h c a s e s m a s we have j u s t s e e n m f ( u ) yields a root; in the f o r m e r case, it is always the o n e we already know, r o o t a, existing by a s s u m p t i o n ; in the latter case, it is the luckily g u e s s e d r o o t x. In either case, the above e x p r e s s i o n f ( u ) yields only roots o f the e q u a t i o n F(x) = 0, a n d it yields all roots of this e q u a t i o n b e c a u s e it already p r o d u c e s any d e s i r e d x of these roots by the ass u m p t i o n u -- x. O u r f ( u ) is thus n o t only sufficient for the r e q u i r e m e n t s which are

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i n c l u d e d in the n o t i o n of the g e n e r a l r o o t 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 n e g a t i o n of F(u) m o r e conveniently by F(u), instead of by F(u), t h e n we have the t h e o r e m : {F(a) = 0} :(= ({F(x) = 0} = ~ [x = a" 1 ;F(u) ; 1 + u{0 ~ F(u) j- 0}]) u

12)

- - w h i c h , 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 a

IF(a) = 0} = 1

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 f u r t h e r r e m a r k that the g e n e r a l solution x = f ( u ) , which we f o u n d by 11) a n d gave in 12), corresponds to itself according to the principle of duality (by c o n t r a p o s i t i o n ) . Namely, the expression of our x =f(u) formed by dual correspondence from 11) would be x = (a + 0 ~ F j- 0)(u + 1 9F; 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.

Page 167

Thus, each solution i s - - a f t e r a is o n c e c h o s e n for the k n o w n r o o t or p a r t i c u l a r solution 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. T h e 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" ( b e l o n g i n g to r o o t a) of e q u a t i o n F(x) = O. T h e "rigorous" solution is o n e of the forms of the " g e n e r a l " solution. T h e p r o o f of the existence of the g e n e r a l f o r m of the solution (by establishing it explicitly) now also yields the missing a r g u m e n t s for what we c l a i m e d u n d e r the "first" a n d "third" c l a i m s - - a t least as far as we i n t e n d e d : that it be "theoretically" possible that the g e n e r a l r o o t x of 1) can only be r e p r e s e n t e d in the f o r m of 2), w h e r e u is arbitrary a n d f also fulfills the adventive r e q u i r e m e n t 4). But t h e r e is m o r e . We may be sure of the e x i s t e n c e of a g e n e r a l solution of such a character, n o t only in theory, b u t also we can in

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p r a c t i c e establish i t - - a t least as a " r i g o r o u s " s o l u t i o n . T h e p r o b l e m o f a c o m p l e t e s o l u t i o n o f 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, o r very special r o o t a, of this

equation. A n d we can always discover o n e s u c h r o o t - - t h u s especially in all p r o b l e m s with w h i c h o u r t h e o r y has to deal. In g e n e r a l , I have to f o r g o p r o v i n g t h e s e claims h e r e , w h i c h r e g a r d "practice." B u t I w o u l d 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 solutions. 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 rejoice in h a v i n g a c q u i r e d t h e very g e n e r a l l y o b t a i n e d s o l u t i o n 12), this j o y is c o n s i d e r a b l y t o n e d d o w n , we e v e n t u r n q u i e t , w h e n we l o o k at it m o r e closely a n d l e a r n m o r e a b o u t t h e n a t u r e o f s u c h " r i g o r o u s " solutions. It d o e s 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 o f d e s i r e d p a r t i c u l a r s o l u t i o n s o r roots for a series o f arbitrarily a s s u m e d values o f its indefinite 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 hit a r o o t o f e q u a t i o n 1) with t h e a s s u m e d v a l u e m i t only p o i n t s a g a i n a n d a g a i n to t h e l o n g k n o w n , 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 "bori n g " m r o o t a. T h e d e s i r e to d i s c o v e r all roots o f e q u a t i o n 1) with t h e h e l p o f e x p r e s s i o n 11) for t h e r i g o r o u s s o l u t i o n really leads us to prove for all possible relatives u w h e t h e r they m a y be satisfied for t h e e q u a t i o n . T h e " r i g o r o u s " s o l u t i o n is t h e r e f o r e n o t yet a satisfactory f o r m o f t h e g e n e r a l solution; it solves t h e p r o b l e m only in an e m e r g e n c y - - / ~ la rigueur~and that is why I gave it its n a m e b e c a u s e it was n e c e s s a r y to d i s t i n g u i s h it f r o m o t h e r m o r e p r o m i s i n g f o r m s o f t h e g e n e r a l s o l u t i o n . B u t we g o t s o m e hints a b o u t t h e f o r m in w h i c h we have to f i n d t h e g e n e r a l s o l u t i o n o f an e q u a t i o n ; it also t a u g h t us t h a t t h e c o m p l e t e

FROM PEIRCE TO SKOLEM

275

s o l u t i o n o f e q u a t i o n 1) F(x) = 0 exists in t h e f o r m 2) x = f ( u ) . A n d this r e m a i n s a last r e s o r t if we c a n n o t find a " b e t t e r " f o r m o f t h e g e n e r a l r o o t for a g i v e n e q u a t i o n , for all cases w h e n we n e e d a n 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 t h e e x a m i n a t i o n . It m a y n o t b e easy to e s t a b l i s h a n o t i o n o f w h a t c o n s t i t u t e s "a satisfactory" f o r m a n d o f w h a t m a y e v e n t u a l l y be c a l l e d " t h e best" f o r m o f a g e n e r a l s o l u t i o n ; m o s t likely, it will o n l y e m e r g e g r a d u a l l y o u t o f t h e p r a c t i c e o f o u r science. Page 169 At least, we can, in g e n e r a l , justify o r motivate "the first adventive requirement" for t h e g e n e r a l s o l u t i o n . In a n a l o g y with t h e e x e r c i s e a l r e a d y available f r o m a r i t h m e t i c a l analysis, we can call t h e " i n d e t e r m i n a t e a r g u m e n t " u o f t h e g e n e r a l s o l u t i o n 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 expression 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 c o m m o n characteristic of both kinds of parameters i s u t o our justification--their unrestricted arbitrariness. We have already emphasized that we can only consider u as an arbitrary relative when the unknown x is d e t e r m i n e d by the r e q u i r e m e n t 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 p a r a m e t e r u will be subject to certain restrictions, and that it can even turn out to be absolutely d e t e r m i n e d in individual cases; it can, for example, h a p p e n that u =0 has to be taken to yield the root x - - 0 m a s 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 m e n t i o n e d 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 t h e t h e o r e t i c a l r e q u i r e m e n t s o n f(u), w h i c h a r e i n c l u d e d in n o t i o n o f t h e g e n e r a l s o l u t i o n , are e x h a u s t e d with 3), t h e r e is still one requirement--4)--in r e l a t i o n to this p a r a m e t e r u, w h i c h is req u i r e d in practice. H o w c a n we k n o w this u o r s u c h a u, w h i c h yields a d e f i n i t e , p e r h a p s a l r e a d y k n o w n , a given o r desired r o o t x? Systematically, o n e s u c h u c o u l d be g a i n e d by t h e s o l u t i o n o f e q u a t i o n

2) f(u)

=X

f o r t h e u n k n o w n u. B u t this s o l u t i o n p r o b l e m m a y n o t s e l d o m b e m u c h m o r e difficult t h a n t h e o n e u n d e r 1), t h e s o l u t i o n o f w h i c h was exp r e s s e d in e q u a t i o n 2).

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It is an u n d e r s t a n d a b l e wish to know for each u n k n o w n x that satisfies

Page 170 the e q u a t i o n F(x) = 0 i m m e d i a t e l y one u--at l e a s t - - w h i c h would yield, if substituted in f(u), this x. This d e m a n d c a n n o t be satisfied in a s i m p l e r a n d b e t t e 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 w h e n the g e n e r a l solution is so a r r a n g e d 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 g e n e r a l solution is also justified f r o m a s e c o n d a n d a third perspective. T h e s e c o n d is the check or p r o o f of the correctness of a solution that has b e e n f o u n d (or also a root) for e q u a t i o n 1). T h e " g e n e r a l solution" s h o u l d also take care of this c h e c k for each special or p a r t i c u l a r value o f the g e n e r a l root; it s h o u l d spare us the p r o o f by substituting the x, f o u n d as x =f(u), in the polynomial F(x) of the e q u a t i o n to be solved, a n d offer certain g u a r a n t e e s of its c o r r e c t n e s s in itself; it s h o u l d also show the following. In o r d e r to get various or even all roots f r o m 1), we have to assume, as to be substituted or as b e i n g substituted in the e x p r e s s i o n f(u), others, in p r i n c i p l e 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 W i t h o u t c o n t r a d i c t i n g the n o t i o n 3) of the g e n e r a l solution of 1), f(u) can be so c o n s t i t u t e d that, w h e n we substitute a r o o t x~ for u, any o t h e r r o o t x 2 =f(x~) results; a n d if we substitute x 2, again a d i f f e r e n t root, x 3 - f ( x 2 ) , will c o m e out, a n d so on. W h e t h e r a value taken for u is p e r h a p s not a root of e q u a t i o n F(x) = 0 c a n n o t be seen in this case w i t h o u t m a k i n g a direct proof: substituting it into the p o l y n o m i a l F(x) of o u r e q u a t i o n 1) to c h e c k w h e t h e r it will vanish. O n e of the m a j o r goals of the g e n e r a l solution, n a m e l y to save us o n c e a n d for all f r o m giving this proof, thus evaporates into thin air. We m i g h t as well ren o u n c e the g e n e r a l solution a n d be c o n t e n t with s e p a r a t i n g the relatives empirically into two classes by processing t h e m individually into those u for which F(u) ~ 0 will result, a n d into those u that will be called x for which we obtain F(u) = O. A p a r a b l e will illustrate the point. T r a n s p o r t a t i o n by railway would Page 171 n o t h e l p us if the train passed the d e s i r e d station w i t h o u t s t o p p i n g or if the stations were not identified. If we have correctly a s s u m e d or guessed a r o o t x, which may be of p a r t i c u l a r interest to us, p e r h a p s o b t a i n e d it f r o m reflections of still d u b i o u s value, then the g e n e r a l solution has to tell us that this is the c o r r e c t root. We have now r e a c h e d the desired final destination, but we have to know that we are already there; the train may n o t c o n t i n u e to a n o t h e r root. In a d d i t i o n to the p r i m a r y or m i n i m a l r e q u i r e m e n t s of the g e n e r a l solution, which are i n c l u d e d in the n o t i o n of a g e n e r a l solution, we have a s e c o n d a r y or adventive r e q u i r e m e n t because of two reasons which have already b e e n m e n t i o n e d : that t h e g e n e r a l solution thus yields every

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r o o t that was happily guessed; a n d that its expression, w h e n it is substituted for the u in f(u), gives us this r o o t itself, reproduces it. This m e a n s f(u) has to be so constituted, if it may be called a satisfactory g e n e r a l solution, that {F(x) =0} :(= {f(x) = x}.

13)

And, m o r e o v e r , since the converse p r o p o s i t i o n a l s u b s u m p t i o n - - a s c l a i m e d for u = x a c c o r d i n g to 3 ) m i s valid, we may also give this subs u m p t i o n the f o r m of equation 4). This a d d i t i o n a l or adventive r e q u i r e m e n t 4) of the g e n e r a l solution suffices, in turn, if, for s o m e function, f is satisfied, b u t not yet to justify this f u n c t i o n as suitable to r e p r e s e n t the g e n e r a l solution of 1); it only g u a r a n t e e s that the f u n c t i o n f(u) e n c o m p a s s e s all roots x a n d leaves o p e n w h e t h e r it can assume or yield o t h e r s as r o o t values; it only tells us that " P r o o f 2" has to be correct. O n the o t h e r h a n d , we have seen that 4) is n o 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 c h a r a c t e r i z e a "not initially unsatisfiable" g e n e r a l solution, we o u g h t to write the p r o d u c t of p r o p o s i t i o n s 3) a n d 4) or t h e i r n e g a t i o n as double equation {F(x) = 0} = E Ix =f(u)} = {f(x) = x}. u

14)

For all solutions of special p r o b l e m s which we stipulate f r o m 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, a n d o u r specification of solutions s h o u l d i n d e e d satisfy t h e m (if n o t otherwise r e m a r k e d ) a n d claim t h e m to be satisfied. It would be too t r o u b l e s o m e to a c c o u n t for this fact with the explicit a d d e n d u m of f(x) = x---especially w h e r e f(u) has a c o m p l i c a t e d e x p r e s s i o n m a n d we shall thus limit ourselves to e x p r e s s i n g the solution in the f o r m 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.

Page 172

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 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. u

T h e "rigorous" solution has shown that we can always satisfy the ad-

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S C H R O D E R ' S LECTURE V

ventive r e q u i r e m e n t 4) as well as 3) with t h e c o n s t r u c t i o n o f a s u i t a b l e f u n c t i o n f(u). B u t 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 tog 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 p o i n t s to a t h i r d p e r s p e c t i v e , w h i c h m o t i v a t e s t h e use o f t h e a d v e n t i v e r e q u i r e m e n t : t h a t we have to a i m at f l a m i n g o u r p r o b l e m as a m o r e d e f i n i t e o n e , to m a k e t h e n o t i o n o f " o n e " g e n e r a l s o l u t i o n p r e c i s e in s u c h a way t h a t we c a n s p e a k o f "the" g e n e r a l s o l u t i o n o f a specific p r o b l e m , o r o f "its" s o l u t i o n in an u n c h a n g i n g way. A l r e a d y in t h e i d e n t i t y calculus we c o u l d i n d i c a t e t h e f u n c t i o n s t h a t a r e s u i t a b l e for all values. F o r e x a m p l e , cu + ~2 is o n e s u c h f u n c t i o n , s i t u a t e d 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 infinite set. This m e a n s t h a t we have to a d m i t t h a t t h e r e are also f u n c t i o n s in t h e 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 r e c a p a b l e o f t a k i n g o n t h e value o f any d e s i r e d relative b e t w e e n 0 a n d 1 (inclusive), acc o r d i n g to t h e value w h i c h we give to u. I n d e e d , t h e r e is an i n f i n i t e n u m b e r o f f u n c t i o n s ~(u) with this p r o p e r t y : th(u) - ~ o r ~ o r u a r e f u r t h e r m o r e (the simplest) e x a m p l e s o f t h e m . . . If ~(u) is thus suitable for all values ("varying w i t h o u t limits"), t h a t is to say, x = f ( u ) is a g e n e r a l s o l u t i o n o f 1) in t h e earlier, " b r o a d e r " s e n s e ( o n l y l i m i t e d by 3), t h e n obviously x =j~O(u)}

15)

Page 173 b e c o m e s a g a i n "a g e n e r a l s o l u t i o n " 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 ~ w h i c h 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 designated by ~(u), will not satisfy it; less still ~F(x) = x, but certainly only 'F(~) = x ~ a n a l o g o u s l y '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 (abovementioned) rigorous proof that 4) cannot follow from 3), because we are now in a position to specify functionsfwhich 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).

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FROM PEIRCE TO SKOLEM

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 r e p r e s e n t the general root x of 1). T h e i n d e t e r m i n a t e n e s s of the notion of the general solution of 1) is somewhat controlled by the adventive r e q u i r e m e n t 4 ) - - b y which in fact the solutions, m e n t i o n e d 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. T h a t this r e q u i r e m e n t 4), with the adjunct 3) that is included the notion of a general solution, is not yet sufficient to d e t e r m i n e 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) a n d 4) and yet are essentially different, will be seen in special solution p r o b l e m s in the t h e o r y . - 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}]), u

{F(1) = 0}=~- [{F(x) = 0 } = E { x = u + u

1 ;F(u);1]].

16)

17)

So m u c h for the solution problem in general. T h e elimination problem is, without fail, associated with it and culminates in the r e q u i r e m e n t to eliminate from every equation 1) or subs u m p t i o n F(x) =~-0 one relative x. If we are able to eliminate any desired relative, then we are able to eliminate several relatives in any sequence and thus also any one system of relativesmsimultaneous in their eff e c 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 eliminate from any one equation

f(u) = 0 a relative u It seems useful to tackle the p r o b l e m in the formal, r a t h e r general version; to form immediately the resultant of the elimination of u from the equation of the form f(u)

= X.

Since the latter could be easily b r o u g h t to the predicate 0, it is clear

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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 p r o b l e m of elimination (in a m o r e special form). In this e x t e n d e d version, the elimination p r o b l e m appears as the i m m e d i a t e converse, inverse of a pure solution problem, and it has the advantage of letting us see that with every pure solution p r o b l e m a certain elimination p r o b l e m is also solved, and vice versa. T h e f o r m e r allowed us to p r o c e e d from the equation F(x) = 0 to its Page 175 general solution x-f(u); the latter d e m a n d s that the reverse path be taken! If we have found the solution of 1) with 3), we have the subsumptionmfollowing a f o r t i o r i from 8 ) m If(u) = x} =(= IF(x) =0}

18)

a n d the solution to the p r o b l e m of the elimination of u from the left side of the equation, namely, from 2); and i n d e e d the right side, that is, equation 1 ) m b e c a u s e of 9 ) m i s 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 c o n s e q u e n c e from the fact, plausible from the beginning, that very different initial equations will yield the same resultants, just as occasionally very different premises yield the same conclusions. To obtain the latter is the goal of the elimination problem. And conversely, we could posit obtaining all general solutions to a given equation 1) (as the total proposition 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 se.m

At the beginning of this section we assumed (what can always be a d d e d 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 elimination of x would Translator's note: English in text.

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not yield any "resultant." In practice, the opposite is usually the case. And if we have now found for that case the general s c h e m e by which the solution always has to be a p p r o a c h e d , we still have to deal with this case. We have to tax the patience of the r e a d e r 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 p a r a m e t e r s t h o u g h t to be given, such as polynomial coefficients, for example, possibly also of o t h e r unknowns. Page 176 T h e 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 u n d e t e r m i n e d relatives which occur in 1), besides x, is either 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 equation c a n n o t be solved, is {F(x)= 0 } - 0, or the equation F(x)= 0, for every m e a n i n g which we may attach to x, is absurd. O u r scheme t h e r e u p o n proves to be the propositional equivalence 0 - 0. In the second case, we have (R = 0) = 1. T h e n the condition is satisfied according to which we have justified scheme 3), that is to say, equation 1) is solvable. O u r 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=Br. To conclude this discussion, I would like to say a word a b o u t the

methods for solving the two problems. We can also r e p r e s e n t these problems which are associated with equation 1) as the analogous p r o b l e m for the coefficients of the unknowns, respectively, of the eliminant x, by calculating or e x p a n d i n g , for each suffix ij, the left side of the condition to be satisfied

{ F(x) }o = 0

20)

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SCHRODER'S LECTURE V

a c c o r d i n g to the stipulations of w 3. T h e n we only have to calculate, respectively, to eliminate, the general coefficient Xhk----or better, all Xhk m a s an u n k n o w n f r o m the e q u a t i o n ; t h e n the p r o b l e m p r e s e n t s itself Page 177 as o n e o f the p u r e p r o p o s i t i o n a l calculus. Already in the identity calculus, even m o r e for this calculus, the m e t h o d s of solution a n d elimin a t i o n were b r o u g h t to a certain stage of p e r f e c t i o n . T h e y were develo p e d in extenso a n d cultivated into m e t h o d s that c o u l d be satisfactorily wielded. Nevertheless, it would be w r o n g to t h i n k that every p r o b l e m can be solved in o u r d i s c i p l i n e m a n d this for the r e a s o n that .... b e c a u s e we have only focused on certain a n d limited sets of u n k n o w n s , respectively e l i m i n a n t s , a n d b e c a u s e in fact this m e t h o d a p p e a r s sufficient or fairly qualified only for the calculation o f or e l i m i n a t i o n of these sets. As a rule, the universe o f discourse is to be a s s u m e d to be infinite, a n d 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 o f u n k n o w n s a n d eliminants; even if the universe of d i s c o u r s e 11 consists only of a few e l e m e n t s as i n d i v i d u a l s ~ l e t us say, t h r e e or m o r e m t h e calculations to be c a r r i e d o u t a c c o r d i n g to k n o w n m e t h o d s will reveal themselves to be scarcely feasible in practice, with the n u m b e r o f u n k n o w n s increasing by the square. Finally, even if the p r o b l e m c o u l d be solved for the coefficients, the reverse c o n c l u s i o n back to the relative itself, for which relation, respectively, relations, we i n q u i r e d , is n o t that easy to do. It is easy to a c q u i r e a m a s t e r y o f the basic principles of o u r discipline, b u t its two f u n d a m e n t a l p r o b l e m s have to be called difficult. So far, t h e r e is no method to solve t h e m in general. In our next lecture we will propose such a solution for a group of 512 problems. 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.

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w 13. Continuation. Iteration. Limiting Values and Convergence. Power. R a t h e r g e n e r a l are t h e two classes o f s o l u t i o n p r o b l e m s , in w h i c h t h e 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 e q u i v a l e n t l y in e i t h e r o f t h e two f o l l o w i n g ways:

x ~(x),

~(x) 4= x.

T h e s e are cases in w h i c h t h e 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--Ohas t h e f a c t o r x, o r Y - - t h e n its cofactor, r e s p e c t i v e l y t h e f a c t o r itself, can be p u t o n 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 s o l u t i o n o f the p r o b l e m in b o t h cases can always be given in the f o r m o f t h e often infinitely iterated function o f an a r b i t r a r y relative 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 e x p r e s s i o n . In fact, t h e two t h e o r e m s are valid: {x =(= 4~(x)} = ~ {x =f=(u)}, w h e r e f(u) = uq~(u),

{~b(x) :(=x} = E {x = f = ( u ) } , u

w h e r e f(u) = u + ~b(u).

1)

T h e e x p l a n a t i o n a n d p r o o f o f t h e s e 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 e m a r k s . First, we will d e f i n e t h e "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 o r " f u n c t i o n " in t h e usual s e n s e o f t h e a l g e b r a o f b i n a r y r e l a t i v e s ) - - f i r s t for all " e x p o n e n t s " r t h a t a r e (finite) natural numbers. This d e f i n i t i o n has to be m a d e in the usual way "by i n d u c t i o n , " by w h i c h we m e a n n a m e l y t h a t

f~

=u,

in g e n e r a l " --to

f ' ( u ) = f ( u ) , f 2 ( u ) =j{f(u)} . . . . f~+l(u) - f{ f (u)}

2)

this we only have to r e m a r k that t h e symbols w h i c h a p p e a r as

"exponents" in o u r theory, O, 1,2 . . . . . r, r + 1 . . . . , n e v e r o u g h t to be und e r s t o o d as relatives, b u t always o n l y 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 n u m b e r [Anzahl], namely, to justify the so-called "definition by induction" as a definition, to demonstrate its efficacy as one which Page 179 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 rem e m b e r 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 mentioned--just as it occasionally happened in the course of the previous volumes, and, incidentally, as it is customary

SCHRODER'S LECTURE V

284

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 lectures, which do not follow Euclid's ideal of an absolutely rigorous, step-by-step constructionDas, for example, Mr. Dedekind's m o n o g r a p h 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 o r d e r 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 o f 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:

f"+ 1(u) =f'{f(u)},

3)

f " {f" (u)} = f .... (u) =f"{f" (u)}

4)

( f ' ) " (u) = f ..... (u) = (f")"' (u)

5)

as well as in general

Page 180 m a n d if we wish also

D a s indeed will be immediately clear to those who already understand the concept of n u m b e r 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 b e defined, provided that t h e n a m e is c a p a b l e o f a d e f i n i t i o n w h i c h is b a s e d o n t h e n o t i o n o f f• a n d is m o t i v a t e d by t h e b e h a v i o r o f this relative for all a n d , in p a r t i c u l a r , for infinitely i n c r e a s i n g i t e r a t i o n e x p o n e n t s X. T h e c o n d i t i o n f o r this last case will be c a l l e d t h e convergence condition for fX(u), n a m e l y f o r i n f i n i t e l y i n c r e a s i n g ~. In g e n e r a l , if t h e value o f a relative u• is determinate for t h e i n f i n i t e

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

285

series o f n a t u r a l n u m b e r s 3, = 0, 1,2, 3 ..... for e x a m p l e , actually given for arbitrarily m a n y relatives o f the "series" UO, U 1, U 2, U3, . . . ,

conceptually fixed for the rest by a law or principle, t h e n we have to say for an infinitely increasing i n d e x ;~ (as natural n u m b e r ) in g e n e r a l that u• "diverges" a n d that the symbol u~ has n o m e a n i n g ; b u t t h e r e is also a class of cases in which we can say that the g e n e r a l t e r m u• o f o u r series "converges" b e c a u s e it "strives" toward a d e t e r m i n a t e , fixed relative a n d thus toward a relative to be d e s i g n a t e d u= as the "limit." T h e latter case occurs if, a n d only if, for every position o n the matrix o f 1~ m a r k e d by a suffix ij---or, in o t h e r words, the m a t r i x 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.

Page 181

A position ij o f the matrix o f the relative u x with variable ;k shall be called a final [endgiiltig], "definitely" occupied position o f this variable relative, if t h e r e is such a value n o f ;k such that for all ~ > n the position in u• turns o u t to be o c c u p i e d ; it shall be called a final unoccupied or definitely empty position, if t h e r e is such a n u m b e r n that the position in u• r e m a i n s u n o c c u p i e d for all X > n. By using this f o r m of expression, we can say m o r e briefly: u• shall be called convergent with increasing X if, for every position o f its matrix, 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 definitely to r e m a i n empty. By the limiting value [Grenzwert] (limes) o f u• (for lim ;k = oc) t h e n we u n d e r s t a n d that relative which in the "definitely o c c u p i e d " positions o f u• has "filled circles"; the "definitely u n o c c u p i e d " positions o f u• are empty. A n d this limiting value we d e s i g n a t e u= for short. In g e n e r a l , however, t h e r e are positions ij which b e l o n g n e i t h e r to the definitely o c c u p i e d n o r to the definitely u n o c c u p i e d positions o f the u• d e p e n d i n g o n ~; there, for e a c h i n d e x n, h o w e v e r g r e a t it is, t h e r e is always a n u m b e r m > n such that, if the position is o c c u p i e d in u,,, it a p p e a r s again as u n o c c u p i e d in u,,, a n d conversely. Such positions in u• oscillating with i n c r e a s i n g ;k ( a l t h o u g h n o t necessarily in r e g u l a r variation), which are s o m e t i m e s o c c u p i e d a n d sometimes u n o c c u p i e d , may be called "oscillatory occupied" (or, likewise, uno c c u p i e d ) or (finally) oscillatory* positions of u• W h e t h e r t h e r e is only o n e such position or w h e t h e r t h e r e are several, n o definite n o t i o n (such as that of a relative) can be associated with * Not: "oscillating"; the positions themselves do not fluctuate, only their o c c u p a n t , the "filled circle" oscillates (blinks, scintiUates) between p r e s e n c e a n d absence.

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SCHRODER'S LECTURE V

t h e sign 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 i g u r e 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 i v e r g e s . In o u r d i s c i p l i n e , d i v e r g e n c e c a n always o n l y t a k e p l a c e as "oscillatory." A l t h o u g h this symbol is meaningless, we can nevertheless associate a definite m e a n i n g to it with subsumptions in which it occurs as subject or as predic a t e - a l t h o u g h I do not want to attach great i m p o r t a n c e to this point at this time. But we can write

in o r d e r 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 u n o c c u p i e d 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 u n o c c u p i e d 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 e n c o m p a s s i n g or m a x i m u m value of a, which still satisfies the r e q u i r e m e n 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 e n c o m p a s s i n g or m i n i m u m value of b, which still satisfies the r e q u i r e m e n 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• continuously oscillates between these two limits, after a sufficiently large X, and can only pass t h r o u g h or assume "intermediate values" between them. [It may be that too m u c h has been said with the last corollary, a l t h o u g h it will often be t r u e - - i n particular always with a finite n u m b e r of elements 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 l o n g e r 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 property that, after X has e x c e e d e d it, all n a m e d positions have found their final occupation and can never again be empty positions in u x. But if the set of tile positions which we take into consideration increases indefinitely, it is problematic, and future m o r e subtle e x a m i n a t i o n s will have to decide w h e t h e r or not this greatest of all (smallest) values n (which b e l o n g to each position) is pushed farther away in the n u m b e r sequence, and

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TO

287

SKOLEM

the sequence of values n, itself of infinite growth, does not include any value as the greatest one. T h e n we can indicate for every individual position a n u m b e r )~ = 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 h a v e to s e p a r a t e clearly t h e p r o b l e m o f c o n v e r g e n c e , o r diverg e n c e , of the series itself from t h e p r o b l e m o f 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• o f o u r series, if t h e t e r m s a r e t h o u g h t to b e c o n 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 t h e six s p e c i e s ) . If t h e t e r m s o f t h e series a r e j o i n e d to e a c h o t h e r by a n identity species, t h e n we g e t a n "infinite product" o r an "infinite sum" ( o r "series" in t h e n a r r o w s e n s e ) . T h e n , t h e r e s u l t o f c o m b i n i n g t h e first 3, + 1 t e r m s is Vk

u 0 u 1 u 2 ...

U x = u 0 -~- u 1 +

u k ,

u 2 - - 1 - ' ' ' ~- u x ,

6)

t h a t is, " t h e p r o d u c t i o n a l factor," respectively, t h e s o - c a l l e d " s u m m a t o r y t e r m " o f t h e series, is t h a t g e n e r a l t e r m a b o u t w h o s e c o n v e r g e n c e we a r e c o n c e r n e d in t h e l a t t e r case. H e r e t h e 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. E v e n w h e n t h e g e n e r a l t e r m u• is d i v e r g e n t : we have in o u r d i s c i p l i n e 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 s u m s (series) with d i v e r g e n t t e r m s - - t h e s a m e w o u l d have to be e x p l a i n e d in a n 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 v a l u e in t h e d o m a i n o f b i n a r y 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 j o i n e d to the product with increasing 3,; each occupied position of the Ux 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 (rl

- n (u,j) -- n u,),

u ,j -

(u,)) -

u,)

is d e f i n e d - - f o r every choice, for example, a series, of values u as extension of H, of Z - - t h e r e are only two possibilities for a determinate position ij, namely: Left: Either there is a n u m b e r 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.

288

SCHRODER'S LECTURE V

Left and right show that all positions of the matrix of U• are either definitely occupied or definitely unoccupied, and a third solution, oscillator3, 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 Hu u and Zu u, in whatever way the 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 Eu u).

Page 184

For relative infinite p r o d u c t s a n d sums, a similar p r o p o s i t i o n of the s a m e generality does not hold. We can easily illustrate this point, for e x a m p l e , with the relative p r o d u c t in case of always equal factors. a relative product of factors that are all equal is called a "power." We d e f i n e the power (u;)• or (; u) • when X r e p r e s e n t s a natural n u m b e r , simply as u • (u to the p o w e r ;k), e i t h e r "by i n d u c t i o n " ("recursion") by m e a n s of the stipulation: U 1

=

U,

U 2

=

U ,9 U ,

.-.

, U~,+ 1

=

U X

;u,

8)

or " i n d e p e n d e n t l y " as u•

;u ;u;""

;u.

9)

T h e r e follows the known propositions for the powers in arithmetic: 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 uncomfortable with this, he should accept u x only as a "stenographic code," a conventional sign for the purpose of simplification. T h e dual c o u n t e r p a r t to the p o w e r is the relative s u m 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 h e m "iterates" or "iterative sums." If we wish to push the analogy with the n o t a t i o n in a r i t h m e t i c constructions, t h e n

FROM

PEIRCE

TO

289

SKOLEM

t h e a b b r e v i a t i o n s p r e v i o u s l y i n t r o d u c e d - - w h i c h a r e p a r a l l e l to w r i t i n g u • f o r (u ; ) • h a v e to b e c h a n g e d , w h e r e t h e i t e r a t i o n e x p o n e n t ~, has to b e p u t as a " m u l t i p l i e r " b e h i n d t h e u. We w o u l d t h e n h a v e to distinguish three multiplications---counting the arithmetical one, f o u r - - a n d t h e 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 a r e d e f i n i t e l y t o o m u c h ("All g o o d t h i n g s a r e t h r e e " ) . T h e r e a d e r m a y write d o w n o r f i g u r e o u t t h e d u a l c o u n t e r p a r t s to t h e a b o v e laws o f power. W e c a n t h u s claim: t h e p o w e r u • d i v e r g e s 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 t h e f o l l o w i n g a s s u m p t i o n b e l o n g s to t h e u n i v e r s e o f d i s c o u r s e 1~ o f o n l y t h r e e e l e m e n t s : 0

u=

9

0

9o 9 0

9

0

for w h i c h

~i=

0

9

9o 0

9

Page 185

and

we

now

have u; u

= u 2 = u, u2;

u -- u 3 - u ; s o

u x

oscillates,

regularly, w i t h o u t

fluctuates

e n d , f r o m o n e to t h e o t h e r o f t h e two v a l u e s u a n d ~i b e c a u s e we h a v e U 2~ = Z l ,

U 2 X + 1 __ U ,

a n d u • is d i v e r g e n t , t h e s y m b o l u,0 h e r e is

meaningless.

Of course, we could arbitrarily attribute to this meaningless n a m e any chosen meaning. In whatever way we would make this choice, rational considerations c a n n o t be found for it, the introduction of such a n a m e would not grant any advantage; on the contrary, it would do much damage. This n a m e 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 c u m b e r s o m e exceptions which necessarily occur and which we have not e n c o u n t e r e d here. Meaningless names in a discipline are, as it were, by-products of a certain (notation) industry. At times, they present valuable raw material which can be processed in a n o t h e r industry and thus finds a valuable usage in the overall econo 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 e x t e n d e d domains of numbers. But the circumstances are not always so favorable; m u c h waste has to be discarded. N o w we h a v e o0

~k 1

U X=

U "+" U 2 -t-" U 3 ~'- U 4 "-t-"'"

29 ~

SCHRODER'S

LECTURE

V

as an obvious e x a m p l e o f 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 g e n e r a l t e r m . It has for o u r u above the s u m u + ~i - 1. F o r t h e c o n v e r g e n c e o f t h e p o w e r u • o f a relative u---with an e x p o n e n t X i n c r e a s i n g w i t h o u t e n d - - t h e necessary and sufficient c o n d i t i o n s are n o t yet k n o w n . But s o m e c i r c u m s t a n c e s can be p r o v e d as sufficient conditions. So x • has to c o n v e r g e for 3, = ~, if x has t h e p r o p e r t y t h a 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 t h a t x • =(=x• in the last, t h a t x • : ~ x • has to h o l d for e a c h X ( h o w e v e r g r e a t ) . In t h a t case the empty positions will be finally c o n s e r v e d by cont i n u a l relative m u l t i p l i c a t i o n with x, b e c a u s e w h e r e v e r x • has an e m p t y p o s i t i o n , also x • m u s t have o n e ; t h e p o w e r c o n v e r g e s t h e n "decreasingly" Page 186 t o w a r d a fixed limit. In this o n e , the s a m e is valid for t h e "filled circles," a n d t h e p o w e r c o n v e r g e s "increasingly" to a limit. In particular, if x is o f t h e f o r m o f 1' + a, t h e n 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 e a c h m e a n i n g of a. Also, w h e n the base x of a p o w e r x • is o f t h e f o r m a ; 6 ( t h a t is to say, t a k e n for a, at the s a m e time also of the f o r m d;a), this p o w e r has to c o n v e r g e - - a n d this b a s e d o n the p r o p o s i t i o n : a ; ~ :(= a ; ~ ; a ; ~ ,

a ~ a ~ a~t a ~: a ~ ~,

w h i c h we will identify later as a special case (for b = 4) o f a g e n e r a l p r o p o s i t i o n - - 2 1 ) of w 1 8 - - i n t h e m e a n t i m e , we p r o v e it by the coeffic i e n t e v i d e n c e , with t h e r e m a r k t h a t the t e r m s o f 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 d e t o u r , we n o w r e t u r n to t h e i t e r a t i o n of the f u n c t i o n s 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 w h i c h "the r times (rth) iteration o f a f u n c t i o n f(u)," namely, fr(u) for lim r = 00 c o n v e r g e s , t h a t is to say, "in general"for every =

argument u. An e x a m p l e f r o m the identity calculus will c o n f i r m this p o i n t . W h e n we a s s u m e

f(u) = au + b, for w h i c h fZ(u) = a(au + b) + b = au + b, t h u s fe(u) = f l u ) , 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 g e n e r a l : fr(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 g e n e r a l , for every a r g u m e n t , its s e c o n d i t e r a t i o n is e q u a l to t h e first (or the f u n c t i o n itself) m a y be called "invariant." All i t e r a t i o n s o f s u c h a f u n c t i o n , to b e g i n f r o m t h e z e r o t h ab, are t h e n e q u a l to it, as can be easily seen: each invariantfunction

FROM PEIRCE TO SKOLEM

291

remains unchanged by iteration, a n d also the f u n c t i o n which is iterated infinitely often is then no o t h e r than itself. u itself is also an invariant function of u. In general, the rth iteration of a s o m e h o w given f u n c t i o n f(u) m u s t diverge with increasing r. This point, too, the identity calculus can illustrate. T h e m o s t g e n e r a l f u n c t i o n of u which can be f o r m e d with the species of this discipline is f(u) = au + bfi. T h u s we have f{f(u)} = a(au + bgt) + b(du + b~i); therePage 187 fore fZ(u) = (a + b)u + abfi = (a + b)u + ab. This is different from f(u) in g e n e r a l m a s , for e x a m p l e , the a s s u m p t i o n b = d easily shows, w h e r e for f(u) = au + gt~i we now have f 2 ( u ) = u = etc. f~ O n the o t h e r h a n d , we have again in the above g e n e r a 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 divergent a n d f~176 is meaningless, e x c e p 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 ) , a n d this f u n c t i o n is invariant. If t h e r e is a pair of n u m b e r s m, n such that for every definite f u n c t i o n f(u) we have, for ever?, u,

fro+, (U) =fro(U) , we call the f u n c t i o n a "periodically (or oscillating) iterate, with an iteration period n" or, in short, "with the iteration p e r i o d n" in case n is at the same time the smallest n u m b e r of the n a m e d p r o p e r t y (which such a pair of n u m b e r s m, n has). T h e r e f o r e , the notion of an invariant function is s u b s u m e d u n d e r that of a periodically iterating function of period I. While the iterations of such functions are c o n v e r g e n t , each periodically iterating function, which has a p e r i o d n > 1, has to be divergentiterating. 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.

292

SCHRC)DER'S LECTURE V

general function in the identity calculus we h a v e s e e n t h a t it has to be, 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

if it is n o t i n v a r i a n t ,

For the two meanings o f f ( u ) m e n t i o n e d in our T h e o r e m 1, we now have

f~+' (u) =f~(u)4~{f~(u)},

Page 188

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 s u m m a n d : "r of all up to now"; the law of formation of the iterating function is easily overlooked, although the expressions b e c o m e rapidly m o r e 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)], ... a n d on the right f(u) = u + r f~(u) = u + r

f2(u) = u + r +r

+ r

+ r

+ r

+ r

+ r r

+ r

"".

Although the names for "all up to now" increase in length, the difficulties or efforts in calculation do not increase at all. T h e formation o f f ~+l(u) from f~(u), already obtained, remains as easy as before, and requires no m o r e work than the calculation of the function 4), respectively r itself for any given argument. T h e indefinitely c o n t i n u e d 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 proposition proved above); f = ( u ) has a very definite value. "Proof 1" for our T h e o r e m 1 requires us to show that it specifies a root of the solvable subsumption, whatever value of the arbitrary par a m e t e r u may be chosen. "Proof 2" requires us to show that if we have from the b e g i n n i n g x~(x),

r

~x,

then also f~ = x must hold. T h e latter is easy, in view of the fact that the assumptions can be written as equivalences: x = x~(x),

thus:

x + r

= x,

293

FROM PEIRCE TO SKOLEM

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

Page 189

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. Theretore, we had to determine the relative f=(x), so that the same is true for the latter. P r o o f 2 is thus certain, and it also a p p e a r s certain that o u r solution 1) will yield all roots of the p r o b l e m . T h e first is not that simple, namely, to show "Proof 1" or bring the p r o o f that o u r solution 1) always yields only roots of the p r o b l e m (for every u). If we accept the proposition which every m a t h e m a t i c i a n knows, that if f =(u) has a m e a n i n g , namely fr(u) converges for infinitely increasing r, t h e n we have f=+'(u), u n d e r s t o o d as f{f=(u)}, = f = ( u ) , a n d so the p r o o f is easily obtained, because we have f~(u) =f{f~(u)}= therefore f~(u) ~-4~{f~(u)}; ~b{f=(u)} =(= f=(u)ch{f=(u)} ~-4~{f~(u)}, f'~(u) + ~{f=(u)} =f{f=(u)} = f = ( u ) , t h e r e f o r e ~{f~(u)} ~--f=(u), thus for x =f'~(u) P r o o f 1 is i n d e e d true, namely, for every u, x@4)(x), respectively ~(x) ~ x . But that "proposition" itself is not that easy to c o n f i r m for o u r discipline. Before discussing it in its generality, I will limit myself to the p r e s e n t c a s e - - f o r example, the left s i d e - - a n d say: By the left side of s c h e m e 12), taken for u n b o u n d e d l y increasing iteration e x p o n e n 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 a n d has to be c o n t a i n e d in it. In o t h e r words, w h e n f o r m i n g the infinite p r o d u c t f r o m which we had to gain f=(u), in addition to the s e q u e n c e of factors w i t h o u t e n d 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 f u r t h e 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) itself--q.e.d. (?) This reflection is certainly unassailable as soon as o u r 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 o u r factor s e q u e n c e c a n n o t continue to be differ f r o m each o t h e r endlessly; a n d finally the p r o d u c t has to be constant; namely, it will r e p r o d u c e itself tautologically w h e n furt h e r factors are added. In this f=(u), 4~{f=(u)} really occurs as factor.

294

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SCHRt~DER'S LECTURE V

S c h e m e 1), a c c o r d i n g to which we can c o n s t r u e g e n e r a l solutions satisfactorily for n u m e r o u s individual p r o b l e m s , is t h e r e f o r e justified as a c o r r e c t s c h e m e o f solution, at least for every finite universe of discourse--and with this, we have g a i n e d a lot! But I m a i n t a i n 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 n o t completely satisfactory. W h o e v e r shares my scepticism n e e d d o n o m o r e t h a n trust the special solutions based o n s c h e m e s only within the m e n t i o n e d limitations.

Appendix 5: Schr6der's Lecture IX

Introduction At the e n d 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. T h e first must be solved for plus, given the constant relation successor. T h e seco n d must be solved for times, given plus and successor as constants. S c h r 6 d e r points out informally how these arise as the smallest set of o r d e r e d triples satisfying conditions that can be read from such equations. In the ninth lecture, he does the same thing for inductive definitions in general. H e r e he presents the definitions for primitive recursion, which of course includes those for plus and times. He c o m p l i m e n t s D e d e k i n d as the first person to see this m e t h o d , and criticizes Frege a n d others for not appreciating what Dedekind had done. Recursion equations, defining a function in terms of simpler functions, are, as D e d e k i n d realized, implicit conditions that must be proved to have a solution. T h a t is, the set of o r d e r e d triples defining, for example, multiplication, must be defined in some direct way. D e d e k i n d ' s theory of chains was developed for this purpose. Schr6der realized that D e d e k i n d ' s key notion was the transitive closure of a binary relation, the smallest transitive relation containing the given relation, defined as the intersection of all transitive relations containing the given relation. At that time it was not realized that the axiom of infinity was n e e d e d to show that there is at least one relation that is a set (not a class), and transitive, a n d c o n t a i n i n g the given one. This m e t h o d of a r g u m e n t was Dedekind's, who implicitly used but did not m e n t i o n the axiom of infinity. Without it, the integers could be a p r o p e r class, and the transitive closure of a relation a p r o p e r class. This actually h a p p e n s in the ranked universe of level o m e g a (a.k.a. the A c k e r m a n n model), which contains only finite sets. 295

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However, S c h r 6 d e r wished to r e f o r m u l a t e these n o t i o n s a n d prove D e d e k i n d ' s results entirely within the calculus of relatives. H e r e inductive definitions of functions are t h o u g h t of as implicit solutions of the relational e q u a t i o n form of the r e c u r s i o n clauses, which t h e n m u s t be solved. T h e solution is p r o v i d e d by transitive closures of a p p r o p r i a t e relations, a n d the device for h a n d l i n g transitive closures is S c h r 6 d e r ' s version of D e d e k i n d chains. Since this was S c h r 6 d e r ' s only e l u c i d a t i o n a n d e x t e n s i o n of D e d e k i n d ' s work o n justifying definitions by i n d u c t i o n , 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 D e d e k i n d ' s m o n o g r a p h at h a n d . It seems likely that the purpose of this lecture was to show that the most delicate piece of found a t i o n s work thus far in the history of m a t h e m a t i c s c o u l d be carried o u t neatly in the calculus of relatives. O n e s h o u l d n o t e that D e d e k i n d used the word "system" w h e r e we use the word "set." O n e of the aims of S c h r 6 d e r appears to be to avoid ever m e n t i o n i n g e l e m e n t s of sets, r e p l a c i n g all set a r g u m e n t s 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 circumstance 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, constitutes 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

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the abbreviated form of ~ 1 to ~ 167 and mark his direct s t a t e m e n t s with quotation marks, reserving the right to italicize i m p o r t a n t passages. T h e mathematical world has generally recognized the i m p o r t a n c e of this work: it went rapidly out of print and was then r e p r i n t e d again in u n c h a n g e d form; however, s o m e individual mathematicians have p r o f o u n d l y u n d e r e s t i m a t e d it, both with respect to its possibilities and to its m e r i t s m t h e most blatant case Page 347

b e i n g the negative review by the well-known editor of a m a t h e m a t i c a l j o u r n a l , Mr. R. H o p p e , who only saw an "intellectual exercise" in D e d e k i n d ' s essay and to w h o m the goal that "the writer i n t e n d e d to gain r e m a i n e d obscure." Similarly, the achievements of Mr. D e d e k i n d have n o t b e e n sufficiently recognized and a p p r e c i a t e d by philosophers or those leaning toward p h i l o s o p h y (Frege 1893; Husserl 1891). T h e a u t h o r of this b o o k t h e r e f o r e takes satisfaction in putting t h e m in their right and due light. In o r d e r to show the general t r e n d of Mr. D e d e k i n d ' s work, we let him speak for himself:

Page 348

What can be proved, o u g h t not to be believed without p r o o f in science. A l t h o u g h this d e m a n d seems clear e n o u g h , 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 n u m b e r s , as the most r e c e n t publications show.* By calling arithmetic (algebra, analysis) only a part of logic, I show that I consider the c o n c e p t of n u m b e r s as totally i n d e p e n d e n t of any c o n c e p t or any idea of space a n d time, and that I consider it as the i m m e d i a t e result of the p u r e laws of thought. My main response to the question asked in the title of this essay is: n u m b e r s are free creations of the h u m a n mind; they serve as a m e d i u m to conceptualize with ease and clarity the differences between things. T h r o u g h the purely logical structure of the science of n u m b e r s and t h r o u g h the c o n t i n u o u s realm of n u m b e r s g a i n e d by it, we are in a position to e x a m i n e o u r ideas of space a n d time, by c o m p a r i n g t h e m to the realm of n u m b e r s created in o u r mind.** If o n e 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 o u r 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 n u m b e r s has to be based on this foundation alone, a basis which is essential in o t h e r 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.

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A f t e r giving a s h o r t h i s t o r i c a l b a c k g r o u n d o f 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] ( , ~ 6 4 ) , the concept of the "number" [Anzahl] of things ( , ~ 1 6 1 ) , the proof that the mode of proof known by the name of mathematical induction ( o r t h e 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 t h a t the definition by induction (or recursion) is definite and free of contradictions ( ,~ 126):

Page 349

Anybody who possesses what is called c o m m o n sense can understand 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 b e g i n n e r 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 experienced 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 unintended exercise, which begins in early childhood, and the related formation of j u d g m e n t s 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, attempting to establish the science of numbers on a uniform basis, ought to e n c o u n t e r 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.

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So m u c h 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; a m o n g them are the propositions: that in every finite set of n u m b e r s there is a greatest and least n u m b e r ( ~ 1 1 4 ) ; that if a part of a system is infinite, the whole system has to be infinite ( , ~ 6 8 ) - - a n d so on. Mr. Dedekind has thus successfully c o n c e r n e d 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). T h e gap is particularly large at that point where the science of arithmetic o u g h t to originate from general logic, where it o u g h t to be rooted in o r d e r to branch off in different directions. We m i s s e d - - a n d actually are still missing in p a r t - - t h e connection of that discipline of arithmetic to the algebra of relatives, which includes the theory of single-valued assignment [eindeutigen Zuordhung] or mapping [Abbildung] as its particular or special branch. I can hardly blame myself or any other expositor of arithmetic for this gap when I imagine, on the one hand, how advanced the d e v e l o p m e n t of logical calculation had to be in order to produce the missing connection smoothly, and, on the o t h e r 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. Page 350 T h e final goal of the work is to reach a strictly logical definition of the relative concept "number of-" ["Anzahl von-'] from which all propositions regarding this concept can be derived purely deductively. We may also mention that considerable "benefit" of this essay which consists in destroying the nourishing g r o u n d 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 determination 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. O u r book will prove this point. F u r t h e r m o r e , we want to stress the fine point that Dedekind's essay makes by introducing ordinal n u m b e r s before cardinal numbers; and this considerably earlier. This, incidentally, corresponds to the historical d e v e l o p m e n t of a series of areas of m a g n i t u d e 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 introduction 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 impressions, or in the field of economics and its n u m e r o u s value determinations, or in aesthetics and its comparisonsmwe are able to range in a certain order, grade, stage, sequence the objects falling u n d e r a concept, without being able to measure them in absolute terms, long before we can represent t h e m as being

300

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equal to a " n u m b e r " of a measuring unit. Being "comparable," they are by far not "measurable" q u a n t i t i e s u " q u a n t 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 d e t e r m i n e which o n e of t h e m can be called "harder"; nevertheless, we c a n n o t c o n n e c t any c o n c e p t to the assertion of double hardness, of "being twice as hard." T h e same is true for the ordinal number, as the original and simpler concept, and the cardinal number, as the derived and less simple concept. So m u c h about Dedekind's essay in general. E n t e r i n g into a revision of o u r theory from the s t a n d p o i n t of that theory, I must now distinguish three parts of Dedekind's essay, which I also must keep strictly apart in the discussion. T h e "first part" consists of Dedekind's w 1, subtitled "Systems of Elements," comprising ~ 1 to ~ 2 0 , be it definitions or propositions. This first part is essentially only an exposition of the most elementary (but for the a u t h o r essential) concepts and propositions of the identity calculus as a calculus with ("classes" or) "Gebieten"; this last w o r d - - w h i c h I mainly use in volume l u i s replaced by "systems" in Dedekind; I also prefer it here in volume 3. T h e d~nitions introduce: the "system" of "elements" (our "individuals" of the first universe of discourse); f u r t h e r m o r e , 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 s u b o r d i n a t i o n [Unterordnung] of the "proper" part [echter Teil] to the whole; finally, the identity p r o d u c t (called the "commonality" [Gemeinheit] in Dedekind) and the identity sum of two or m o r e systems ( D e d e k i n d calls the latter the " c o m p o u n d " system consisting of them), c is called "common part" [ Gemeinteil] of a and b, if c=(c--aand c=Ec-b;likewise i is called "common" e l e m e n t of a and b, if i=(=a and i=(c-:b ( ~ 17). T h e propositions are the well-known and very close corollaries of those which we are already entitled to use not only for ( ~ ' s ) "systems," but also for binary

relatives. This first part can thus be o m i t t e d - - f o r us; instead of drawing on its propositions, 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 propositions) 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 discipline which is not dry, but rather rich in its applications. Otherwise one c a n n o t say from my voluminous volume 1 that a shortcut has been achieved. We have further to m e n t i o n that the s c h e m e of the s u b s u m p t i o n inference, stipulated as "principle II" in o u r volume 1, is stated as a "proposition" with a " p r o o f ' in ~ 7 . Within the m e a n i n g of the author, who generally uses "argum e n t a t i o n regarding individuals (elements)," this is v a l i d E b u t we are also entitled to present the scheme from o u r point of d e p a r t u r e loc. cit. as unprovable. A (real) expressed, continually tion to .~9,

duality has to run t h r o u g h the first part. But this is n o w h e r e clearly probably because of the circumstance m e n t i o n e d (that he argues with regard to individuals); and the dually c o r r e s p o n d i n g proposifor example, is not explicitly stated. But Dedekind at least d o c u m e n t s

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3ol

the duality by stating propositions as neighbors; they follow each o t h e r closely. Whenever the opportunity presents itself, I will stress this duality with greater clarity by juxtaposing the corresponding propositions or formulas, as I have d o n e 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: ~ 3 6 - - ~ 6 3 . 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 ~ 2 6 - 3 5 , and finally ~ w 5-14 from ~ 6 4 - 1 6 7 . 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, u n d e r s t o o d 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 u n d e r s t o o d by an assignment occasionally also "multivalued," not the least an assignment possibly also remaining u n d o n e (not occurring, "un-valued")--in which case the term is synonymous 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 incorporation 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 argumentation 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), although it has to be used with the broader meaning m e n t i o n e d above.

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Nevertheless, some small deviations will be unavoidable. We shall never omit the m a p p i n g rule [Abbildungsprinzip] in the formulas, and will suppress it less often in the text; thus we will have to say for D e d e k i n d ' 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 D e d e k i n d ' s b0 - "the chain of b":

a0; b = "the a-chain of b."

By a 0 = "the a-chain" or "the chain of a" we u n d e r s t a n d s o m e t h i n g essentially different from what D e d e k i n d would have to mean, namely a certain relative (derived from a), which does not at all explicitly appear in D e d e k i n d ' s chain t h e o r y m s i m i l a r l y by b0. And t h e r e f o r e we translate D e d e k i n d ' s b0 = "the image chain of b" by "a00" b = "the a-image chain of b," w h e r e 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 m e a n i n g as a certain relative derived alone from a. T h e r e a d e r now holds the key to translate one presentation of the chain theory to the other. As can be seen, o u r m e t h o d of notation is m o r e expressive. This is generally only b o u g h t at the expense of length or d e t a i l m a n d often d e g e n e r a t e s into pedantry. However, the p r o c e d u r e also has its advantages; in particular, we have to admit that the small sacrifice to g r e a t e r detail is o u t w e i g h e d by o t h e r advantages; o u r presentation of the chain theory is second to n o n e with respect to c l a r i t y m n o t even the originator's, a master of precision and conciseness. After these preliminary remarks, necessary in research literature to elucidate the c o n t i n u o u s transition from o n e theory to another, we could now begin in medias res, if I did not think it necessary to raise o n e m o r e point in advance. While there is, as m e n t i o n e d , a real duality in the "first part" of D e d e k i n d ' s essay, t h e r e is also a duality in the second part; however, I would call it a pseudo or mock duality. This is a p p a r e n t in the fact that the propositions r e f e r r i n g to

Page 354 images or chains of sums and of products a p p e a r in pairs with analogous wording and follow immediately in ~ . While o n e proposition of such a pair states an

equation, the o t h e r conspicuously only states an inclusion and only passes into an e q u a t i o n when the mapping, c o n s i d e r e d singled-valued [gegenseitig eindeutig]by Dedekind, is taken as invertible [eindeutig]. I will separate such "pseudodual" propositions with a double strike. I c o n s i d e r the elucidation of this mock duality as a f u r t h e r gain which springs out of o u r r e v i s i o n ~ a l t h o u g h it responds merely to an aesthetic r e q u i r e m e n t of the intellect (which c a n n o t possibly be satisfied with the startling inaccuracy m e n t i o n e d before). F u r t h e r m o r e , the merit of the real duality will be revealed, and this, t o g e t h e r

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SKOLEM

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 s e q u e n c e of all definitions a n d propositions constituting D e d e k i n d ' s chain theory, in the notational language of our algebra and without c o m m e n t a r y . 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. T h e letters of the lowercase latin a l p h a b e t signify any binary relative, a n d all formulas have universal validity in the algebra. I g r o u p t h e m according to opportunity: ~,~22. ~ ~

~36. ,~37. ~,~38. ~,~39.

(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)

'~40. '~41.

(a; c + b @ c ) ~ : ( a ; b:~--c). {a; ( b + c) :(=c} :(= Eu (a; u:~--cu)(b:~--u) = Eu ( a ; u + b:~--u)(a; u:~--c).

"~42. ,~43.

(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)

Page 355

,~44.

Definition of a o ; b , the "a-chain of b": ao ; b =

II

u,(a ; u + b ~ u)

u = IIu (u + a; u + b:(=u).

~45. ~46. ~47. ,~48.

b:~-ao ; b. a; ao ; b:~-ao ; b. (a; c + b:~--c) :~--(a o ; b:~--c). ( a ; x + b ~ - x ) IIu {(a; u + b @ u ) =(=(x:(=u)} = (x = a 0 ; b).

,~50.

a; b=(=a; a0; b. a ; b:~--ao ; b. ( a ; b :(=b) = (a 0 ; b = b).

• .~51. ".~52. ",~53.

~ ,~55.

x~56. ~57.

( 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).

3 o)

4 o)

T h e o r e m and Definition: a; a 0 ; b = ao ; a; b (= a00; b defined).

304 .~58. ,~59. ",~60. ~.~61. ~

S C H R I ~ D E R ' S L E C T U R E IX

ao ; b = b + aoo ; b. T h e o r e m o f m a t h e m a t i c a l induction: [a; (a,, ; b)c + b=~--c} =(c--(a,, ; b=r E x p l a n a t i o n o f 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 r o p o s i t i o n ,~51 is only given as p r e l i m i n a r y s u b s u m p t i o n . T h e s e thirty n u m b e r e d p r o p o s i t i o n s form, t o g e t h e r with their proofs, the c o n t e n t o f D e d e k i n d ' s theory of chains. We will have to get to know this t h e o r y t h o r o u g h l y ; thus we will discuss the p r o p o s i t i o n s at leisure, so to speak. A l t h o u g h s o m e of these p r o p o s i t i o n s may be used occasionally later, the "main purposg' o f stating t h e m , a n d for us h e r e the only p u r p o s e o f listing t h e m - - n e v e r to be lost sight of, is to p r e p a r e a n d m a k e possible the proof o f the theorem of mathematical induction ~ 5 9 , which contains no circular argument; that is to say, w h e r e the p r o o f p r o c e d u r e for i n d u c t i o n i t s e l f - - n o t the least for a "definition by i n d u c t i o n " - - i s n e v e r u s e d o n 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 m e n t i o n e d m a n d 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 r o p o s i t i o n is written, as we see, by m e a n s o f only nine letters. With the e x c e p t i o n of those signs which we know f r o m the g e n e r a l t h e o r y o f relatives, only one special sign is used a 0, a n d this always in the relation a0 ; b---a sign which is peculiar to this special b r a n c h of o u r discipline, the "chain theory." To establish p r o p o s i t i o n ",~59, a definition of a0, or, at the s a m e time, ao;b, will have to p r e c e d e ; a n d this d e f i n i t i o n m g i v e n by D e d e k i n d with ",~44 (or also in the f o r m o f , ~ 4 8 ) m r e p r e s e n t s the p u n c t u m saliens, the starting point, o f the whole theory. T h e c o n s t r u c t i o n o f the c h a i n t h e o r y d e p e n d s considerably o n this choice. Before talking a b o u t the difficulties which have to be o v e r c o m e with this d e f i n i t i o n (restricted only by the m a i n p u r p o s e we have in view), I w o u l d like to weave in s o m e r e m a r k s which a i m - - a c t u a l l y f r o m a purely

FROM PEIRCE TO SKOLEM

305

e x t e r n a l p o i n t of v i e w B a t r e d u c i n g (by m o r e than half) the system of p r o p o s i t i o n s that will be c o n s i d e r e d later. With the definition ~ 3 6 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 sufficiently 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. Page 357

F r o m the r e m a i n i n g propositions, it seems that e i g h t can be o m i t t e d , which I have m a r k e d with a small circle: ,~22, 23, 24, 38, 39, 54, 61, 62. T h e s e can be u n d e r s t o o d f r o m the g e n e r a l p r o p o s i t i o n s of o u r a l g e b r a 1), 4), a n d 5) of w 6; that is to say, they c o r r e s p o n d exactly to the t h r e e p r o p o s i t i o n s of Peirce, or are only special cases, p a r t i c u l a r applications, of t h e m m ' , ~ 5 4 is based on the a s s u m p t i o n that a 0 finds or has f o u n d its definition as a binary relative. If this a s s u m p t i o n refers only to "a0 ; b," we have to k e e p the last t h r e e of the e i g h t propositions. But the first five do n o t deserve to be n u m b e r e d a n d m e n t i o n e d as special propositions. 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 o f - . " In this sense, we may note: "~22.

If b is c o n t a i n e d in c, t h e n 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. ~24.

'.~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 a n d 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 p r o d u c t . 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.

3o6

S C H R O D E R ' S L E C T U R E IX

6 it can also be inferred from a =6=1 in the form of a" 1 :(:: 1 91 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;bfrom the premiss and is compared to schema ~37. .~54. .~61. .~62. Page 358

The a-chain of a part is a part of the a-chain of the whole. The a-chain of a sum is equal to the sum of the a-chains of its summands. The a-chain of a product is a common part of the a-chains of its factors.

F r o m the same p o i n t of view, ,~56 also a p p e a r s as too obvious an i n f e r e n c e f r o m ,~53 to be stated in a s e p a r a t e p r o p o s i t i o n . A similar p o i n t ( m e n t i o n e d previously in a d i f f e r e n t context) is true for ~ 5 4 , which s e e m s to be given afortiori with ".~52 a n d 53. W h e n e v e r we talk a b o u t the c o m p l e x of formulas of the chain theory, the r e a d e r s h o u l d c o n s i d e r the t h r e e lines of g r o u p (4~ c o n t a i n i n g the five p r o p o s i t i o n s . ~ 5 2 - , ~ 5 6 , as taken away a n d 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). ~55. (b@ao;c) : ( = ( a ; b @ a , , ; c ) . 60 ) - - w h i c h contains only t h r e e propositions, since of the t h r e e propositional s u b s u m p t i o n s in the first line only the first two n e e d to be proved. Finally, also ,~49 as m e r e corollary to ",~45 can be e l i m i n a t e d . 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 introduction.

the

"mock-duality" of which

I spoke

in

the

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

(bc'")'~b'c'"', '~23. (b+c+'")'=b'+c'+'", []"~24. ,~61. (b+c+"')0=b0 +c0 +''', ]] '~62. (bc"')o@boco'", 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 pseudoduality 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

signmthat 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 w a n t to a r r a n g e the c h a i n t h e o r y in its s i m p l e s t f o r m , while a d h e r i n g closely to D e d e k i n d , a n d fit it into o u r g e n e r a l d i s c i p l i n e , we m a i n t a i n o r r e m e m b e r only t h e d e f i n i t i o n o f . ~ 3 7 f r o m t h e w h o l e g r o u p (1~ w h i c h , incidentally, we i n c o r p o r a t e d in w 22 u n d e r 5) i n t o o u r discipline. N o w we will have to deal o n c e a n d for all with t h e f o u r p r o p o s i t i o n s o f g r o u p (2 ~ w h i c h p r e c e d e t h e " 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 observations which begin with ~ 4 4 and which serve to support their purpose, thus showing their relatedness. ' ~ 4 2 , 43 state: The sum, respectively the product, of chains with r e s p e c t to a relative a is a chain with r e s p e c 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+""

=(eb+c+'",

resp.

a" b " a" c ' " =g~--b " c ' " ,

resp.

a ; b c " " ~ - a;b" a ; c ' " .

and then use the schema 4) or 5) of w 6: a ; b + a ; c + "" = a ; ( b + c + " ' ) ,

The conclusion o f t h e left proposition is then more o f a 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 generalized 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)

m e a n s : The image of a chain part is a part of this chain, m o r e precisely: t h e a-image a ; b o f t h e p a r t b o f a c h a i n c with r e s p e c t to a is c o n t a i n e d in this c h a i n c. The proof is given by relatively premultiplying on both sides by a the second

SCHR(~DER'S LECTURE IX

3o8

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 t h e p r o p o s i t i o n reads after splitting t h e s t a t e m e n t s o n b o t h sides, and 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 aimage is contained in c; in o t h e r words, b is a p a r t o f u a n d its a-image is p a 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 f o u r p r o p o s i t i o n s e x p l a i n e d h e r e w i t h can still be c o u n t e d a m o n g t h e "introduction" of the c h a i n theory, w h i c h n o w really begins---with t h e punctum saliens ",~44--and, if o n e wants, c o n c l u d e s a l r e a d y with ".~59. T h e r e r e m a i n to be s t u d i e d 1 4 - - I say fourteen!--definitions o r p r o p o sitions (or 15, if o n e c o u n t s the c o r o l l a r y d e f i n i t i o n to p r o p o s i t i o n ~ 5 7 s e p a r a t e l y ) , w h i c h is n o t m o r e t h a n h a l f a page. We will n o w look at this series o f p r o p o s i t i o n s from two very different standpoints---one c o u l d say with a l m o s t p e r f e c t p r e c i s i o n : we c o n s i d e r t h e m in two d i f f e r e n t d i r e c t i o n s , to a n d from, o r "forward" a n d "backward." T h e "backward" p a t h is far s h o r t e r a n d , f u r t h e r m o r e , easier, especially since it d o e s n o t have to be r u n t h r o u g h c o m p l e t e l y ; t h e 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 t h e " i n t r o d u c t i o n , " can a l r e a d y be o m i t t e d . F o r didactic reasons, it s e e m s best to us to take this p a t h first. O n t h e o t h e r h a n d , only the "forward" p a t h is c o n s i s t e n t with t h e m a i n p u r p o s e o f t h e c h a i n theory! In addition, Mr. Dedekind also r e c o m m e n d e d to his readers to take the backward p a t h m p a g e 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)

309

FROM PEIRCE TO SKOLEM

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). The backward path. T h e f o l l o w i n g m o d u s procendi will l e a d us to t h e goal, if we m e r e l y w a n t to prove t h e f o r m u l a . ~ 5 9 as a g e n e r a l p r o p o s i t i o n in t h e t h e o r y o f b i n a r y relatives, ignoring its significance as t h e "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 h e s i t a t e to state its f o r m a t i o n law by m e a n s o f t h e "proof b y m a t h e m a t i c a l i n d u c t i o n , " w h e n infinite 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 i d in p r e v i o u s l e c t u r e s w h e n a s o l u t i o n c o u l d n o t be given in c l o s e d f o r m ( a n d as we will d o a g a i n in t h e f u t u r e , a l t h o u g h t h e n with strict j u s t i f i c a t i o n ) . We o m i t t h e w e i g h t y p r o p o s i t i o n s , ~ 4 4 a n d 48 f r o m D e d e k i n d ' s c h a i n t h e o r y (to o b t a i n t h e m f r o m t h e 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 f e r to t h e n e x t sections) a n d d e f i n e t h e "a-image chain" aoo a n d t h e "a-chain" a 0 o r " c h a i n o f an a r b i t r a r y relative a" as follows: aoo=a+

a;a+

a;a;a+'",

a o = 1' + aoo,

all = a(aJ- a)(aJ- aJ- a ) ' " ,

1)

al = O' all.

2)

N o w t h e f o l l o w i n g is natural" 1' :(= a o,

a :(= a(,(, :~- a,,,

a I ~ 0',

3)

a 1 ~ al, :(= a,

4)

a n d , f u r t h e r m o r e , t h e following are valid: a;aoo=aoo;aoo=aoo;a:(=aoo,

a ; a o = aoo = a o ; a :(= a o,

a o ; a o = ao,

all :(=a J-all = a l l c t a l l

=all,a,

5)

a 1 :(= a ~ a I = all = a I ~ a,

6)

al d" al = a l ,

7)

as c a n be easily seen, by s u b s t i t u t i n g t h e series set d o w n for aoo a n d ao a n d t h e n a p p l y i n g relative m u l t i p l i c a t i o n . Page 362

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

SCHR~)DER'S LECTURE IX

310

term of one series has to be j o i n e d with every term of the other. In both cases, the sum results from the first term of the terms a d d i n g up to at0 (which could also be called a000, and so on). With 6), we also have a'ao=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 consideration again, but one can refer the proposition back to 5), without o p e r a t i n g again with the infinite seriesmas follows: at at 9

~

(l'+aoo)

9

(l'+aoo)

=1' l ' + a o o ,

,1

/

+ 1 'aoo + a o o ' a o o I

= 1 / + a o o + aoo" a o o = 1 ~ + a o o = a t ,

because the third term of the last line is absorbed by the previous term due to the s u b s u m p t i o n aoo'aoo ~ aoo proved with 5). O n e should not u n d e r s t a n d aoo as the "chain of at," that is to say, (at) o, which would be false. T h e following propositions anticipate such misunderstandings: (a0) = a,,,

(al) l = al,

(a~,,,)0 = a,, = (a0)00,

(all) 1 = a I = (al)ll,

(a~,,,)0,, = a,,,,

(all) l~ = a l l"

8)

Thus, the "chain of the chain ofa" is n o t h i n g less than the "chain ofa." It would be silly to represent it with a d o u b l e suffix 00 which we want to keep free for o t h e r notational u s e . r a T h e reader can easily u n d e r s t a n d propositions 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'+a00)

+a

0+a

0+'''

= a 0.

T h e operations for taking chains or image chains can always be immediately e x e c u t e d with an image chain or chain. To p r o v e . ~ 5 9 , 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 a n d ",~47.

.~45, ,~55,

~ 4 5 m e a n s 9each relative b is p a r t o f the a-chain o f itself a n d is u n d e r s t o o d from the proposition"

Page 363

b =~--ao ; b,

am d" b =~-b,

b=~--b ; a o,

b d- al =~--b ,

9)

f r o m 3), obviously. By relative m u l t i p l i c a t i o n with b o n b o t h sides, we obtain l';b=(=a0; b therefore

b:~--ao; b,

L i k e w i s e , c o n s i d e r i n g 6) a n d 4), we h a v e

q.e.d.

3al

FROM PEIRCE TO SKOLEM

(b~-a,, ;c) = ( = ( a ; b ~ a ; a , ,

; c = a,, o ; c ~ a o ;c),

a n d with it h a v e p r o v e d .~55, o r t h e p r o p o s i t i o n :

{ (b~--a~ ( b =~-c ; oF) =(= (b ; a a~- c ; a o ) ,

(alj'c~--b)=~--(aljca~-aj'b)' (c j- a I a~- b ) a~--( c j- a I a~- b d- a ) ,

10)

t h a t is, the a-image of a part b of the a-chain of c is also a part of the a-chain

ofc. Finally, t h e p r o p o s i t i o n , ~ 4 7 b e l o n g s to t h e g r o u p : (a;c+

b:~-c)~-(ao;b@c),

(b + c; a:~-c) :~-(b; a o =(~--c),

{c:~-(a~c)b}:~-(c:g~--alj-b), {c:~--b(cj- a)} =~-(c:~-bd- al),

11)

a n d its p r e m i s s splits i n t o (a; c=~--c)(b ~-c). It t h u s 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. F o r proof we d e d u c e f r o m t h e s e c o n d p r e m i s s , with a c o n s t a n t view to t h e first, t h e i n f i n i t e series o f i n f e r e n c e s :

b=~--c,

a; b a~-a ; c,

therefore

a; b ~--c,

a; a; b=~--a ; c,

therefore

a ; a ; b~--c,

a; a ; a ; b=~c--a ; c,

therefore

a;a;a;b=~--c,

a n d so o n .

Subsequent adding of the left-hand inferences gives--considering b = 1' ; / u - t h e c o n c l u s i o n a o ; b ~ - c . * W e c a n see t h a t with t h e "and so on" t h e "inference from 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 r d e 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 n m w h i c h is already the case for n = 1 and 2rowe then show by means of the inferences:

a;a";b=(~-a;c=g~--c Page 364

that also a"+l;b=g~--c has to be valid. Now the inference a"; b=g~-c is valid for each 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 (consequently also for n = 2, and again n = 3 and so on in infinitum). T h e above p r o o f rests on these partial i n f e r e n c e s . m Incidentally, one can also give a proof of the whole proposition in one stroke by means of an infinite series of equivalent propositional transformations. 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 e p e t i t i o n o f the o n e m a d e with 3 4 0 , a l t h o u g h it d o e s n o t have to be f o r m a l i z e d as a p r o p o s i t i o n h e r e .

SCHRC)DER'S LECTURE IX

312

(b+ a;c=~-c) --{b+ a ; ( b + c) =(=c} = ( b + a ; b + a;c =(~-c).

12)

T h e hypothesis of the proposition 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 p e r m i t t e d a c c o r d i n g to the principium identitatis of the propositional calculus. T h e propositional s u b s u m p t i o n passes into a propositional equivalence or equation because 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 t a t e m e n t 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 proposition 12) gives us the right to rewrite its statem e n t s 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 r e p l a c e m e n t , if it is consistently d o n e with b, is exactly the same as when d o n e only with the last b; thus we easily gain the following as equivalent to proposition 12): (b + a; b + a Z ; 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 omission of the propositional factor ( a ; c . ~ - - c ) ~ p r e s e n t s the conclusions constituting o u r proposition ~ 4 7 when e x t e n d e d to an infinite set of terms. To justify this "observation" in general, the following p r o o f suffices: if (1' + a + a 2 + a :~ + "'" + a n ) ' b = f " ( b ) , then f " ( b + a; b) =f"+'(b)

Page 365

must h o l d m w h e r e finally f=(b) = ao;b. This p r o o f is subject to no difficulties. Signifying by f l ( b ) =fib) = b + a ; b = ( l ' + a ) ; b here and noting that (1' + a)"; (1' + a) = (1' + a) "+l must h o l d - - c f , the "inductive" or "recursive" definition of power, given in w 13 by 8 ) m o n e can further simplify the p r o o f by showing by m e a n s 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 o n e obtains by relatively multiplying both sides with 1' + a on the right: (1' + a) "+l = 1' + a + "'" + a " + a "+1, with a tautological repetition of all terms between the first and the last.

FROM PEIRCE TO SKOLEM

3x3

T h e n also a,, = (1'+ a) =

15)

is o b t a i n e d . m As l o n g as t h e p r o o f by i n d u c t i o n o b t a i n s its j u s t i f i c a t i o n o n l y f r o m p r o p o s i t i o n ,~59, w h i c h o n its p a r t is only d u e to t h e p r o o f o f , ~ 4 7 , t h e p r o o f o f ~ 5 9 (with w h i c h we will d e a l later) is a circle; its o n l y v a l u e is to give us t h e c o n f i r m a t i o n : if this i n d u c t i v e 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 e d in this one c a s e m a t least in . ~ 4 7 m i t can c l a i m f o r m a l o r general validity for e a c h case o f its a p p l i c a t i o n . Thus all other 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 understood from 4); likewise ~ 4 9 from a;b,~-aoo;b, and ~ 5 0 ; ~ 5 7 from 6), ~ 5 8 from 2). ~ 5 2 is understood 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) D a c c o r d i n g to ~ 4 7 , 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 ~ 5 1 , 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~-bhas to hold. So easy was it to take the backward p a t h D w h e r e even the inferences were a luxury! T h e proposition of mathematical induction ~ 5 9 b e l o n g s to t h e g r o u p : {a ; (a 0 ;b)c + b ~ c} =(=(a0 9b ~= c), {b+c(b;a~,)'a~-c}=g~-(b'ao=g~--c),

~ [c =(= {a ~ (a, 0~ b + c)}b] =(= (c =(= a, ct b),

[ [c=(~--b{(c+b~a~,)~a}] ~ (c=(~-b~ al). 16)

Page 366

Its p r o o f is to be e s t a b l i s h e d as follows, given t h a t its p r e m i s s splits into I.

a ; ( a 0 ; b ) c ~-- c

and

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 t h e inference III.

b @ (a,, ; b)c.

314

S C H R O D E R ' S LECTURE IX

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 b m a ; (a 0 ; b)c :~c-ao ; b, and this, c o m b i n e d 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 u n d e r the s c h e m a of the premiss of ".~47 or 11 ) - - w h e r e only c has to be replaced by the c o m p o u n d expression (ao;b)c; taking this proposition 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 p r o o f are later gained in a noncircular, respectively, unobjectionable way, the previous p r o o f Page 367 is rigorous and need not be repeated. T h e proposition , ~ 5 9 ~ f i r s t viewed simply as a t h e o r e m about binary relatives--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 t h e r words, a0; b must be a part of c if b is a part of c, and the aimage of each c o m m o n 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."

3 x5

FROM PEIRCE TO SKOLEM

I n s t e a d o f ,~60, we w o u l d like to e x p l a i n to t h e r e a d e r why proposition . ~ 5 9 o r 16) in fact f o r m s "the scientific foundation for the mode of proof known by the name of mathematical induction (the inference from n to n + 1) " F o r t h a t p u r p o s e it is only n e c e s s a r y to assign t h e f o l l o w i n g interpretation to t h e u n i v e r s e o f d i s c o u r s e 1 a n d t h e l e t t e r relatives a, b, c in t h e proposition. T h e u n i v e r s e o f d i s c o u r s e 11 is c o m p o s e d o f t h e i n d i v i d u a l s i, 2, 3, 4 . . . . o f t h e infinite number, series; we d i s t i n g u i s h its first m e m b e r , o r t h e number one, with t h e dot, 1, f r o m the m o d u l e 1 o f o u r theory. In the universe of discourse 1z or 1, each element of this n u m b e r 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 o f a assignment-or mapping rule, w h i c h leads f r o m o n e to t h e n e x t , a n d t h u s allows all n u m b e r s o r i g i n a t e f r o m t h e first a m o n g t h e m as t h e "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, t h e n u m b e r s following, i are to be t h o u g h t o f as d e f i n e d : t h e " n u m b e r " two as " g r e a t e r by 1 t h a n 1," w h i c h 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 t h e plus signs have to be. u n d e r s t o o d a r i t h m e t i c a l l y - - - o r in t h e n o t a t i o n o f o u r discipline" 2 = a; 1, 3 = a; 2, 4 = a ; 3 . . . . , t h a t is, e a c h n u m b e r is d e f i n e d as t h a t "a-imagd' o f its p r e d e c e s s o r s - - w i t h t h e exc e p t i o n o f t h e base n u m b e r i itself, w h i c h d o e s n o t have a n y p r e d e c e s s o r d u e to t h e previously i m p o s e d r e s t r i c t i o n o n o u r u n i v e r s e o f d i s c o u r s e . 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 t h e totality o r identity calculus ( b u t n o t arithmetical!) s u m o f all individual n u m b e r s , is r e p r e s e n t e d as the a-chain of the base number 1. In o t h e r words, it is the module

1 = a0"l. In any case, t h e 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 h a n t h e totality 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 t h e given n u m b e r ( i n c l u d i n g itself)"

316

S C H R O D E R ' S LECTURE IX

a 0;i=i+

(i+ i)+(i+2)+(i+3)+'"

----on the o t h e 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 ( e x c l u d i n g itself). This assumed, the relative b m e a n s the e l e m e n t 1 itself, t h e r e f o r e ,

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. T h u s ao;b r e p r e s e n t s again the e n t i r e n u m b e r series. Finally, the relative c m e a n s the totality, the "system" of n u m b e r s , which have the definite p r o p e r t y ~, or, in o t h e r words, for which a definite p r o p o s i t i o n ~ , c o n t a i n i n g an indefinite n u m b e r n, is valid. In o r d e r to prove that the p r o p o s i t i o n ~ is valid for all n u m b e r s , that is, that ao;b:~--c, we simply show, a c c o r d i n g to .~59: First, that the p r o p o s i t i o n is valid for the n u m b e r b = i, that, consequently, b ~=c; Second, that this p r o p o s i t i o n m u s t also be valid for the i m a g e a of every n u m b e r (n) for which o u r p r o p o s i t i o n ~ is a p p l i c a b l e E w h i c h e v e r is, n u m b e r s a p p e a r i n c l u d e d in the expression ( a o ; b ) c n t h a t a; (a0; b ) c ~ c m u s t hold. In simple terms, this m e a n s that if a p r o p o s i t i o n is valid for a definite n u m b e r n, it m u s t also be valid for the next higher n u m b e r n + 1. O n the o t h e r hand, i f n t o be even m o r e g e n e r a l E b r e p r e s e n t s a definite n u m b e r m or any system of n u m b e r s which includes m as its smallest n u m b e r , t h e n ao; b r e p r e s e n t s the totality of numbers beginning

with m. In o r d e r to proye that the p r o p o s i t i o n ~ 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 n u m b e r s of system b; actually, it is only necessary to show its validity for the n u m b e r (b =) m, a n d "second" etc. (same w o r d i n g as b e f o r e ) . [The p r o p o s i t i o n n e e d n o t apply for the n u m b e r s below m]. We have thus e x p l a i n e d the definition ,~60 of Mr. D e d e k i n d r e g a r d i n g

3x7

FROM PEIRCE TO SKOLEM

the m e a n i n g a n d e x t e n t of p r o p o s i t i o n ~ 5 9 . To r e c a p i t u l a t e in the words of the a u t h o r : "'.~60. In o r d e r to prove that all e l e m e n t s of the c h a i n ao;b possess a c e r t a i n p r o p e r t y (~, it suffices to show: First: that all e l e m e n t s of b have the p r o p e r t y @. Second: that the same p r o p e r t y b e l o n g s to the i m a g e a; n of each such

element n "of ao ; b" which has the property ~.

Page 370

In o r d e r to r e c o g n i z e .~60 as the verbal t r a n s l a t i o n o f f o r m u l a ~ 5 9 , we only have to see c as the system of all e l e m e n t s which have the p r o p e r t y (~." In this c o n n e c t i o n , the following r e m a r k may n o t be s u p e r f l u o u s , as it highlights the c o m p a r i s o n of ,~59 with ,~47. If we h a d previously left o u t the stipulation "of ao;b," we w o u l d have the verbal t r a n s c r i p t i o n a n d usage of ,~47. I n d e e d , this f o r m u l a can be u s e d to justify fully an assertion, thesis ao;b~--c, b a s e d o n the proofs of b ~--c a n d a; c=~--c. This s e c o n d partial r e q u i r e m e n t of the h y p o t h e s i s goes f u r t h e r with ,~47 t h a n with ~ 5 9 - - b e c a u s e of (ao;b)c =r162 By o m i t t i n g this s t i p u l a t i o n we r e q u i r e m o r e t h a n absolutely necessary to establish the thesis; the t h e o r e m ~ 5 9 suffices for ',~47 with formally fewer c o n d i t i o n s t h a t will have to be proved. If, in s o m e cases, t h e o r e m ,~47 suffices to r e a c h the goal, o n e can g e n e r a l l y only posit .~59 as the logical c o r e of the p r o o f by i n d u c t i o n . T h e "forward path." I have to ask the reader first to accept the name "a0;b" as a simple or unprejudiced, 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 d e f i n e s the "a-chain ofb" as a b i n a r y relative d e p e n d e n t o n a a n d b---as m e n t i o n e d in anticipation--denoted by "a0 ;b"; it is the identity p r o d u c t II with r e s p e c t to u of all the relatives u of the u n i v e r s e o f d i s c o u r s e 12 which fulfill the c o n d i t i o n s in 3 o) for e x t e n s i o n c o n d i t i o n a ; u + b@u, set u n d e r the (first) H-sign. This c o n d i t i o n splits into

a; u @ u

and

b=C-u,

the first r e q u i r e s that u be a "chain" with r e s p e c t to a, the last that u also i n c l u d e s b as a p a r t of itself; thus o n e can e x p r e s s the d e f i n i t i o n with the words:

SCHRODER'S LECTURE IX

318

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 p r o d u c t 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 condition. " F o r this very important n o t i o n t h e f o l l o w i n g p r o p o s i t i o n s a r e valid." b:~-ao;b. [Verbally e x p r e s s e d in 9).] ~45. Proof. Because, according to the second part of the extension condition, b is contained in every factor u of product I I u ~ i s a " c o m m o n part" of these f a c t o r s ~ i t

must also be contained in their products. T h e s c h e m e of the identity calculus to be used for this inference would be IIu (b =6 u) = (b =6~ IIu u) ---compare 3 . ) of volume 1, used for an infinite n u m b e r of factors, here in e), page 39. ~46.

a ; (a0; b) ~ = a 0 ; b. The "a-chain" of any relative is a "chain with

respect to a." Proof. According to the first part of the extension condition of o u r IIu, each factor u of this p r o d u c t is a "chain with respect to a," and t h u s - - a c c o r d i n g to "~43--also the p r o d u c t IIu of it. .~47.

(a ; c:~--c)(b:~--c) :(=(a,, ; b:~--c).

Proof. According to the conditions of this proposition, c represents a value of u which fulfills the extension condition. It thus falls u n d e r the factors of o u r IIu, and since an identity p r o d u c t has to be included in its factors, we have IIu~-c; the conclusion seems justified. ~ 4 8 is not expressed as "proposition" in D e d e k i n d ~ i n addition to the external 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 easily 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 u n i n t e r e s t i n g way, how to formulate the definition ~ 4 4 of a0;b; it can be left out in o u r course. We have clad it in o u r notational language and will justify it in this form as equivalent to "~44. In o r d e r to save space, we would like to abbreviate the frequently used expression II

),.( a ; u + b , ~ u)

u

by

IIu

NB

Page 372 ~ t o allude to the extension condition a; u + b~--u merely by a NB (nota bene), and u n d e r s t a n d by rI,, as always, a p r o d u c t which possesses full extension over all possible relatives u.

FROM PEIRCE TO SKOLEM

3 X9

F u r t h e r m o r e , 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, i n t r o d u c i n g 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 propositional 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) propositional schema: (x :(= ~ u ) - 8

(x ~r u) =H{NB =(= (x :(= u)} =H,S,

by rightly marking the extension condition NB = (a; u + b:g~-u)

after (instead of under) the H-sign, as a condition to be fulfilled by u; this is permissible since, not a relative, but rather a proposition stands as the factor after the H. We thus know that (x=(c-- H ul - H s

and

R=(c- H S.

As a c o n s e q u e n c e of R ( ~ 4 4 ) we had already proved the propositions ~ 4 5 : b=(=x and ,~46: a;x=gc--x, which, when combined, give us the conclusion

R =(c--( a ; x + b =(c-x) . If we c o m b i n e this with the above propositional subsumption, we have R=I~--L (whereby the t h e o r e m ,~48 appears as subsumption from right to left derived from ~ 4 4 ) . On the o t h e r hand, the following conforms to ~ 4 7 :

x + b :+ x) Page 373

+ x).

which has to be presented not as a c o n s e q u e n c e of ,~44, but as obvious from it; according to the premiss, the x on the left has to be, as a value fulfilling the extension condition "NB" of u, an effective factor of this IIu; this is why it is subordinated. This last subsumption, multiplied with the above

SCHRC)DER'S LECTURE IX

320

u) gives us the conclusion: L=(=R, whereby L = R is proved.

a;b=(c-a; (a0;b) is u n d e r s t o o d by relative p r e m u l t i p l i c a t i o n .~49. with a o n b o t h sides f r o m ~ 4 5 ; it o u g h t n o t to be r e g i s t e r e d as an i m m e d i a t e corollary t h e r e o f B a s m e n t i o n e d b e f o r e ; j u s t as o n e d o e s n o t m e n t i o n in g e o m e t r y , for e x a m p l e , the p r o p o s i t i o n c 2" = (a z + b Z ) " m n o t to m e n t i o n n c 2 = na 2 + n b Z B n e x t to the P y t h a g o r e a n t h e o r e m . .~50. a; b =~--ao ; 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.

(a ; b=~--b) = (ao ; b = b). I f b is a chain with respect to a, then b is ,~51. the a-chain of itself, and conversely. P r o o f . . ~ 4 7 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=(~--ao ;c) =~--(ao ; b=~--ao ; 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) =(=a0 ; 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 ~ 4 6 , 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 ~ 5 2 afortiori:

(b =ge:c) =(~ (a ; b =g~--ao ; c). At this point we can supply the p r o o f of p r o p o s i t i o n ~ 5 9 , m e n t i o n e d o n p a g e 3 6 6 - - - a l t h o u g h its significance as a basis for the p r o p o s i t i o n of m a t h e m a t i c a l i n d u c t i o n will only be u n d e r s t a n d a b l e later. T h e p r o p o Page 374 sition is thus rigorous, namely, w i t h o u t the circle criticized above, p r o v e d a n d may thus be used f r o m now on. But in the m e a n t i m e we m u s t c o m p l e t e o u r "forward path." ",~57. Proposition a n d Definition. It is

FROM

PEIRCE

321

TO SKOLEM

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 c a n t h e r e f o r e d e n o t e s u c h a r e l a t i v e by a00; b (with t h e s a m e r e s e r v a t i o n as o n p. 3 7 0 ) , a n d call it e i t h e r t h e a-chain

image o r t h e 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 c o m b i n e d as

a;{b+ a 0; (a;b)} :(=a 0; (a;b). This r e q u i r e m e n t now has the form of the premiss of',~41 where only cseems to be r e p r e s e n t e d by the right-hand expression. According to this proposition, there is a u m b y the way, the expression in braces on the left-side would be o n e - - w h i c h has the properties indicated after the sum sign--or, if we p r o c e e d according to propositional calculus, the following conclusion is valid: E (a; u + b =(=u){a; u :~- a 0 ; (a; b)}, which, by .~47, tt

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 propositional addition, the sum sign could have b e e n omitted, as long as the general s u m m a n d is constant with respect to the s u m m a t i o n variable u - - w h i c h creates the impression that the sum :(= in the consequences had b e e n inverted with its summands!]. Thus the last successive subsumption is proved. On the o t h e r hand, ".~46 gives the consequence, premultiplied on both sides by a, that reads, when c o m b i n e d with ~ 5 0 :

a ; {a ; (ao ; b)} + a ; b =g~--a ; (ao ; b) and shows the form of premiss "~47, whereby only the b there is r e p r e s e n t e d by a;b here, and c on the right side. According to the schema of the conclusion of this proposition, we have a0; (a;b) :(=a; (a0 ; b), Page 375 which represents the converse of the subsumption proved previously. Thus, the equation, which we set out to prove, is justified.

322

SCHRODER'S LECTURE IX

ao; b = b + aoo; b, t h a t is, the a-chain of a relative b consists of .~58. 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 t h e basis o f p r o p o s i t i o n ,~58, we c a n now, as we h a v e s h o w n o n p a g e 361 u n d e r 0), give a b e l a t e d j u s t i f i c a t i o n o f t h e c o m b i n a t i o n o f t h e t e r m s a0; b a n d aoo ;b; a 0 a n d a00 c o u l d , be e x p l a i n e d as relat i v e s m a l t h o u g h in t h e f o r m o f i n f i n i t e series, w h o s e f o r m a t i o n p r i n c i p l e c a n o n l y be j u s t i f i e d by m e a n s o f t h e 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 n o w o b t a i n e d its full civic rights in o u r d i s c i p l i n e ! T h u s , o u r " f o r w a r d p a t h " has e s s e n t i a l l y r e a c h e d an e n d . 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. For~61.

a 0;(b+

c + ''') = a 0 ; b + a

a;(ao;b+ao;c+'")

0;c+''"

+ (b+c+'"):~-a

o n e has 0;b+a

0;c+'''

- - a s 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 a0;(b+c+'") Page 376

=l~a0;b+a0;c+'".

On the other hand, because b~-b + c + "", c~-b + c + "", according to ".~52, 53), afortiori schema ,~54 is

ao ; b=~-ao; (b + c+ ""), which can be reduced to

ao ; C ~ a0; (b + c + '") . . . .

323

FROM PEIRCE TO SKOLEM

a0;b+a0;c+'-',~-a0;(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 " "

a ; ( a 0 ; b " a 0 ; c ' " ) : ( = a ; ( a 0;b)

is

9a ; ( 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 has to be

b=(~--a o ; b , c=(~--a o ; C, " "

b c " " ,~- a o ; b " a o ; C " " .

If we combine this with the previous, we have a; (ao;b"

ao;c'")

+ bc'"=(~-ao;b"

ao;c'",

and the assertion follows immediately from the schema of .~47. q.e.d. We have r e a c h e d

our

goal.

Let us discuss it a n d the p a t h 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 t h a t the fatal o b s t a c l e of the circle, p r o v i n g the p r o p o s i t i o n of m a t h e m a t i c a l i n d u c t i o n , has been circumnavigated. 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 r e a c h the s a m e goal o n a less artful a n d s h o r t e r path. We have especially to ask ourselves w h e t h e r it is necessary to base the c h a i n t h e o r y with ~ 4 4 o n the d e f i n i t i o n of such a c o m p l e x relative as a0;b? It seems m u c h s i m p l e r to stipulate a d e f i n i t i o n o f a 0 itself; c o m p a r e d to this p r o c e d u r e , the previous o n e has to be called a r o u n d a b o u t way. If i n d e e d we s u c c e e d in simplifying the c h a i n t h e o r y in this sense, I Page 377 have to e x p l a i n why I led the r e a d e r a l o n g this l e n g t h y p a t h a n d did n o t p r e s e n t my simple solution right 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

S C H R ( ) D E R ' 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 k e d m a n d which shows it in its fullest generality, exceeding Dedekind's own claims. T h e r e b y I wanted to reveal the advantages of our system of notation, which is m u c h m o r e expressive a n d extensive; I am convinced that it alone will prevail and maintain a ruling position on the global m a r k e t of science. On the o t h e r hand, the beauty of the theory should b e c o m e a p p a r e n 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 h e o r y - - w h e t h e r one similar to my own or any othermwill soon det h r o n e Dedekind's, there remains much in it which is of m o r e than historical value. Especially, the main point: the formulation and p r o o f of ~ 5 9 (and 60), as r e p r e s e n t e d by 16) on page 366, will survive unchanged. A f u r t h e r simplification of the "formulation" of the proposition of induction is not thinkable; increased simplification of this " p r o o f " - - l i m Page 378 ited as s u c h - - d o e s not seem advisable or possible, since it is already very simple. T h e "main point" thus remains u n t o u c h e d by simplification, which can only apply to the abbreviation of the long and arduous path which leads from the basic definitions in D e d e k i n d to the confirmation of the three lemmas ",~45, 47, and 55, on which rests the p r o o f of ,~59, and which then has to continue until the e n d of the theory is reached. And if i n d e e d we m a n a g e to make do with fewer and m u c h simpler propositions, if we get m o r e quickly and easily to o u r goal with propositions involving~instead of two or t h r e e - - a t most one or two relatives, D e d e k i n d ' s theory nevertheless maintains its advantage of usually operating with m o r e general propositions, a m o n g t h e m propositions that may still prove serviceable for o t h e r goals, even if, for purposes of our i m m e d i a t e goal, they can be dispensed with. O f course, they would continue 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.

FROM

PEIRCE

TO

325

SKOLEM

S i m p l i f i e d c h a i n theory. F o r d e f i n i t i o n o f t h e "a-chain," a 0, o r of the c h a i n o f a relative a, we use

21)

H u = a o, u,NB

w h e r e t h e e x t e n s i o n c o n d i t i o n "NB" o f t h e p r o d u c t IIu is 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" c o m m o n 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 c o m p o u n d e d 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 u n h e a r d of to ask for those chains with respect to a which as "Aliorelativenegatd' contain the module l ' m t o then to take their product. Since t h e e x t e n s i o n c o n d i t i o n splits into 1' =(= u

22),~

and

a; u =~u,

22)t 3

m a k i n g it possible to apply t h e s c h e m a t a : I I ( l ' ~ : u) = (1' =(= IIu),

a;IIu=~--II(a; u) ~-IIu,

w h e r e b y t h e final s u b s u m p t i o n is n o t valid as a g e n e r a l f o r m u l a , b u t m u s t be valid only in r e s p e c t to the s e c o n d p a r t 22)~ o f t h e e x t e n s i o n Page 380 c o n d i t i o n , f r o m w h i c h it results by m u l t i p l i c a t i o n o n b o t h sides. W i t h that, however, we have 1' =(= a 0

23)~

and

a ; a 0 ~=a 0,

23)~

w h i c h can be c o n t r a c t e d to 1' + a; a 0 =(=a0.

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=~--ao ; b, t h e r e f o r e *

b~-ao;b, * Repeated from earlier.

9)

326

SCHRt~DER'S LECTURE IX

w h e r e b y p r o p o s i t i o n ~ 4 5 is p r o v e d , which, incidentally, also e n c o m passes the m i n o r p r o p o s i t i o n s : a:~-ao;a

and

a 0 : ~ : a 0 ; a 0.

It f u r t h e r follows in r e f e r e n c e 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 w h i c h p r o p o s i t i o n ",~55 is won. O f the t h r e e i n d i s p e n s a b l e p r o p o s i t i o n s t h a t we n e e d to p r o v e ,~59, we a l r e a d y 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 simplifications 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 afortiori 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 ~ 5 2 (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, a m o u n t 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 ~ 4 4 . 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 o f the t h i r d a n d last o f t h e i n d i s p e n s a b l e p r o p o s i t i o n s , ~ 4 7 , I have to e n r i c h t h e c h a i n t h e o r y by o n e l e m m a . This is

FROM

{

PEIRCE

327

TO SKOLEM

(a;b:~-b) ={a;(beb):~--be~}, (b;a:~--b) = {(/~eb);a~--~eb},

(b@aeb) (b~--bea)

=(b;b@aeb;b), = (~;b~--~;bea)

24)

- - i n words: if b is a chain with respect to a, t h e n be/~ is also a chain with respect to a, a n d 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,

a n d from the last s u b s u m p t i o n it follows by transposition 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-be [). Conversely, the latter s u b s u m p t i o n can be rewritten equivalently as the previous o n e and itas the first one; that is, the final series of conclusions is c o m p l e t e l y reversible, q.e.d. O n this basis it is p o s s i b l e - - i n analogy to . ~ 4 1 - - t o establish yet ano t h e r l e m m a . But it is not exactly imperative to have f o r m u l a t e d it as such or even to r e m e m b e r it. Much better would be to b l e n d its p r o o f with that of the following m a i n proposition ('.~51). It states: (a; b:~--b) =(= E (1' + a; u =(=u)(u ; b:~--b),

25)

u

in words: If 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. A n d 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 c o n d i t i o n 1':~-u. Second, a c c o r d i n g to the previous l e m m a 24), it also fulfills the c o n d i t i o n a; u :(=u, as well as, third, a c c o r d i n g to 9) of w 17, the c o n d i t i o n u ; b:~--b, since (be/~) ; b and b are even equal. C o r r e s p o n d ingly, in 25) the last inclusion sign can be written 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 ' w

+ (be/~)w].

26)

328

SCHR()DER'S LECTURE IX

This m e a n s if 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 p a t h takes us via ,~51 to ,~47. To p r o v e t h e p r o p o s i t i o n .~51, o r (a; b:~-b) = (a o ; b = b),

(b =~-a j- b)

(b; a =(=b) = (b; % ---b),

(b=gc--bJ- a) = (b ~

= ( a I 0"

b = b),

a 1 =

b),

27)

it is essential, f r o m the p r e m i s s a; b:~--b, to derive 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." Now, certainly, u

a,,'b=(II '

u)" b=(c- II (u" b) =dc--u" b,

9 u,NB

'

u,NB

'

'

a n d it is so r e g a r d l e s s o f which relative satisfying the e x t e n s i o n 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 b e c a u s e o f p r o p o s i t i o n 5) o f w 6 a n d b e c a u s e t h e p r o d u c t m u s t be c o n t a i n e d in e a c h 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 that a m o n g t h e s e factors t h e r e will be at least o n e that is p a r t o f b, namely, t h a t t h e r e is a u t h a t satisfies N B - - i n the f o r m b j-~---for which u;b=dc--b, a n d in fact t h e r e f o r e it also follows ao; b :~--b afortiori. 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--(ao ; 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 o f .~47 or t (a; c + b=~--c) =(c--(ao ; b:~--c) can finally be a c c o m p l i s h e d q u i t e 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.

11)

FROM

PEIRCE

TO

329

SKOLEM

With this, all of the p r e c o n d i t i o n s of the n o n c i r c u l a r p r o o f of .~59, 60 have b e e n gained.---cf, p a g e 366. It r e m a i n s to simplify ~ 5 8 , or prove the proposition" 28)

a o = 1' + a ; a o

- - a r e c u r s i o n f r o m which the p o w e r 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. T h e r e f o r e , u is, in any case, of the f o r m u=l'+a;a0+v, a n d by p u t t i n g in the e x t e n s i o n c o n d i t i o n for v in 23), 1' + a + a ; a ; a o + a ; v = ~ - - l ' + a ; a o + v,

which, however, i m m e d i a t e l y reduces to NB0 =) a;v=~-l'

+ a ; a o + v,

because the first t h r e e e l e m e n t s of the subject can be c a n c e l e d o u t as obviously already c o n t a i n e d in the predicate; t h e n a = ~ - - a ; a o follows directly (as a ; 1' =(= a; a0) f r o m 23)~ a n d a; a; a0 = a ; (a; a0) =(=a; a 0 f r o m 21)t~. This allows us to c o n c l u d e a 0=

11 ( l ' + a ' a

v,NB0

'

0+v) =l'+a'a

'

0+

11 v = l ' + a ' a

~:,NB0

'

0,

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 p r o p o s i t i o n s as well, w i t h o u t using the p o w e r series for a0:* 7)

ao ; a o = a

and

6)

a ;ao = ao

(which d e f i n e s = a00).

T h e first one, 7), results as a c o n c l u s i o n f r o m 23)t~ a c c o r d i n g to the Page 384 s c h e m a of 27) or ,~51, the latter by a s s u m i n g b - a 0. It is, so to speak, the core of p r o p o s i t i o n ,~53: (b=~--ao;c) = ~ - ( a o ; b = ~ - - a o ; c ) , f r o m which it r e s u l t e d for b - a 0, c = 1', where, then, because the p r e m i s e is valid, the c o n c l u s i o n m u s t be valid too. Conversely, ".~53 results easily f r o m it, with (b@a

o

;c) @ ( a o ; b=~--a o ;a,, ; c=(~--ao ;c).

To prove the [The former on 7) from the Thus we also

latter 6) we already have a" a 0 = a 0 and a o ' a = a o. was 23)~, and the latter follows, for example, for b= a 0 based second proposition on the left in 27) backwards.] 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,

a c c o r d i n g 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, the use of series expansion as follows:

without

a00" a00 = a" a0" a0" a = a" a0" a=(=a0" a = a00. Those, finally, for whom it is i m p o r t a n t to go from o u r simpler point of d e p a r t u r e to derive 21) as a t h e o r e m from the general definition ~ 4 4 as well will do best to turn to the considerations following u n d e r 11) of w 24; that is, they n e e d only verify the general solution we gave for the proposition a ; x + b=g~--x as such (which again, without using the infinite series, is feasible on the basis of already established p r o p o s i t i o n s m s e e there) and from that take the p r o d u c 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 i m m e d i a t e use. Finally, t h e n o t i o n s o f t h e a - c h a i n a n d t h e a - i m a g e c h a i n o f b m a y b e d e m o n s t r a t e d with a p a i r o f figures. Just in reference to c o m m e n t s 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 r d e r to have s o m e t h i n g to think about, o t h e r than by the known n u m b e r s " such a step might not seem entirely superfluous. I had Figures 22 and 23 p r e p a r e d , however, while I was still using D e d e k i n d ' s notation, and so must ask the r e a d e r to imagine the letters as having b e e n c h a n g e d somewhat, namely, to consider the small a, a t, b, bt, c, ct as having b e e n 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, bt, b", b', and c, so that the letter a remains free for the relative to be chosen as the m a p p i n g rule. T h e a c c e n t s h e r e a r e to b e u n d e r s t o o d the schema b' = a " b,

b " = a ; b' = a ; a " b = a 2

* In Dedekind 1888 (footnote 1).

as a b b r e v i a t i o n s a c c o r d i n g to

.

b,

b at = a ; b " = a ~ " b,

FROM PEIRCE TO SKOLEM

331

FIG. 22

c h o s e n ad hoc f o r t h e p u r p o s e o f s i m p l i f y i n g t h e e n t r i e s to b e m a d e in the figure. It should not be disturbing to anyone, but simply m o r e instructive, if from now on we make use of a n o t h e r m e t h o d of geometric d e m o n s t r a t i o n than the one illustrated in w 4. For the first universe of discourse to be c o n s i d e r e d 1 ~, this time let us keep in mind the entire collection of points in a sector of a circle, or the corresponding a n g u l a r 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 u n d e r s t a n d 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 " m a p p i n g " from our theory as o p p o s e d to the m o r e restricted circle of ideas in D e d e k i n d ' s text--with the precise intention of showing that it already has a substratum projecting broadly over the n u m b e r system. In the context of the three sectors to be seen in Figure 92, this m a p p i n g is also an invertible one, thus in ~ ' s terminology "similar" (or " d i s t i n c t " ) m b u t a different o n e in the sector going to the left than in the two going to the right. In the latter, the a-image of (an e l e m e n t - r e p r e s e n t i n g ) point A is the point

A'= a;A, which lies half as far from the c e n t e r 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 portion of a concentric 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 o u t e r m o s t q u a d r a n g l e b inward, the a-image chain aoo;b of the same. T h e d i m e n s i o n s of

SCHRODER'S LECTURE IX

33 2

C

R

J

~t

2?

]a.. A~r FIG. 23

b have been chosen here so that each of the objects appears to be disjoint from its image. For the sector r u n n i n g down and to the right, b is chosen so that the image is adjacent to its object. T h e a-chain of b here is the entire shaded sector, a n d the i n n e r half m e a s u r e d radially must manifest the a-image chain of b (regardless of its image chain). T h e success would be similar in the case of an object exc e e d i n g its image. For the sector or angular surface going down and to the left, is the limit of the o u t e r infinite series of successively larger shaded quadrangles, and such, however, that the i n n e r m o s t one is l a c k i n g m w h e r e b y it does not matter that divergence would exist here in the series of measures (in the sense

ao;b

aoo;b

of area) of these shaded surfaces. In Figure 23 the assignment selected as m a p p i n g rule a is single-valued, but it is not invertible. As universe of discourse I i (or Dedekind's system), we see h e r e the point system of a s e g m e n t of a circle c o m p o s e d of two sixths of the circle or sextants

Ill.

I and H and a half of such, Let the following be the m a p p i n g rule a. T h e 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"

II.

[Rutschkante]MR (without

flipping it over),

until it coincides with T h e image of a point C (c in the figure) in 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 over the folding edge MF until it coincides with the adjacent

II.

III

III

half of T h e image B ' = of a point B in H (respectively b' of the b in the figure), in contrast, is d e t e r m i n e d similarly to [that described] above as that point which

a;B

lies hallway from the m i d p o i n t on the same ray as B.

FROM PEIRCE TO SKOLEM

333

T h e r e u p o n it is obvious that what we are w o r k i n g with is a m a p p i n g o f the e n t i r e d o m a i n of discourse or system 1 ~ to itself.

Page 387

In the s h a d e d p o r t i o n of Figure 23 t h e n is the a-chain a 0 ; (b + c) of the system b + c of points m a k i n g up the area of both circles b a n d c (A a n d B i n the figure), a n d we r e c o m m e n d that the r e a d e r look at the four s e g m e n t s into which the "edges" MF, M R divide these circles a n d their images, etc. It will t h e n be easy to check the validity of o t h e r of the propositions such as ~ 4 5 . . . . ~ 5 6 f r o m how they look by i m a g i n i n g a region of points that is s o m e h o w limited 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 of 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 r o b l e m 1. To d e t e r m i n e the most g e n e r a l relative with r e s p e c t to which a given b is a chain, that is, to solve the s u b s u m p t i o n :

x ; b=g~-b. This p r o b l e m is only a special case of the o n e we solved in w 17 with the first inversion t h e o r e m . Accordingly, the solution 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 arbitrary u; that is, the p r o p o s i t i o n 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 well 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. P r o b l e m 2. To d e t e r m i n e the most g e n e r a l relative with r e s p e c t to which a given a is a chain; that is, to solve the s u b s u m p t i o n for x:.

a; x=~-x. We have already given the solution to this p r o b l e m in 5) of w 22 (page 325), finding two expressions as the g e n e r a l root: x = a0 ; u a n d x = a ~j- u, which can easily be r e c o g n i z e d as essentially different; namely, a l t h o u g h these two terms b e c o m e s 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 different values for arbitrary u. For otherwise, the following would have to be valid of u-- 1' for any given a: a0; 1'= a 0 = 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 n o t have to be =~= 1'. If we may be allowed o n c e again to point out the most i m p o r t a n t aspect for us of 5) of w 22, we may write the proposition:

334

SCHR6DVR'S LECTURE IX (a; x=gex) = ~ (x = a,, ; u) = E ( x = a , j- u). u

2)

u

This p r o b l e m has probably provided us with the most natural o p p o r t u n i t y to i n t r o d u c e tile c o n c e p t a 0 of the a-chain, as well as that of the a - d u a l chain [a-

Gekett]: the u n k n o w n must be the a-chain of some relative umsimilarly the a-barconverse dual chain to one such u n d e f i n e d relative. As particular roots for any given a, only x--0 and x = 1 can be given from the g r o u p of modules; that is, it can be said (in conformity with ~ 3 8 ) Page 388

that the

entire 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 relative--any o n e 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 particular 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 t c . - - 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 of a, like the a-image chain, accordingly, is also always a "chain with respect to a. "Etc. We already did the two proofs with both solution forms 2) on page 331; for the first one, P r o o f 1 resulted in the proposition a; a 0 =(=a, the c o n s e q u e n c e s of which have also b e e n registered already in 6), page 361. For the s e c o n d solution torm, P r o o f 1 has already b e e n d o n e at the b o t t o m of page 331; but we have not yet registered as a proposition the fact that it is a solution. If we say b for the u that appears there, then we get the p r o o f of the proposition: 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,

a,. a+a,,

a~, + - a , , s a.

4)

To derive the c o r r e s p o n d i n g 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.) ...:~ Having made

it p o s s i b l e , w i t h t h e s u f f i x e s 0, 00, 1, 11, to p r o d u c e

e x t r e m e l y concise n a m e s for e x p r e s s i o n s that, while having the f o r m of infinite expansions parent formation solution

nevertheless manifest the simplest and most translaws, w e will a g r e e f o r t h e s a k e o f b r e v i t y to say o f a

of a problem

t h a t it is p r e s e n t e d

in " h a l f - c l o s e d f o r m "

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 ) , p r e s s i o n is c o m p o s e d

(or

as s o o n as its ex-

by m e a n s o f a finite c o l l e c t i o n o f o p e r a t i o n s

of

* The conversion rules for e4~,a,,0, a I, and all (pp. 388-389) have been omitted here.

FROM PEIRCE TO SKOLEM

335

t h e six s p e c i e s a n d t h e d i s t r i b u t i o n o f s u c h suffixes. T h e s e f o r m s t h a t we h a v e d e s i g n a t e d " h a l f - c l o s e d " f o r b r e v i t y ' s s a k e a r e t h e r e f o r e in ess e n c e t h e o n e s t h a t o c c u r i n s i d e t h e c i r c l e o f o p e r a t i o n s o f t h e six s p e c i e s as i n f i n i t e e x p a n s i o n s , w h i l e o n t h e 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 forms after a d j o i n i n g the n o t i o n of the a-chain. W e m o v e n o w to t h e s t e p o f g e n e r a l i z i n g t h e last p r o b l e m . P r o b l e m 3. Solve f o r x t h e s u b s u m p t i o n a; x + b:~-x.

6)

T h e p r o m i n e n t role this plays in various of the D e d e k i n d p r o p o s i t i o n s ~ 4 0 , 41, 44, 47, 48---seems by itself e n o u g h to bring up the p r o b l e m . It can be solved in several ways, and we want to follow to the e n d all of them, insofar as they lead to interesting results and c o m p a r i n g them appears to be methodologically instructive. T h a t objective leads us to the following four forms of the solution, Page 390 written out separately: = E I x = a(, ; (b + u)}

a;x + b@x)

u

7)

= E [x = a0 ;{u + g~); (d~ ~f a)b}]

8)

=Elx=alj-(a0;b+u)}

9)

u

u

= Eu [x = a o ; b

+ u{a 1 ~ ( a 0 ; b + u)],

10)

o f w h i c h t h e first is to b e c a l l e d " t h e best." On the derivation. Since x = 1 satisfies the r e q u i r e m e n t of the p r o b l e m , 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 proposition 1) of w 13: (a" x + b=~-x) = E {x =f~(u)}, u

where f(u) -- b + u + a" 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 iterated 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 9b + a 0 "u = a o ' ( 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. A l t h o u g h unnecessary, we shall also verify it directly. T h a t Proof 1 is correct is based solely on the propositions 1' :(=a 0 and a" ao=~-a o. Proof 2 coincides with the p r o o f of the proposition:

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 ~ 4 5 ) , while the first parts of the forward propositional subsumption can be justified accordingly from 13) of w 23 (or from ~ 4 0 ) , in that b=g~--x) ~-{a" (b+ x)=(= x} =(=la" (b + x)=g~-b+ x}

(a'x+

={a0"(b+x)-b+x--x} according to ,~51, and because b =(= x must be valid, the reverse direction is a l r e a d y p r o v e d by Proof 1 (using it for u = x), q.e.d. S o l u t i o n 7), w h i c h is n o w j u s t i f i e d , p u t s us in a p o s i t i o n to c o m p l e t e , in t h e m o s t e l e g a n t way, t h e final o u t s t a n d i n g s t e p in t h e b a c k w a r d c o u r s e o f t h e i n v e s t i g a t i o n in w 23, n a m e l y , to arrive h e u r i s t i c a l l y at D e d e k i n d ' s d e f i n i t i o n , ~ 4 4 o f t h e a-chain o f b. It follows" II

x,(a;x+l,~x)

Page 391

=IIa0;(b+u) u

=%;b+IIa0;u=a0;b, u

in t h a t II a 0 ; u = 0 is o b v i o u s l y valid, as s h o w i n g t h e f a c t o r a0 ; 0 ( = 0 ) . u

W i t h that, we have n o w c o m e all t h e way to t h e e n d by t h e " b a c k w a r d p a t h , " as it is c a l l e d in w 23 .... 4 Page 402

O n e m o r e w o r d in c o n c l u s i o n . On ~ 6 3 . 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. T h e n 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 aimage 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, u n d e r the conditions given above, it must be that a o 9[~c=(c--c.

4 Schr6der's three other solution forms for Problem :3 have been omitted here.

38)

FROM PEIRCE TO SKOLEM

337

[More precisely, if even b C c, t h e n a0;/~c C c; the /~c is D e d e k i n d ' s U.] We recall that we have these propositions 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 from the previous material. 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 s e c o n d claim proved. A third claim is merely a repetition of the second. If, namely, a0;/~c = d

Page 403

is given as an abbreviation, 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 a n d c o m e s out as d=(c--c, which makes up the s e c o n d claim. [A fourth claim asserts that b--a;d+

ccl,

a n d a fifth (and final) claim says that, if, in addition to e v e r y t h i n g else, b= a;c, t h e n it m u s t also be that cd ~ : a ; cd. T h e s e two claims, at all events, are not as generally valid as the others above for any given binary relative. T h e y s h o u l d have b e e n o m i t t e d f r o m 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 d e l i n e a t i o n of the same. A n d this is not the place to check their validity for single-valued m a p p i n g s a a n d systems b,c.]m

Last of a l l m s c i e n c e 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 valid for the chains are the propositions ( a : ( = l ' ) = (a,, = 1'),

(0' :(=a) = (a~ = 0 ' ) ,

or

(0' + a) l = 0',

(l~z) 0 = 1',

which can be proved with utmost ease from their formation 6) o f w 22, p a g e 325. If a:~--l', t h e n a ; a : ~ : l ' ; 1 ' = 1', etc. A n d a 0 = (0~)0, T h e c h a i n o f a is t h e r e f o r e p a r t [ A l i o t e i l ] o f a.

a I = (1' + a) 1.

39) laws; s e e

40)

the s a m e as the c h a i n o f O'a, t h a t is, t h e alio-

Various types of proofs are possible, the simplest based on 15), page 365, with a,, = (1' + a) ~~= (1' + 0~z)= = (0~),,.

With the solution of the solution problems,

of courseDin

harmony

33 8

SCHRODER'S LECTURE IX

with p a g e 1 7 4 f f . - - a few e l i m i n a t i o n p r o b l e m s h a v e also b e e n solved. Namely, in t h e f o l l o w i n g 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), (x,, = a) :(=(a 0 = a),

41)

(a - 1' + y)(y;y:~--y) :(=(a 0 - a)

42)

t h e r i g h t side r e p r e s e n t s t h e full r e s u l t a n t s o f t h e e l i m i n a t i o n o f x o r y, respectively, f r o m t h e left side. A direct proof is also easy: Since, according to 5), page 361" x00" Xoo:(e:Xoo, by inserting this, a" a:g~-:afollows 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. Page 404

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 recognizable from first glance as the complete one, because then x = a suffices. Regarding the second subsumption 42), we may rewrite its second premise equivalently, 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 i s p o s a l for r e p r e s e n t i n g all transitive relatives, in t h e f o r m o f 34), p a g e 339, a r e c l o s e d e x p r e s s i o n s , a n d t h e q u e s t i o n arises as to t h e e x t e n t to w h i c h t h e l a t t e r are o f value for c a l c u l a t i n g i m a g e c h a i n s . As r e g a r d s d e t e r m i n i n g t h e i m a g e c h a i n a00 for a given relative a, this d o e s n o t a p p e a r p o s s i b l e so far. If t h e issue, however, has o n l y to d o with s p e c i f y i n g t h o s e a m o n g t h e b i n a r y relatives t h a t are, respectively, i m a g e c h a i n s o r c h a i n s at all, t h e s a m e results w o u l d 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 t h e f o r m o f an i n f i n i t e e x p a n s i o n : U00

=

12 q- U2 -t- U3 -q- ""

,

U0 =

1'+

U00.

T h i s b e c o m e s an easy m a t t e r , however, if we take for u00 t h e 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 t h e s c h e m a c i t e d above, a n d e v a l u a t e i t - - w h i c h has a c l o s e d 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 p r o d u c t (universal quantifier) over domains of the first o r d e r [Stufe] (quantifiers over individuals) and of the second o r d e r (quantifiers over all binary relatives on the domain, his version of seco n d - o r d e r 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 operations on formulas. This lecture thus has a highly algebraic flavor, even t h o u g h it expresses m o r e 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 m e a n t that he regarded a universal existential prefix as a p r o d u c t of sums or a greatest lower b o u n d of least u p p e r 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, where f r a n g e s over all funche writes (qx)(3y)rh(x, y) as (u tions 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 u p p e r b o u n d Ef on the right is over a c o n t i n u u m 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-tofollow p r o o f of the L6wenheim-Skolem theorem. We have used a simpler notation 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 S c h r 6 d e r got 339

SCHRC)DER'S LECTURE XI

34 ~

with q u a n t i f i e r s - - f u r t h e r t h a n the early Peirce; in addition, the algebraic p o i n t of view of the c o m p l e t e distributive law a p p e a r s h e r e for the first time in m a t h e m a t i c a l history, which was the device that L 6 w e n h e i m p i c k e d up for his proof.

Eleventh

Lecture

Studies of Elimination,

P r o d u c t a n d S u m Problems

w 29. O n 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 Evaluation.

Page 491

On the Inversion Problem.

O n p a g e 190 of "Note B" (Peirce 1883; see also Peirce 1880, p. 55), Peirce r e m a r k s that in the relative algebra t h e r e are a n u m b e r of "curious 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) }{bo'- (u + c) },

a" bc = II (au" b + ati" c),

a d- (b + c) - E {(a + u) a- b]{(a + zi) a'- c},

u

~

1)

u

w h e r e the II a n d E as identical p r o d u c t a n d identical sum, respectively, e x t e n d over all relatives of the universe o f discourse 12. . . . 1 Page 497

Peirce's p r o p o s i t i o n s 1), a n d o u r e x t e n s i o n o f t h e m , f o r m the "startu p stock" of p r o p o s i t i o n s a n d m e t h o d s which allow us to evaluate s u m s E as well as products II in o u r discipline. It is often useful to be able to give the identity p r o d u c t (the intersection [Gemeinheit]) of all relatives x which fulfill a certain c o n d i t i o n , for e x a m p l e , roots o f a given e q u a t i o n ~ a s we have already shown in the n i n t h lecture. Likewise, the q u e s t i o n r e g a r d i n g the identity sum o f all roots can be of i m p o r t a n c e . For this reason alone, the art o f determ i n i n g the sum a n d p r o d u c t ~ w h i c h is n o t so e a s y - - d e s e r v e s to be cultivated a n d d e v e l o p e d systematically. T h a t this is f u n d a m e n t a l for e l i m i n a t i o n p r o b l e m s , a n d for i n f e r e n c e in g e n e r a l , was shown at the e n d o f w 28. I shall now p r e s e n t a series o f my own findings which aim at i n c r e a s i n g this capital (science); they no l o n g e r relate to Peirce's publications, b u t are nevertheless relevant. We are d e a l i n g with sums E; a n d p r o d u c t s II which have "absolute" e x t e n s i o n , i.e., over the entire d o m a i n o f discourse. D e p e n d i n g o n w h e t h e r the i n d e x is an e l e m e n t symbol i or j, etc., a n d its e x t e n s i o n is the first d o m a i n of discourse 1', or w h e t h e r it a p p e a r s as the s u m or p r o d u c t variable of a binary relative u o f any given type with the s e c o n d ' Schr6der's proofs of Peirce's "curious development formulae" (pp. 521-536) have been omitted here.

341

FROM PEIRCE TO SKOLEM

d o m a i n o f discourse 12 as its e x t e n s i o n , we can distinguish two o r d e r s [Stufe] in s u m a n d p r o d u c t d e t e r m i n a t i o n . Even t h o u g h Peirce's p r o p o s i t i o n s 1) already b e l o n g to the s e c o n d o r d e r [zweite Stufe], we want, first, to c o n s i d e 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 b e g i n with, the r e a d e r can easily prove these g r o u p s o f little p r o p ositions by m e a n s of the coefficient evidence:

n i = n ; = n ~ = n ~ = 0,

Ei = E ~ = E i ' = E ~ = 1,

E i i = 1' = l-I(/+ ~) = H(~ + ;),

Ei~ = Ei-/" = 0 ' = II(~+ ~),

~ [ = 1,

II(i + i) = 0.

7)

8)

9,)

T h e i n d e x is a s s u m e d 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 time being, J as the r e p r e s e n t a t i v e o f o n e o f the f o u r e l e m e n t relatives i, i, i, ;, we can s h o r t e n f o r m u l a 7) as follows: EJ=l,

nj=o.

Because II~(i)J~--IIJ, etc., it is clear that m o r e g e n e r a l l y we m u s t have {4~(i) + J} = 1,

II 4~(i)J = o.

Thus, we d o n o t have to write o u t the p r o d u c t f o r m u l a s for e x p r e s s i o n s such as IIi a;i" i, IIii" i;b, b e c a u s e we i m m e d i a t e l y r e c o g n i z e t h e m to be e q u a l to 0. Etc. P r o p o s i t i o n s such as E 0 ij'= 1,

II 0 ( ~ + ] ) =0,

which refer to d o u b l e o r m u l t i p l e sums o r p r o d u c t s , we d o n o t want to take into c o n s i d e r a t i o n for the time being. R e g a r d i n g f o r m u l a 7), the following p r o p o s i t i o n c o m e s f r o m 29) o f w 25:

342

"

SCHRODER'S LECTURE XI

{

- "/

Z i a ; i " i = Z~(aj. i)i

H,(a;i+~)

=Hi(ao ~+~)

IIi(~+ i;a) =II,(~+~j-

Z i i " i, a - Z~i( [j- a)

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) = ahkFinally, these can be i n f e r r e d m i n the form of r.,~a'i, i;l'= a" 1~, for examp l e m f r o m the more general proposition 14) which we will give later. F u r t h e r m o r e , we are able to e v a l u a t e Z i a n d IIi for t h e 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 relative a a n d a relative o f i, w h i c h w e r e d i s c u s s e d Page 499 in f o r m u l a s 21) to 23) o f w 25. T h e results in q u e s t i o n are e x p r e s s e d by 32 f o r m u l a s w h i c h , 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 t h e o p e r a t i o n s c o m b i n i n g t h e 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 t h e 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 = Ei; a = Z~'a

= Z i ; a = Z ~ ; a = Z ( ~ j - a),

a j - 0 = H(aj- i) = II(aj- ~) = 1-I(a 0, i) = I I ( a ~ ~) = H a ; i,

12)

0 0' a = H(i 0~ a) = H(~o~ a) = H ( i o ~ a) = II( ~j- a) = I I i ; a,

{

Ei(a~i)

=(aj-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 i d e m t h a t i ; a - i " 1 ; a b y 21) o f w 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 m a n n e r (because of i';1 - 1) as Z,a;i=Z;a;i"

i;l=a;1,

II~a;i=II~(a;i+ i';0) = a3-0,

from the more general proposition 14), which will follow.

FROM PEIRCE TO SKOLEM

343

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 i m p o r t a n t , it seems to me, are the following groups of propositions: 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 g o o d idea to m e m o r i z e the first formulas on the right 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. T h e dual to 14) is v

II~a;i.

,..

i;b=(aj-O)(O~b),

E~(a;i+i;b)

=a;l+l;b,

because of 12), where, on the left, we must H~a; i 9II~i'; b, etc. While we now can also easily evaluate

have

II~a;i.

15) i;b=

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 2 5 - - i t is not at all true for II~a; ~b, and, in general, we still look at the large majority of sum a n d p r o d u c t expressions with confusion. T h e r e f o r e it seems advisable, first, to have at o u r disposal as completely as possible the simplest sum a n d p r o d u c t formulas from the ':start-up stock" of o u r discipline, and, second, to learn m e t h o d s to refer a given s u m m a t i o n p r o b l e m , etc., back to the p r o b l e m s solved with the simple formulas from the "start-up stock." C o n c e r n i n g the first task, we believe we should state, discuss, and

344

S C H R O D E R ' S LECTURE XI

p r o v e at the very least the following set of p r o p o s i t i o n s , as duals a n d s u p p l e m e n t s o f 10)"

{

E ; a ; ~ . i'= a;O',

H;(aj-i+~) = aj-l',

-

I];i" ;; a = 0 ' ; a,

,.,

II; ( i + i'o~ a) = l'j- a,

16)

,.,

Eia; i ; i = ~;(aj- ~ ) i = a ; 0 ' ,

II,(a ; i + ~') = II,(a j- ~ + i) = a j- 1',

E,~" i; a = E,~(~0~ a) = 0 ' ; a,

1-I;(i + i; a) = l-I;(/+ ~

{

a) = 1'~ a,

H , ( a ; ~+ ~) =a;O',

E; (aj- i)i = aj- 1',

H ; ( ~ + ;; a) = 0", a,

E ; i ( i ; a) = 1'~ a,

17)

18)

v

Z;a;['i=a;1,

H , ( a j - i + i) = aj-O,

~;~" i ; a = 1 ;a,

l-I; (i + i'd" a) = 0 d" a,

w h e r e , w h e n w i t h o u t t h e asterisk, r e a d a ; 0 ' ; 0 '

{

E,(a~i)~=(a~l');O', _

,.,

~;i(ij-a) =0'; (l'~a),

Page 501

for a ; 1 . Etc.,

II, (a ; ~ + / ) = a i 0 ' ~ 1', ~,

II;(i+i;a)

19.)

= l'~0';a,

20)

In the 32 f o r m u l a s 10) a n d 16) to 20), we will find the ~ a n d II o f all ( b i n a r y identity) p r o d u c t s a n d sums which can be f o r m e d f r o m a ; i o r a;~, as well as f r o m a j - ~ o r a j-i and ; o r ~, e t c . - - a s l o n g as at least t h o s e E, II that can be r e d u c e d to 0 o r 1 at o n e g l a n c e are n o t t a k e n into c o n s i d e r a t i o n . 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 f o r m u l a s stated, f o r m u l a 18) a p p e a r s as especially r e m a r k a b l e b e c a u s e it t e a c h e s us to r e p r e s e n t c e r t a i n relative p r o d u c t s , such as a ; 0 ' , as identity products, while, in g e n e r a l , this is only c a r r i e d o u t in t h e f o r m of an identity 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).

FROM P E I R C E T O SKOLEM

345

For formulas 19) and 20), we appeal to the coefficient evidence, w h e r e u p o n the left side is, respectively, Lhk=Ei(a'[)hk t^k ~ = EiE t amtu, - -tk^ = E t ahl~,iOliOik , , 7

!

=

(a'O' "O')hk,

=

Lhk=~;IIt(aht+ ilk)Zkh=F.,iIIt(aht+ lt;)01k R^k,

q.e.d.

As d u a 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 likewise have to cite the f o l l o w i n g p r o p o s i t i o n s w h i c h 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: = (aj-l') ; (l'j-b),

E,(aj-i)(ij-b)

...

II,(a;~+~;b)

II,(a ; ~ + i; b) = II,(a ; i +

= (a o" 1') ;b,

= a;O'~

b,

,1.

~,(a j- ;)( i ~ b) = E,a ; i " ( ij- b)

...

II,(a ; i+ ;; b) = R ( a ~ ~+ ~; b)

= a ; (l'c~ b),

{ {

= aj-O'

E , a ; i" i'; b = E,a; ~. (~j. b) ] -_ _J ~ , a ; i" ~;b = ~,(a ~t i)" i ;b

IIi(ao~i+~j-

b) = I I , ( a j - i +

I2,a ; ~. ~; b = a ; 1 ; b,

b) 22)

", b,

= a;O"b,

i';b) }

I-I, (a j- i + i ~ b) = lI, (a ; i + ij- b)

Page 502

~

..,

B,(aj- i)( i ~ b) = r.,(aj, i) 9 i; b

21)

=a;O'j-O';b,

1' = a j-

j- b,

II,(a j- i + i'0~ b) = a ~ 0 ~ b.

23)

24*)

T h e II, o f the g e n e r a l t e r m s o n the left a n d the ~i o f t h o s e o n the r i g h t are easy to state a c c o r d i n g to 11) a n d 13) since the g e n e r a l t e r m s are 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 ' ,

w h e r e u p o n we can then rewrite { a at i = a at l t at i= (aat 1')"i, iatb=~at

l'atb=i;(l'atb),

_

(as well as a a t i = a ' i ,

.

;atb=i;b),

a " ~ = a " O" i , -

.

;;b=i;O"b

25)

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 t h e 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 right) 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

34 6

SCHR6DVR'S LECTURE XI

c o m b i n i n g relatives with e l e m e n t r e l a t i o n s , relative a d d i t i o n is always d i s p e n s a b l e , namely, it c a n be p l a y e d o u t as a relative m u l t i p l i c a t i o n ( e v e n 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 n a t u r a l l y t e r m i n a t e in i d e n t i t y a d d i t i o n . Basically, o n e o n l y n e e d s to l e a r n to c a l c u l a t e well with t h e 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 Page 503

I n s t e a d o f e s t a b l i s h i n g still m o r e f o r m u l a s , we n o w w a n t to d e m o n strate, with a series of s m a l l e r p r o b l e m s , h o w we c a n d e t e r m i n e , with t h e 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 variously s t a t e d p r o d ucts a n d sums. F o r t h a 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 t h e g r o u p s o f a s s o c i a t e d f o r m u l a s . F r o m t h e way we t r e a t t h e s e e x a m p l e s , the r e a d e r will at least be able to g e t an a b s t r a c t i d e a o f o u r m e t h o d s , a n d m o r e so in t h e f o l l o w i n g sections. 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)].

Page 504

Problem 2. We are looking for x = H i (a; i'+ f; b).

Solution: x = r I i ( a ; 1 9f+ [. 1;b) =(a;1 + 1 ;b)(a;1 + IIih(IIii'+ 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)IIi(a;i+ i) = (a~0 + 1 ;b)IIi(a+ 1 ' ) ; i = x=a~ ( a ~ 0 + 1;b){(a+ l ' ) ~ 0 } = a ( a + l ' ) ~ 0 + { ( a + l ' ) z t 0 } " 1;b; therefore 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+ ~), 1 "a=IIi(0"i+

i'a) =II;(~+ i'a),

a j- 0 = Ei (a j- ~) i', Oj-a=Eii(~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 understood 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} u n d e r the previously solved problem, and, therefore, has to be

x = a j - O + {(a~ 1' + l'),:t0l" 1 ;b. We can also determine 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.

FROM PEIRCE TO SKOLEM

347

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) 91 ; 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=(aj-O+l;b){(a+O')~O}=a&O+l~a;l'l;b=aj-O+l'a;l'l;b=aj 0+l~t'l" l'b. Page 505

-

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 t h r o u g h the double product: x = H;Hj(a;j + j'; i + i; b) = Ilfla;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. T h e two results coincide because of the second formula of proposition 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 p r o b l e m 3: x = I ;a" 1 ;b. As n u m e r o u s as t h e p r o b l e m s a r e w h i c h c a n be s o l v e d in this way, we still c a n n o t discover, for e x a m p l e , Ili a;~b. L e t us t u r n n o w to t h e s u m a n d p r o d u c t p r o b l e m s o f t h e s e c o n d o r d e r [zweiten Stufe]. A n i m p o r t a n t p r o b l e m o f a g e n e r a l c h a r a c t e r is: to determine the "intersection"[ Gemeinheit] H, as well as the ( c o m m o n o r t o t a l ) "domain" [Bereich] ~ of all binary relatives x which fulfill a given condition--for e x a m p l e , t h e e q u a t i o n F(x) = 0 - - a s t h e i r " r o o t s . " It makes sense to d e n o t e these two unknowns (as p r o d u c t and as sum) by P and S. But the p r o d u c t is the Subject and the sum is the Predicate to each of the roots, so that this d e n o t a t i o n could be misleading. I will therefore call them P and Q. By marking the extension condition under the signs H, ~, the m a t h e m a t i c i a n would be inclined to write

P=IIx

Q=2x

{F(x) =0}

{F(x) =0}.

x

x

But our discipline has the advantage that the extension condition can be inc o r p o r a t e d into the H, E-expressions themselves. How this can be d o n e shall be explained shortly with an obvious extension of the problem. T h e p r o b l e m c a n b e 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

348

SCHRODER'S LECTURE XI

o r a d d a given f u n c t i o n ~(x) o f t h e roots, i n s t e a d o f t h e r o o t x itself. We c o u l d t h e r e f o r e l o o k for P = II r

Page 506

Q =g r

x

x

{F(x) = 0}

{F(x) = 0}.

We give II a n d ~ a b s o l u t e e x t e n s i o n m o v e r all p o s s i b l e relatives x o f t h e s e c o n d d o m a i n o f d i s c o u r s e . N o w we o n l y h a v e to m a k e t h e g e n e r a l t e r m n e u t r a l w h e n x d o e s not fulfill t h e e x t e n s i o n c o n d i t i o n F(x) = O, t h a t is, we o n l y have to w o r r y that, in e a c h s u c h case, t h e g e n e r a l t e r m o f 1-I a n d E is n e u t r a l , n a m e l y e q u a 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 t h e o t h e r h a n d , t h e t e r m ~(x) m u s t a c t u a l l y a p p e a r o r be effective for every x w h i c h fulfills t h e e x t e n s i o n condition. We c a n d o this by w r i t i n g P = 11 { ~ ( x ) +

F(x)

= 0l,

Q = E

x

Depending

~b(x){F(x) =

33)

0}.

x

on whether

x is r o o t o r n o t , t h e f a c t o r p r o p o s i t i o n

F(x) = 0 in Qwill in fact have t h e t r u t h v a l u e 1 o r 0; o n t h e o t h e r h a n d , t h e n e g a t i o n w h i c h a p p e a r s as a s u m m a n d etc.

in P will b e e q u a l 0 o r 1,

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)], x

Q = ~ ~(x) F(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 g e n e r a l , we can n o w r e p l a c e , in 33), t h e p r o p o s i t i o n a l t e r m , acc o r d i n g to t h e s c h e m e o f w 11, by a b i n a r y relative, t h a t is to say, a d i s t i n g u i s h e d relative w h i c h t h e n takes o n t h e v a l u e 0 o r 1. T h e n we have m

F(x) = 0 = {F(x) 4: 0} = 1 ;F(x) ; 1,

{F(x) = 0} - 0 ~ F(x) ~ 0.

T h e n it follows P = II {O(x) + 1 ; F(x) ; 1 l, x

Q = 1~O(x){0 e F(x) ~ 0l.

34)

x

H e r e u can be w r i t t e n for x. In t h e l o w e r p o r t i o n o f t h e p r o b l e m , w h i c h i n t e r e s t e d us initially, we have, in p a r t i c u l a r ,

FROM PEIRCE TO SKOLEM

349

II{x + F(x) =0} = H{U + 1 ;F(x) 91}, x

u

{F(x) = 0} = g u{0 e F(u) e 0}. x

35)

u

If we could evaluate H, respectively g, taken over u with the absolute e x t e n s i o n (over all binary relatives), for any given f u n c t i o n of u, we would be able, with s c h e m e s 34), 35), to obtain the P a n d Q in quest i o n - - e v e n w i t h o u t knowing or d e t e r m i n i n g the r o o t x [of the e q u a t i o n

F(x) Page 507

= 0].

If, however, we know the g e n e r a l r o o t or the solution of this conditional e q u a t i o n in the f o r m of

x =f(u), we are in an even b e t t e r position to find the solution of o u r p r o b l e m , as we have m o r e i m m e d i a t e l y a n d simply P = H ~{f(u)}, u

Q = ~ ~{f(u)},

36)

as well as for the lower p o r t i o n of 35) of o u r p r o b l e m :

II,, {x + F(x) = 0} = H f(u),

r,x x{F(x) = 0} = ~f(u).

37)

A c c o r d i n g to the idea of the g e n e r a l solution, we have t h e n 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 r o b l e m in its full and u n l i m i t e d generality is n o t known.* But to discover such a m e t h o d is the ideal o f this t h e o r y which will p e r h a p s never be possible. Probably we will only be able to solve it in stages a n d thus a p p r o a c h this faraway goal slowly. For the time being, we can only set a definite goal for ourselves and, in o r d e r to reach it, create a part of the m e t h o d . Such a practical g o a l - - a n d , in fact, the closest o n e f r o m a systematic p o i n t of view--is the d e t e r m i n a t i o n of H a n d E of all roots of o n e of o u r t h r e e e l e m e n t a r y inversion p r o b l e m s . Page 508

Since x= u(aj-~) is the general root of the subsumption x" b 4= a, and, of course, we must have IIu =0, u

* Compare the end of this section.

Yu=l, u

35 ~

SCHRODER'S LECTURE XI

and since u =0 and u = 1 themselves figure u n d e r the values over which u extends, 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.

W h e n e v e r 0 belongs to the roots x of the given condition, 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 l o n g e r here. For example, IIu'a=0,

Eu'a=l'a

u

u

is immediate. T h u s , t h e r e r e m a i n s to b e s o l v e d t h e i n v e r s i o n p r o b l e m f o r t h e (ext e n d e d ) s e c o n d case a n d f o r t h e t h i r d case, a n d h e r e it will b e imp o r t a n t u i f we o n l y d e a l with o n e r e p r e s e n t a t i v e f o r 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 h e f o r m o f x = II [u + a{(b + (t) j- c} ; d].

38)

u

In o r d e r to r e a l i z e this g o a l , w h i c h c a n o n l y b e a p p r o a c h e d we n e e d to t a c k l e a s e r i e s o f p r e l i m i n a r y p r o b l e m s .

in s t a g e s ,

Pro b l e m 8. We are looking for: x = II (u + 7i" a). u

According to Peirce's proposition 1) or 4), we can immediately state x = 1 9al', a n d so x = II (u" 1' + ,i" a). u

Because of 1 9i1~= 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: II(u+a" u

~'b) =a'l'b=a"

l'bl',

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 I u = a + O = a, and the II of the second, which falls u n d e r the above schema. P ro b l e m 9. We are looking for: x = II ( u ' a + (t r b). u

According to 14), we can r e p r e s e n t tij-b as a product (over z), by which we have won the game! Because we can now c o n c l u de

35 x

FROM PEIRCE TO SKOLEM

x = I I { u ' a + II;(~i" i + i;b)] u

•II;{II (u" a + ~i'i) + i;b}-II;(1 "ai+ i;b) u

Page 509

a c c o r d i n g to 4), a n d further: x=II~(i;a+ i;b) =I'I;/; ( a + b). By the last f o r m u l a 12), we have thus found: x --0 ~ (a + b) m w h i c h is, curiously e n o u g h , symmetric with respect to a a n d b. If we assume that a -- 1' and t h e n take a for b, t h e n we have f o u n d , in particular, I I ( u + 7i,y a) = 0 ~ ( a + u

1').

[However, if we were to assume b =0', e x c h a n g i n g u for ti, we would get the result of p r o b l e m 8 again, a l t h o u g h in the s o m e w h a t d i f f e r e n t f o r m of 0j-

(a+ 0').] As corollary of this result, we now also know I I { u + a(~i~ b)} =a{Oj- ( b + 1')}, u

which can be i n f e r r e d in a similar way as above. With exactly the same m e t h o d , we can also solve an e x p a n s i o n of the previous problem. P r o b l e m 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 + u

u

fi'i)}

=YI~(b" i + i;c+ 1 "ai) =II;{b" i + i ; ( c + a)} = b,y (a + c) by 14), 4), a n d 14). In particular, we have now f o u n d 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 m e a n s of a I~ over u, so we can now also r e p r e s e n t a0,- (b + c)malt h o u g h its f o r m is u n s y m m e t r i c a l m b y a H over u. P r o b l e m 11.* We are l o o k i n g for x -- H{u + mi" b}. u

We will not s u c c e e d in solving this p r o b l e m in the same way as the two previous ones because we c a n n o t r e p r e s e n t the relative p r o d u c t of the s e c o n d t e r m of the g e n e r a l factor as H; but only as Ei, a c c o r d i n g to 14)" however, a I] c a n n o t be t r a n s f e r r e d from b e h i n d a II to the f r o n t of it. In the case of b =0', we can refer to 18). T h e r e f o r e , we must, in factmsolve only the simplest case:

n(u+ a . 0 ' ) : n , l : + n ( , , + a.;)l-n;(~+h =n;~=0 u

u

a c c o r d i n g to 7) a n d p r o b l e m 8 - - w h i c h we have already u n d e r s t o o d on page * Because of u = 1u" 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 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 m i n d 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, aft'bO' + a" 1 "bl',

u+ aft'b=u+

Page 510

suppressing 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 right-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 l a c e m e n t of the parentheses which are different from those in the corollary to p r o b l e m 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 t h e o r e m 1, but, because of au =6- u, we must have y =6= x. T h e same lower limit can also be d e t e r m i n e d 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 propositional s c h e m e o) on page 41. Because we now have Iluhk=0 , we get, as we will show in the next problem, II(uh, + fth,)= l't,, SO follows E, ah, b,,l'~,= (a" l'b)hk=C--Xhk. [It is conceivable to set an u p p e r limit for the lower limit for x. T h e reasoning goes thus: aft;b=6--a;b" ft;b, therefore x=6--a;b" 1 ; b l ' = a ; b ; l ' b , and the proposition 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 a r e 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 i c u l t o n e ! Its s o l u t i o n will o n l y b e p o s s i b l e f o r c e r t a i n s p e c i a l cases, s u c h as I I { u + (7i0~0) ;b} =0, tt

II{u+ u

(720~ a ) ; l ' b } =00~ (a0' + bl'),

f o r w h i c h we g e t t h e r e s u l t easily f r o m w h a t we h a v e 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 h e g e n e r a l f a c t o r splits i n t o (7i0~0) 9 1 ;b

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

353

o r (~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 t h e g e n e r a l f a c t o r itself a n d its II split up. This last r e s u l t is o b t a i n e d by m u l t i p l y i n g t h e c o m b i n a t i o n 0 0 ~ (a + 1') with 1 ;bl', w h i c h 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 a m o n g the u (that is, a m o n g 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 t h e g e n e r a l case we can (again) find two limits b e t w e e n w h i c h t h e u n k n o w n ( b u t fully d e t e r m i n e d ) relative x m u s t be. To d e t e r m i n e t h e s e limits b e f o r e h a n d is w o r t h t h e t r o u b l e for two r e a s o n s . First, t h e y give u s m a s in t h e p r e v i o u s special c a s e s m a v a l u a b l e way o f c h e c k i n g t h e e x a c t value o f x w h i c h we shall find later with c o m p l e t e l y n e w m e t h o d s . Also, we can t h e r e b y d i s c o v e r 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 + (~ij-a);i" i;b}, a n d since, by t h e p r o p o s i t i o n a l s c h e m e 0), p a g e 41, we have EII :(= l i E , we m u s t g e t u

E ~ I I { u + (Tiz~ a ) ; i } I I ( u + u

u

i;b) = ~ i ;

(~izj 1') 9i;b=F,~i;(gtztl')b:~--x

- - - 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, w h i c h gives us t h e lower limit. In o r d e r to find t h e 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]. B u t 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 t h e 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 w o u l d , o n t h e c o n t r a r y , a m o u n t to d i s p l a c i n g t h e m a n d w o u l d 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, w h i c h = II{a ; b}. T h u s , It

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 b e c a u s e o f i ; b = i . 1 ; b a n d 1 ; i 1 ' - i ; first t e r m can be c h a n g e d into i" 1" b a n d we obtain: x ~: II~i; (1' + a;b)

,,,

1'-i,

the

9 (1 ; b + fl~i;a;b)

={0ff(l'+a;b)}(1;b+0j-a;b) =0ea;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 s e c o n d o n e , can be o m i t t e d b e c a u s e 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 s e c o n d t e r m r e m a i n s as the upper limit that we s o u g h t a n d is c o n f i r m e d by the whole expression, which m u s t be

1;(dj-l')b~x~{O~.(a;b+l')}.

1;b=0j-(a;b+l;b.

1').

In the last e x p r e s s i o n for x [Translator's note: x = II [u + {IIi(~; i + i; a)} ;b]], we c a n n o t pull the /I~ o u t o f the braces by an e q u i v a l e n t t r a n s f o r m a t i o n , and, by the s a m e token, we c a n n o t p u s h the E; in f r o n t Page 512 o f the II, in the first e x p r e s s i o n for x [Translator's note: x = HE~{u + (~ 0~ a) ; i" i; b}], as was asserted above, in o r d e r to find the lower limit. This c o u l d only s u c c e e d with a c o u r a g e o u s p r o c e d u r e : T h e m e t h o d would be to o p e r a t e with infinite (or u n l i m i t e d ) multiple p r o d u c t s H; even with o n e whose H-sign c o u l d possibly f o r m a continu u m (in case we would write it d o w n in detail); for e x a m p l e , 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 p r o d u c t s a n d sums we may also w i t h o u t hesitation transfer a n d apply the i n f e r e n c e rules which are g u a r a n t e e d by the p r o p o sitional s c h e m e based o n the dictum de omni. This is p r o b a b l y the first time in m a t h e m a t i c s that this has b e e n d o n e . I will t h e r e f o r e g u i d e the s t u d e n t heuristically a l o n g the p a t h o n which the m e t h o d first o c c u r r e d to me. I first tried to e x t e n d the p a r t i c u l a r case y o f o u r p r o b l e m - - w h e r e the solution was f o u n d - - b y trying to o b t a i n (as a s u b p r o b l e m ) : "

u

u

"

u

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 + u

tim( Uhm "~ a,,,,) +

1-I,,(~h,, + a,q)}

= II,,,,{ami + a,q + II (Uhk + ftnm + ~h,,)}" u

And

now

II (Uh, + Uhm + Uh,,) = l'mk + l',,k, u

FROM PEIRCE TO SKOLEM

355

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, a n d 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, =

w h e r e b y we have f o u n d ;

As we now f o u n d a solution to P r o b l e m 12 easily in the case w h e n b = i is an e l e m e n t , as well as w h e n b = i + j r e p r e s e n t s a system of two e l e m e n t s , we c a n n o t foresee why it s h o u l d n o t also work in the case w h e r e b = b ; 1 is a system a n d t h e r e f o r e a sum of any n u m b e r of e l e m e n t s that could possibly as points c o n t i n u o u s l y fill a line. We i m m e d i a t e l y observe that the investigation only has to be g e n e r a l i z e quantitatively, and, i n d e e d , we will find Page 513

H { u + (ftj-a) ;b;1} = l ;f~; (Sj- l'), u

H{u + ((tj-a) ;1} = l ; (Sj- l'). u

L o o k i n g back at Zhk, we observe that o u r i n f e r e n c e s would n o t have b e e n possible if w e - - w h i c h s e e m e d feasible at f i r s t - - h a d used the s a m e letter m for the i n d e x n of the last H as for the previous H. If, however, all terms are p r o d u c t s II in a sum with m u t u a l l y indep e n d e n t indices, n a m e d i n d e p e n d e n t l y , t h e n 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 r e p r e s e n t e d symbolically by g. We now i n t e n d to d e d u c e the f o r m u l a on the left, o n p a g e 512 (bott o m ) (of which the o n e on the right is m e r e l y 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 r o u n d s to formalize o u r p r o c e d u r e . For that p u r p o s e it is c o n v e n i e n t to use for the system b;1 = b the f o r m u l a o b t a i n e d f r o m w 27 as b = E~b~i or, shorter, ~ii; w h e r e we only have to k e e p in m i n d that the sum over i, the ~ which does n o t have the i n d e x a d d e d as d e p e n d e n t suffix, but w h e r e it a p p e a r s (ad hoc) written u n d e r n e a t h , that this s u m does n o t have the full, b u t a s o m e h o w given (limited or u n l i m i t e d ) e x t e n s i o n f r o m the d o m a i n of discourse 11 of the e l e m e n t s . For s = I I { u + (~2o~a);E;}, t h e r e f o r e Shk=II{u + E;(~20~ a);i}hk u

=HlUhk + E,({tJ- a)h,} = IIluhk + Z,flm(5h,, + am,)}. u

u

W i t h o u t a new idea, we c a n n o t c o n t i n u e b e c a u s e we c a n n o t b r i n g the II m b e f o r e the ~i a n d t h e r e b y b e f o r e the II. T h e idea (already a l l u d e d to) which will get us f u r t h e r is the o n e which~we now generally f o r m u l a t e

35 6

Page 5 1 4

SCHRC)DER'S

LECTURE

XI

in the m a i n text a n d c o n f r o n 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 latter very m u c h . If we have a E i of a lI m of a g e n e r a l t e r m f(i,m), a n d we wish for s o m e r e a s o n to push the E b e h i n d the II in an equivalent transformation, this is n o t i m m e d i a t e l y possible. Because of ~II :(= II~, we could only do so by drawing weakened conc l u s i o n s - - i f we want to be satisfied with such a p r o c e d u r e . Otherwise, n o t h i n g h i n d e r s us f r o m r e n a m i n g the i n d e x of the IIm in all the o t h e r t e r m s 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 c h a n g e s in "parallel" with i. It a p p e a r s suggestive to take for ~ the letter i itself as suffix for m. D i s r e g a r d i n g the fact that m~ already has a fixed m e a n i n g as relative coefficient of the e l e m e n t m in w 27, it still would n o t 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 c h o o s e for L a symbol which contains the n a m e /-----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 r o n t of o u r Z. We can now justify the i m p o r t a n t f o r m u l a

{ ~,Ilmf(i,m ) = E,IIm,f(i,m,) = II, (IIm,)]],f(i,m,), II,~.f(i,m) = II,~m,f(i,m,) = l-I, (~ml)II,f(i,m,),

39)

by which we have a t t a i n e d o u r goal of having p u s h e d all II in front of

the ~. To facilitate the printing, we have set the indices i a n d m as if they, as e l e m e n t s , would have full e x t e n s i o n over 11. I a m sure this is permissible. But it is not at all necessary for the p r o o f of o u r s c h e m e . T h e indices i a n d m may have any given e x t e n s i o n s in l ~ - - p r e s u p p o s i n g of course that the e x t e n s i o n of m is i n d e p e n d e n t of i, the s a m e for e a c h i a n d is also t r a n s f e r r e d to each m,, that is, associated to e a c h of these (a limitation of which even the last p a r t of each of the d o u b l e s c h e m a t a is i n d e p e n d e n t ) . In o t h e r words, we may also write Ei for Zi or 1-Im for I-I m .

O u r s c h e m a would r e m a i n in force even if the two indices, or o n e of t h e m , were not e l e m e n t letters but would have their e x t e n s i o n in 12 as a u or v. But we do n o t want to discuss this possibility here. T h e last part of o u r s c h e m e n e e d s a f u r t h e r e x p l a n a t i o n , but has to be understood first. If L (in parallel with i) would have to run t h r o u g h 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 r o n t of the last ~ in 39) by writing it in the o r d i n a r y way, explicitly--by n o t m e n t i o n i n g the g e n e r a l t e r m or factor, in the f o r m of

FROM

PEIRCE

TO SKOLEM

357

II,(II,,,,) =IIm II,, IIm.~'''

or

Hmlm2m3.." =

Hiltm

t

a n d d e f i n e it as p r o d u c t symbol for a (possibly u n l i m i t e d ) "multiple p r o d u c t . " And t h e n II,(E,,,) = EmtEm2E,,~...

or

E,,~m2m3... = En, m,

would be n o t h i n g but the s u m m a t i o n symbol to indicate a " multiple s u m . " Since the latter c o r r e s p o n d s , dually, to the m u l t i p l e p r o d u c t , we can Page 515 see that in the new symbol (which is shown to be i n d i s p e n s a b l e for abbreviation h e r e ) the II, may not be transcribed, dually, to E,, but will r e m a i n as II, in the dual c o u n t e r p a r t to the s c h e m e . It r e m a i n s o p e n w h e t h e r the t h e o r y will ever m a k e use of symbols such as E,(IIm, ), E,(Em,), by which only some, any, but at least o n e of the II or E over m, could be set. O f the given r e p r e s e n t a t i o n s or m e t h o d s of expressions, the ones on the right are less good, even misleading, for the r e a s o n that the composite suffix m 1m.2m3"" of a II or E s h o u l d not be a real p r o d u c t ( n e i t h e r an identity n o r a relative o n e ) , but stands conventionally for the "series" m l , m 2 , m 3, "" (cf. p. 24). It is true that o u r II, does not p o i n t to a real p r o d u c t either, but to a succession of signs (of the type i n d i c a t e d after it in p a r e n t h e s e s ) which may also b e c o m e a c o n t i n u u m . If the L in parallel with i has to run t h r o u g h a c o n t i n u u m of values, such as all the points of a line, we can no l o n g e r write the m e a n i n g of II,(IIm,) explicitly. A r i t h m e t i c allows us, however, to n a m e t h e m all a n d differently by assigning each of those points to a real n u m b e r f r o m an interval. For e x a m p l e , 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 p o i n t L. Thus, we can only say with respect to the e x p l a n a t i o n o f o u r symbols: we have to a s s u m e f o r each p o i n t L of the line a II m t 9 T h e o r d e r in which such II over d i f f e r e n t indices are taken (if we want to a s s u m e t h e m in a definite s e q u e n c e , which is n o t always necessary) is of no c o n s e q u e n c e , as is well known. B e c a u s e ~ a c c o r d i n g to the broadly used dictum de omni: what applies at e a c h m for e a c h n necessarily has to apply at each n for each m, etc. To prove o u r s c h e m e 39) on the left-hand side, we want to think, for didactic reasons, of a discrete series of i a n d L, w h e r e we c h o o s e the for the index m of IIm r e f e r r i n g to the values A, n a m e s m l, m2, m3, B, C, ..., or also il, i2, i3 . . . . . of i. T h e n the left side of o u r s c h e m e is . . .

L = I l m l f ( i l , m x ) + Ilmzf(iz,m,2) + I I , , 3 f ( i ~ , m 3 ) +

-

rlm,n rlm,

...

{f(i~,m,) + f ( i z , m 2 ) + f ( i ~ , m 3 ) + ...} = R.

For its proof, we only have to say that the sum of the t e r m s of L,

358

SCHRODER'S LECTURE XI

which, for l-I, p r e c e d e or s u c c e e d a definite m• do n o t c o n t a i n this i n d e x m• and, as a c o n s t a n t with respect to it, can be d e s i g n a t e d as a Page 516 o r b, ad hoc. T h e n it is only necessary for the t r a n s f o r m a t i o n o f L to R for e a c h o f the d i f f e r e n t i a t e d (that is, differently n a m e d ) rn to use the scheme" a + IImf(m) + b = rl m{a + f ( m ) + b}, w h e r e the II m can be e x t e n d e d over the p r e c e d i n g o r the s u c c e e d i n g c o n s t a n t a d d e n d . This very scheme---cf. 26), p a g e 1 0 0 ) - - c o u l d easily be p r o v e d by the initial p r o p o s i t i o n a l s c h e m e 3,), p a g e 40. In a similar way, we would have for the s c h e m e 39), the r i g h t - h a n d side" L = r,m,f(i,,ml)E,,.~f(iz,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 n o t j u s t be justified a n d stated for an arbitrary discrete value series of i a n d L (for which we illustrated it above, so to s p e a k ) - - f o r e x a m p l e , by p r o o f by i n d u c t i o n , b u t we w a n t to use it for all i, L a c c o r d i n g to the i. If the i, L would have to r u n t h r o u g h a c o n t i n u u m o f values, we can a s s u m e for each i x, X of their value, as we have p r o v e d f r o m / 3 ) , p a g e 37, that a c c o r d i n g to 18), p a g e 98, each t e r m o f a "E" can also be r e p r e s e n t e d as a real term o f a (binary) "sum" (in the n a r r o w e s t sense), the o t h e r t e r m o f which, d e s i g n a t e d as a ( i n d e p e n d e n t o f the m app e a r i n g in this term) m u s t to be subject to the s c h e m e Ilmf(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 o u r s c h e m e would be false a n d illusionary if we would replace the i n d e x n a m e m, by m i or by 4)(i). Because, in its last part, F,if(i,m ) would a p p e a r as the g e n e r a l factor o f II, a n d this w o u l d have to have a value totally i n d e p e n d e n t of i since the letter i only f u n c t i o n s as p l a c e h o l d e r for the values which are assigned to it f r o m the e x t e n s i o n o f i. T h e r e f o r e , we can also n o l o n g e r use m i in its e v a l u a t e d expression. (In analogy to a definite integral which is indep e n d e n t o f its i n t e g r a t i o n variables!) T h e r e f o r e , we can totally o m i t the o p e r a t o r Ili(IIm;) p r e c e d i n g the term, a c c o r d i n g to the law of tautology H a = a, a n d o u r s c h e m e would t h e n be very m u c h simplified. T h a t such simplifications are generally n o t admissible c o u l d be illustrated with examples. T h e application o f o u r s c h e m a to o u r p r o b l e m is now: Page 517

Shk = I'Iluhk + II,(IIm,)~-,i(Uhm ' + am, i)} = nn,(nm,)lu,,, + r,(a,,~, + am, i) } = II,(IIm,) H (Uhk + ~,~2,,~, + Z;,am,,) tt

=

am,, + n

+

FROM PEIRCE TO SKOLEM

359

Again, we can easily see that we have to have 40)

IIu (Uhk + I]iuhm,) = I]il'kmt

which, in case even only o n e of the m, is equal to k, m a k e s sense in the f o r m of the e q u a t i o n 1 = 1; in the case, however, that all m, of the sum over i are u n e q u a l to k, to be r e c o g n i z e d in the f o r m of the e q u a t i o n 0 = 0, t h e n a m o n g the admissible values of u there will be o n e such for which uh, = 0, as well as i (and the L c h a n g i n g in parallel with it) a n d 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,, w h e r e we used o u r s c h e m e 39) again in reverse. T h a t is,

S,,k = E,{(I' ~ a) ; ilk,, = 1(1' ~ a) ;F,,iik, , = 1(1' J- a) ;bi,h = 1/~; ( ~

1')1,,,,

w h e r e b y it is f o u n d consistent 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 g e n e r a l p r o b l e m , r a t h e r than the special p r o b l e m 12, which we c h a r a c t e r i z e d in p r o b l e m 8. Before we start, we have to say that by specializing the result of the g e n e r a l investigation, the solution to p r o b l e m 12 can be easily f o u n d as

x=l:(d0

~l')b.

T h a t is, the lower limit for x f o u n d on page 511 r e p r e s e n t s in this case the exact value of this u n k n o w n . T h e r e f o r e , the c h e c k given by the previous d e t e r m i n a t i o n of this limit c o r r e s p o n d s to o u r result. This result also gives us an answer for the p a r t i c u l a r cases of the p r o b l e m settled previously. Thus, it is i m m e d i a t e for a - 0. If, on the o t h e r h a n d , we have bl' for b, we have to u n d e r s t a n d 1 ; (~-

Page 518

l ' ) b l ' = 0 0~ (a0' + bl').

To that end, we can (and I anticipate a little), a c c o r d i n g to the already stated propositions 47), 46), separate the left-hand side into 1 ; ( ~ 0 ~ 1 ' ) 1 ' - 1 ; (1'0~ a ) l ' = 0 cr (a + 1') a n d 1 ; b l ' - 0 el- (b + 0'), t h e n the product of these two expressions 0 el- (a + l')(b + 0') can be e x p r e s s e d as the r i g h t - h a n d side, q.e.d. If, finally, b ; 1 stands for b, we i m m e d i a t e l y have 1 ; (~0~ l')(b ; 1) 1 ;b; ( ~ 1), q.e.d. At times, o n e obtains interesting propositions even if o n e m a k e s a mistake! I have b e e n led to wrongly c h a n g e (i0~ a) ;b into i'cl- a ; b in the d e d u c t i o n of x by inaccurately r e m e m b e r i n g p r o p o s i t i o n s 27) of w 29,

SCHRODER'S LECTURE XI

36o

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 incorrect result lies correctly between the previous determined limits, and by checking it, one obtains remarkable propositions! Indeed, we must have

1;(dd-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" ( 0 c t l ; b ) = x " 1;b, we only have to represent x=(= x. 1;bofx~=l;bby0~(a;b+l'):(=0j-(1;b+l')=0j-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 proposition of the following pair: 0' 9 1 ; (g~t l ' ) b ~ : a ; b ,

0'

a(l'j-/~)" 1 :,~--a;b,

aj-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'(dj-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 proposition 41) from the coefficient evidence, we have to show, bringing the right side to 0, that 0 ' ' 1 ; ( a c 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 + 10)(d~h + bhj)bhi equals 0, and therefore each term of I; h vanishes, q.e.d. Afortiori, we also have I

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), page 508. It comprises the previous Problems 8 to 12 as special cases--but Problems 9 and 10 not fully, that is, only the m i n o r cases. We have--because of 39) u

FROM

PEIRCE

TO

36x

SKOLEM

x,, k = II {Uhk + I]~II,,ah, (~ih. , + b,,m + C,,~)d~k] tt

= I],(I'Im,)[r, iahi(bhm ' + Cm,i)dik -k- ~ (Uhk -t- F.,iahidik~thm,)] a n d w o n d e r w h i c h v a l u e t h e last II has. At this p o i n t we h a v e to o b s e r v e t h a t m, is n o t c o n s t a n t with r e s p e c t to i, b u t c h a n g e s in p a r a l l e l with i in E;, t e r m by t e r m . If o v e r 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 o v e r L, a n d o u r II vanishes. If, h o w e v e r , o v e r L s o m e m, is e q u a l to k, t h e last I2~ o f t h e f a c t o r 7ibm' (= ~ihk) in all a c c o m p a n y i n g t e r m s will be s u p p r e s s e d in t h e p r e s e n c e o f t h e s u m m a n d Uhk, a n d r, iahidi, l',m ' o c c u r s as a n o n e x p r e s s i v e c o m p o n e n t o f t h e g e n e r a l f a c t o r in o u r lI, to w h i c h t h e 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 t h e 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 possible values f r o m 12 h a v e to be w r i t t e n for u. We m u s t t h e r e f o r e have

II (Uhk + )2iahidik~ih.,, ) u

=

' }] iahidik 1kin,,

43)

w h i c h also gives t h e c o r r e c t value, 0, for t h e p r e v i o u s case. T h e s u m o n t h e r i g h t - h a n d side, o f c o u r s e , c a n n o t be r e d u c e d to a single t e r m acc o r d i n g to s c h e m e 12), p a g e 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 s p e c t to i b u t its m e a n i n g c h a n g e s in p a r a l l e l with i; this s u m c a n h a v e a n y n u m b e r o f effective terms. We t h u s 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 r e v e r s e ( a b o v e it was u s e d in t h e f o r w a r d direction). N o w we c a n write c,,,~ = i~,,, = (i; c)hm = (c;i)mk in a n y way we w a n t a n d also a s s u m e t h e t e r m as t a u t o l o g i c a l l y d o u b l e d , a n d c h o o s e for t h e o n e t h e f o r m e r , for t h e o t h e r t h e l a t t e r f o r m . A f t e r 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 therefore v

..,

x = F,~a;i" {(b + i; c") j- ( c ; i + 1')} 9 i ; d , w h e r e o f t h e two t e r m s i; ~ a n d c;i, t h e o n e o r t h e o t h e r ( b u t n o t b o t h ) c a n be s u p p r e s s e d . We m a y write o u r r e s u l t m o r e s i m p l y as

SCHRI3DER'S LECTURE XI

362

x = ~ a ; i " {brt (c;i+ 1')} 9i;d.

44)

T h e x, however, is not completely r e p r e s e n t e d in closed form, but the II over u has been reduced from the second o r d e r to a I2 over i of Page 520 the first level or order. T h e latter can also be given in a simpler 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 s e p a r a t e d from c a n d a d d e d as i" to b as a s u m m a n d . This can be easily verified by establishing the g e n e r a l coefficient Xhk for the last ~, as a result, we have in fact i m m e d i a t e l y the f o r m e r expression of Xhk----only l is taken for k. O n the o t h e r hand, we can also derive systematically the last expression of x, 45), from the previous, 4 4 ) r u b y passing t h r o u g h a d o u b l e sum. To that end, we write the m i d d l e factor in 44) as (b + i;c) ~t 1' in the f o r m of e~t 1' and choose f r o m the four r e p r e s e n t a t i o n s for e~ 1' which we have a c c o r d i n g to 14) or 17), 16) or 22), a n d 18): e0~ 1'= IIj(e ;j + ]) = II~(ectj+ ]) = IIj(ertj+]) = ~j(e,j)], the last because the two s u m m a t i o n signs can be e x c h a n g e d . 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., a n d 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 r e p r e s e n t a t i o n 45) of x, except for the d e s i g n a t i o n of the s u m m a t i o n variable. [We had to consider the propositions (because i = i ; 1 ) 10) of w 27, t h e n 27) and 26) of w 25, and, finally 14).] T h e r e are m a n y ways to check o u r result. In particular, we want to derive the solution of P r o b l e m 12 already checked. For that purpose, we let a = 1, b = 0 in 45), a n d t h e r e f o r e write a a n d 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). A c c o r d i n g to 26), x = 1 ;{(1'0~ a) ; b}l' = 1 ; 1'{/~; (gj- 1')}, Page 521

because 1~ = l'c'~ Now this r e m a r k a b l e proposition is valid:

FROM PEIRCE TO SKOLEM

363

l'(a; b) ; 1 = a/~; 1,

l'(a j- b) ; 1 = (a =/~) 0~ 0,

1 ; (a; b)l' = 1 ; ~b,

1 ; (aj- b ) l ' - 0 j- ( ~ / + b),

46)

of which the s e c o n d f o r m u l a on the left (necessary here) can be p r o v e d with the coefficient evidence by m e a n s of L o = l~t 1;t~ h au, bhjl ~ = E h ajhbhj = I~h 1 ~h(~/b)h~ = R o 9

This p r o p o s i t i o n belongs to a g r o u p of propositions which refer to relatives of the f o r m of l~z; 1, etc., some of which we have studied u n d e r 24), 25) of w 22 (p. 335), a special case in the f o r m of 30). T h a t requires a l s o - - a s obvious from aiibii = (ab)ii, lh;1

9l'b;1 = lhb; 1,

1;al''

1;bl'=l;abl',

l h ; 1 + l'b;1 = l'(a + b) ; 1, etc.

47)

which can be i m m e d i a t e l y e x t e n d e d to m o r e than two terms. By 46), we have f o u n d the solution of P r o b l e m 12: x = 1 ; (~/~t l')b, as given on page 517. A n d t h e r e f o r e we have also c h e c k e d o u r investigation's m a i n result r I - . . = E~... consisting of setting the values of x f r o m 38), 44), a n d 45) equal to each other: u

,.,

II [u + a{(b + ~i) ;t c} ; d] u

=E~a;i.{bj-(c;i+

1')}" i; d

,.,

= E~i" a { b e (c + i)} ; d.

48)

As a f u r t h e r check, it is left to the s t u d e n 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 s c h e m e of o u r P r o b l e m 13, and from this s c h e m e the results that solve the p r o b l e m . . . . ~ P r o b l e m 19. We may now also decide the question which a p p e a r e d on page 268, namely w h e t h e r the general solution of e q u a t i o n x ; 0 ' = a ; 0 ' given u n d e r 22) c o r r e s p o n d s in essence with that in the first line of 26). T h e answer to this question is positive. This also brings us to propositions which are of s o m e interest. First, we notice that the solution 25), page 269, to x ; 0 ' = a is by using a :t 0 = (a 0~ l ' ) a a n d is f u r t h e r simplified to x = (a# l'){d + u + (z/# 1') ;1},

94)

w h e r e it now consists of seven instead of nine terms w i t h o u t having lost a n y t h i n g in clarity. Instead of the (last) relative factor 1, we could also write 0'--cf. 15), page 229. We can also bring the expression of x in 26), page 269, closer to the o n e in 22) by writing "~Problems 14-18 (pp. 521-538) have been omitted here.

364

SCHR()DER'S

a ; 0 ' o ` 0 = (a;0'o` 1') 9a ; 0 '

LECTURE

XI

(do` l')a = (a'0'o` l')(do` 1')

T h e r e f o r e 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 simpler expression for the general root of e q u a t i o n x" 0 ' = a ; 0 ' , which now consists of nine instead of 10 terms. Page 539 It has the advantage that only a appears in the c o m b i n a t i o n 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 i n d e e d 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 s u b s u m p t i o n s of this e q u a t i o n L = R divide because their predicate is a product, and they are therefore obviously valid as partial conditions 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. T h e latter, b r o u g h t completely to 0, results in a/~(b;0'o` l')(ao ~ 1') 9a{(/~+ d ; 0 ' ) 0~ 1'} ; 0 ' = 0, a n d the factor/~ proves to be irrelevant. If we replace (ao` l ' ) a by a o` 0 we can use the proposition (ao`O) 9ab; c = (ao`O) 9b; c,

97)

which is easily proved based on 24), page 255, a n d (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 without the above proposition, the p r o o f would follow a fortiori from the latter. F u r t h e r m o r e , we can suppress the term d ; 0 ' , and the last factor now appears as the negation of the second but last and makes the p r o o f clear a n d understandable. This is because we must categorically have (ao`0) 9{(d" b + c) o` d } ; e - (ao`0) 9 (co" d) ;e. If we multiply with (a o` 0) o, namely Hha~h in F,klIt (F,,,,di,,bml + cu + dlk)ekj,

98)

365

FROM PEIRCE TO SKOLEM

we eliminate the whole Em because each dim coincides with a factor aim of II h. T h e s u b s u m p t i o n for a/~L is t h e r e f o r e done, a n d 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 a d d e d the subjects. This is clear even without factor/~--after we transcribe the last two terms a n d multiply 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 a p p e a r s as =~= to b ; 0 '

Page 540 on the right. If we suppress identical factors, we m u s t necessarily get c o n t a i n m e n t [ Obergeordnete]. T h e o t h e r part, b r o u g h t to 0 on the right, likewise requires dbla" b(a~ 1 ' ) ; 0 ' # 1'} = 0, w h e r e the factor a can also be omitted, as we can see, a n d this brings b, a for d, /~ to a m o r e accessible proposition: ab :~-{(a + b ; 0 ' ) j- 1'}0',

a(b j- 1') ; 0 ' 3 1' :~: a + b,

99)

etc. In o r d e r to prove it, we have to use the coefficient evidence since n e i t h e r a~--(aj- 1') ;0' n o r 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 s u b s u m p t i o n , we have to show S0 -aob!i:~--PO, where ' P0 = ~J, 0hjIIk (aik + F'lbilOtk' + 1 '

kh) .

To p r o c e e d in a completely analytical way, we adjoin to the universal factor o f l I k the t e r m 0'k~l'kjwhich is = 0 , and we analyze f r o m the s c h e m e a + bc = (a + b)(a + c), evaluating also IIk of the first factor a c c o r d i n g 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 f u r t h e r multiply the general term of the last I~t with 1'0 + 0~i, which is =1, a n d we evaluate the I] t which comes from the first term, a c c o r d i n g to 12x), page 121, we get the s u m m a n d boO~k which can be simplified to b o because of the following l~)k, a n d we can write this term, i n d e p e n d e n t of k, in front of II k. We now have Po = F"J,Ojo(ao'

+ F'tb,Oo){bo'

+ IIk(aik + Et bitO'tk...O~3'+ lkht + lkj) }, .

' = ( a b ) it 9 1 = S O, a n d eveBy multiplying we easily get the terms aobuF, hO~o rything is thus proved. In 99) we could, of course, also add b' 0' left of the subject.

366

S C H R O D E R ' S LECTURE XI

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 m o d u l e 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 o r d e r 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 underlined term must vanish, according to the formula d + (d0~/~) ;/~=d m e n t i o n e d 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, and we only have to show that Page 541

(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, troublesome for our discipline, that the t h e o r e m 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 c a n n o t vouch for its i n d e p e n d e n 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 i n d e p e n d e n t solution, so that in addition to the formula already m e n t i o n e d , we also have the pair of formulas,

FROM

PEIRCE

TO

367

SKOLEM

a(a;b;1);b=a;b, a;(1;a;b)b=a;b,

(a+ a~b~O) j-b=aj-b, aj-(Oj-aztb+ b) =a~b,

100)

b e c a u s e we have i m m e d i a t e l y

a(a;b;1) ;b=a;b;1 9a;b=a;b,

q.e.d.

B o t h g r o u p s o f f o r m u l a s can be c o l l e c t e d in t h e g e n e r a l p r o p o s i t i o n t h a t x = a 9a ; b ; (/~ + w) r e p r e s e n t s a class o f p a r t i c u l a r s o l u t i o n s o f t h e p r o b l e m x; b = a ; b for any a r b i t r a r y w, etc., a class o f c o n s i d e r a b l e generality, yet s i m p l e in its e x p r e s s i o n o f t h e root. P r o b l e m 21. Finally, we also w o u l d like to m a k e a c o n t r i b u t i o n to t h e o f 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 q u e s t i o n s are: W h e n ( t h a t is, u n d e r w h i c h cond i t i o n s for a a n d b) d o e s t h e x, d e t e r m i n e d by t h e r e q u i r e m e n t x; b = a;b, r e m a i n C o m p l e t e l y arbitrary? A n d w h e n is x p e r f e c t l y 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 e q u a 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 . B e c a u s e if we h a v e to have u ; b -- a; b for e a c h u, we m u s t have a; b = 0 - - a s t h e 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 e a c h u, t h u s 1 ; b = 0 o r b = 0, q.e.d. To affirm the s e c o n d q u e s t i o n , 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 sufficient c o n d i t i o n , b u t it is n o t at all necessary. To find t h e n e c e s s a r y a n d sufficient c o n d i t i o n for t h e fact t h a t t h e r e is o n l y o n e r o o t o f t h e e q u a t i o n x; b = a;b, o r t h a t t h e s o l u t i o n 64) is c o n s t a n t in r e l a t i o n to u, we have to go d e e p e r . T h e sufficient o r a d e q u a t e a n d n e c e s s a r y o r r e q u i r e d c o n d i t i o n for t h e fact t h a t a f u n c t i o n f(u) o f a ( r e s t r i c t e d o r u n r e s t r i c t e d ) variable relative u is c o n s t a n t with r e s p e c t to it is

"determination"

Page 5 4 2

~f(u)

= Hf(u)

101)

( w h e r e in t h e first case in p a r e n t h e s e s , t h e E a n d II a r e e x t e n d e d o n l y o v e r t h e d o m a i n o f variability o f u, in t h e s e c o n d - - p r e s e n t - - - c a s e , they have a b s o l u t e e x t e n s i o n ) . If we have f(u) - e for all u o f t h e s a m e value, t h e a b o v e e q u a t i o n is c e r t a i n l y valid b e c a u s e o f ~ e a n d IIe. By t h e s a m e t o k e n , if this is valid, tt u we c a n call t h e c o r r e s p o n d i n g value o f t h e i r two-sided e x p r e s s i o n e ( a n d it b e c o m e s i n d e p e n d e n t o f u b e c a u s e u only a p p e a r s as i n d e x in t h e m ) . B e c a u s e o f f ( u ) ~ E f ( u ) a n d 1-If(u)::(c-f(u) we also g e t f o r e a c h u, f(u) ::~--ea n d e::~--f(u), a n d thus f(u) = e, as we have s h o w n . By a p p l y i n g this s c h e m e ' s 101), we n o w find by c o l l a t i n g 65) with 66): c = cE~..., o r c :~: E~ .... t h a t is, c :~- Z , i . (c; b)lizt (/~ + i)}; 1

102)

368

SCHRI~DER'S

LECTURE

XI

as the necessary a n d sufficient c o n d i t i o n for the fact that r o o t x o f the e q u a t i o n x; b = a ; b is c o m p l e t e l y 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 d e f i n e d as a;b:t/~ has to be e q u a l to a, which results--infinitely m o r e easily than f r o m i t s e l f - - f r o m the observation that also these have to c o i n c i d e b e c a u s e c a n d a are always roots. In any case, the general, k n o w n particular solutions o f the e q u a t i o n have to c o r r e s p o n d , which gives us--cf, p a g e 2 6 0 - - c = a = a" a ; b ; b = c" a ; b ; b a n d leads to the d o u b l e s u b s u m p t i o n :

a; b j- ~:/~--a:~--a; b; b,

103)

o f which the first part has the s t r e n g t h of an e q u a t i o n . If we i g n o r e the m i d d l e term, we get

(a ; b j- ~) ( d j- [~j- ~) = (a ; b ) ( d j- [~) 0~ /~ = Oj-/~=(=O, which converts into /~ct 0 = 0 a n d thus supplies a result for b: b; 1 = 1.

104)

Page 543 It also tells us that b may n o t have an e m p t y row. O b t a i n i n g this result

directly with the e l i m i n a t i o n o f a f r o m 102) may have its difficulties. In 102), we can write a for c. If we do this, after c o n s i d e r i n g all four ways o f writing from 66) a n d r e d u c i n g the right to O, we get f o u r 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 t e r m can be e l i m i n a t e d - - a s e x p r e s s i o n o f the r e q u i r e d c o n d i t i o n . However, it can n o l o n g e 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 r o v e d c = a. T h e sufficient or c o m p l e t e c o n d i t i o n was still e x p r e s s e d in 105), if we r e i n t r o d u c e d c for a a n d t h e n wrote for c its value a; b0~/~). As coefficient, this last r e q u i r e m e n t 105) is m o s t simply written as

ahkIIm{( t~ ~" b)ltm de_ ~,anlOtk~lm}

=0.

105 ~

We can draw m a n y i n f e r e n c e s f r o m 105). T h e r e q u i r e m e n t has to exist a fortiori if we suppress any terms after the Hi or, possibly, a d d s o m e factors. We have, for example" aII~{(50~/~);i + i;/~} =0, i.e., a c c o r d i n g to 14) a(53-/~0~/~)=0

or

a:~--a;b'b

as a c o n f i r m a t i o n of the s e c o n d s u b s u m p t i o_ n 103). It also i m m e d i a t e l y follows o u t o f the last form o f 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.

FROM PEIRCE TO SKOLEM

369

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 f o 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 9a ; 0'(b o~0) - 0

106)

as a further, necessary condition. It is possible to satisfy the first subsumption 103) i n d e p e n d e n t l y 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 f o u n d in a more painful procedure, both requirements 103), involving also 104), can be satisfied in the most general way with i n d e p e n d e n t parameters c~,/3 with the following formula which is easily verifiable, a = o l ; ( / ~ j - O ) + c ~ ; / 3 j - O + (c~;f3j-/~) 91;/3,

b=/3o~0+/3,

107)

and it may be possible to continue to gradually satisfy further partial requirements or subrequirements of the p r o b l e m - - a s 1 0 6 ) - - b y determining the parameters. But as long as we c a n n o t 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 "determination problem" (for the third inversion problem) in this way. We therefore have to leave the problem here as it is. Page 544 To evaluate II~ hardly promises success as long as it did not succeed with m u c h simpler products, such as

x = II~a; ~b = Ilia;~b.

108)

C o n c e r n i n g this last p r o b l e m of the first o r d e r [erster Stufe], with which we return to the main theme of this section and w h i c h - - a l r e a d y for b = 0'---cannot yet be solved, m u c h could be said that is of interest. But we have to abstain and r e c o m m e n d it as a problem, next in difficulty a m o n g the still unsolved problems, for further investigation.

In view of the importance, the fundamental significance for the completion of elimination, for inference in general, which the p r o b l e m of sum and p r o d u c t on the second o r d e r [zweiter Stufe] gained at the end of w 28, we will devote some time to this problem. It will be shown by modifying (in a practically insignificant way) what was said on page 468 that our algebra has also methods for the solution of this p r o b l e m which

37 ~

SCHRODER'S LECTURE XI

are theoretically usable in a g e n e r a l way, but which s e e m 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 m a t h e m a t i c a l c o m p l i c a t i o n s w h e n we try to use t h e m practically. W h e n the e x t e n s i o n of H, E is absolute (over the e n t i r e universe o f discourse 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 e p l a c e a b l e by e a c h o f its relatives. If u can take on any value, t h e n also z~, fi, etc. If, f u r t h e r m o r e , the e x p r e s s i o n o f f ( u ) only a p p e a r s as u a n d fi, b u t n o t z~, u (or reverse), after its p r o p e r r e d u c t i o n by e x e c u t i n g all negations o n c o m p o u n d s u b e x p r e s s i o n s (next to any p a r a m e t e r relatives or constants relating to u), t h e n we k n o w

nf(u,(,)

=tip,o) = nf(~,u),

which we can write in a s i m p l e r way as Hf(u)

- f(O).

110)

T h e n we have f ( 0 ) in o u r rl as the m i n i m a l factor, c o n t a i n e d in all others, the p r o o f of which c o u l d 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 a n d its c o n j u g a t i o n s in c o m b i n a t i o n , or mainly a c c o r d i n g to t h e o r e m 1) of w 6 by c o n s i d e r i n g 0 :~:v. We thus have, for e x a m p l e , i m m e d i a t e l y rl [a{(u + b) j. c} ; d + e; (tf] = a(b j- c) ; d.

Page 545

We leave it to the r e a d e r to state the dually c o r r e s p o n d i n g p r o p o s i t i o n for I2. As a p r o b l e m , the discovery of only such H, E is o f i n t e r e s t in which in the g e n e r a l term f ( u ) n o t only u or ~ b u t also z~o r u a p p e a r essentially. We have solved p r o b l e m s o f this kind in 1) a n d 6). T h e s e can be m a d e into the m o r e g e n e r a l t h e o r e m by m e t h o d s which we will c o n s i d e r later:

u

111)

w h e r e the sums over K a n d X e x t e n d over any series or systems o f suffix values, such as 1,2, 3 . . . . .

371

FROM PEIRCE TO SKOLEM

B e f o r e discussing the m e t h o d s , we want to state s o m e c o n c r e t e exa m p l e s which are i n t e r e s t i n g for their derivations a n d results. P r o b l e m 22. W h e n we look at 5), we are inclined to i n q u i r e after the value o f the following p r o d u c t l-I, after which we i m m e d i a t e l y state

II(au;b+ u

c;gtd) = a d " 1%;1;b1',

II(a;ub+oi'd) u

=bc" 1 ~ ; 1 ; d 1 ' , 112)

which gives the a n s w e r m w h e r e the s e c o n d 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 o f 109). For proof, we call the x the first, s o u g h t for II a n d Uits g e n e r a l factor, a n d we have

x-nU.

.,;- n v,j.

u

U 0 = r,h a~hbhiu~h + Ek qkdkj~ikj" We now "develop" this e x p r e s s i o n for u 0 by m a k i n g the latter ( a n d its n e g a t i o n ) "prominent" or by "bringing it into evidence." F o r that p u r p o s e it is necessary a n d sufficient to let the terms w h e r e u o r ~i are i n d e x e d with this p a r t i c u l a r suffix ij to e m e r g e or to let c o m e o u t w h e r e v e r they can be f o u n d . This can be d o n e by p u r e calculation, in o u r case: by m u l t i p l y i n g the g e n e r a l e l e m e n t o f ~j, with (1=) (l'hj + 0,'0), the ~k with (1=) (l'h, + 0~,,), t h e n resolve t h e m , t h e n apply to the first terms the s c h e m e 12x) f r o m p a g e 121. Now we have i n d e e d :

U0 = aobiiuii + c,d~gt 0 + ~,,O,',~a~hbjou,h + ~kO~kc~kdk~(%, w h e r e we only have in the last two sums u-coefficients which are d i f f e r e n t f r o m e a c h o t h e r a n d f r o m u o. T h e s e may also a p p e a r in H with 0-values u~h = 0 (at h r j), ~ikj = 0 (at k r i), so that only the first two terms can c o n t r i b u t e s o m e t h i n g to the value o f x. T h e actually o c c u r r i n g m i n i m a l 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 linear function" Tt

~u + ~i--c~

Page 546

+ c~ju + o ~ i

m u s t be the p r o d u c t o f its coefficients, t h e r e f o r e be ol/3, since the last two terms vanish i n d e e d w h e n 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 c o n t a i n s at least the t e r m aobjjc~d 0 a n d will, for certain values o f u, no l o n g e r c o m p r i s e 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 a n d

x = ad" 1~; 1 ;bl',

q.e.d.

T h e p r o c e d u r e is certainly impeccable; however, the conditions for developing it so smoothly and simply are almost never as favorable. We can gain a d e e p e r insight into the p r o d u c t m e t h o d which promises general success, with the derivation (of the first) of the following results: P r o b l e m 23. We have to discover that

II {au ; b + c( ~t j- d) ; e} = (arid)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 + ~{a~(u+b)IlcJ-(d;fz+e)}=(a;d+ tt

c)~(d;b+

e),

113)

e),

- - w h e r e the ones standing below each o t h e r are immediately o m i t t e d by e x c h a n g i n g d with d in o n e proposition. Derivation. By formulating the first p r o b l e m again as x = IIU we get

x 0 = II U0 a n d

U0 = Eta,boU,l + ~kc, kek~IIt(~i, + dj, l).

T h e p r o b l e m 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 o n e definite h. We have shown previously how to use Et to that e n d (we multiply the general term by l'u, + O~h). In dual c o r r e s p o n d e n c e , 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), a n d take IIt individually of these two factors, where for the first factor the s c h e m e 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 better to leave in the n o n h o m o g e n o u s state. Prior to writing the specific values of or, 3, "t' which are here not visible, we want to i n t r o d u c e a s o m e w h a t m o r e accessible symbolism that we r e c o m m e n d for all similar problems. A sum of the form Et01h4)(/) represents n o t h i n g m o r e than the sum of all 4~(1) without do(h) and can also be expressed completely by I2~-h4~(l). By analogy, I2~-"-k4)(/) = I; t O~,,Oikch(l)l b e c o m e s the sum o v e r / o f all 4)(1) without 4)(h) and 4)(k), and so forth.

FROM PEIRCE TO SKOLEM

373

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'lh + l',k + r

= /it-n-*

r

'

a n d so forth, w h e r e the expressions o n the left r e p r e s e n t n o t h i n g m o r e t h a n the p r o d u c t o f all 4)(1) w i t h o u t ~(h) a n d 4)(k), respectively, etc. This small m o d i f i c a t i o n of the symbolism which is legitimate in o u r discipline has the a d v a n t a g e that the g e n e r a l t e r m o f E a n d o f 1-I always has the s a m e (a constantly simple) e x p r e s s i o n (whereas in the n o r m a l way of r e p r e s e n t a t i o n , it would grow a n d increase in bulk), as m o r e a n d m o r e terms are o m i t t e d f r o m the sum, factors o f lI; t h e r e f o r e , also the g e n e r a l term, which o f course r e m a i n s the old o n e , d o e s n o t n e e d to be m e n t i o n e d a n d r e p e a t e d . W h e n finally all terms o f Et, all factors o f / I t are e x c l u d e d , that is, e l i m i n a t e d , the f o r m e r will be equal to 0 a n d the latter e q u a l to 1. If we use these results, we now have

a = aihb#,

13-

-h

~ks

,

3" = ~-~h q_ ~kCikekjdhkI-i1 h '

w h e r e the g e n e r a l t e r m o f E t is now a,~Ou ., a n d the g e n e r a l t e r m o f lI t is u~h + dhk, jUSt as in the first expressions o f o u r U0. In w h a t e v e r f o r m 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 linear f u n c t i o n s o f it take U0 as its m i n i m a l value which m u s t be

which can be r e d u c e d slightly a n d results in 3/' + OL~ = ~27 h + ~,, Cikekj(aihbhj + d,,k)/ii -h .

Page 548

We now m a k e a n o t h e r u~h, p r o m i n e n t in this 3' + c~/3, w h e r e we have to a s s u m e h' 4: h b e c a u s e h does n o t o c c u r in it. We n o w 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

= O~ u ih, +

i~l

-

I

uih, + 3" ,

and, again, the m i n i m a l value o f this e x p r e s s i o n in view o f the variables (or f u n c t i o n s of) u~ z is 3" + ol'{3'=

F.,[h-# +

F.,kcikekj(aihbhj + dhk)(aih, b~lj +

dn,,)/i-/h-n'.

This c o u l d have b e e n written i m m e d i a t e l y w i t h o u t the i n t e r m e d i a r y calculation o n the basis o f the o b s e r v a t i o n that 3' + o~j3 with r e s p e c t to u , again gives the same f o r m as u o - - e x c e p t for the a b s e n c e o f o n e t e r m (that is, u p to the e x t e n s i o n ) in E a n d /i over l, a n d e x c e p t for the c i r c u m s t a n c e 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 expression o f 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 complicated in 3' + c~/3 (as we can see it t h e r e now).

374

SCHRODER'S LECTURE XI

This observation is also valid for "y' + a'{3', a n d the law of f o r m a t i o n has b e c o m e clear. If we assume these conclusions to be u n l i m i t e d until all terms of the E t a r e e l i m i n a t e d , thereby, at the same time, all factors of IIt have b e c o m e ineffective, = 1, we have f o u n d x 0 as the last e x p r e s s i o n of the m i n i m a l value, 3,~~176 + a~~176176

Xij = ~kCikekjI-Ih (aihbh~ + dhk) = E~c,JL(a,,, +

~)rI,,(~,,~

+ bhj)ek) = {(a0~ d)c; (dj- b)e} o,

a n d t h e r e f o r e , 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 " d e t e r m i n a t i o n of the limiting value" a n d is c h a r a c t e r i z e d as a kind of "method of exhaustion": the E a n d H over I were gradually "exhausted"--as we would e l i m i n a t e in a p r o b l e m of e l i m i n a t i o n of x the u n l i m i t e d d o u b l e series which c o u l d possibly f o r m a c o n t i n u u m of its coefficient Xhk by progressive e l i m i n a t i o n of o n e after the other. In the same way, we have taken o u t o f c o n s i d e r a t i o n or "annihilated" o n e U~h after the o t h e r by k e e p i n g only what it c a n n o t avoid c o n t r i b u t i n g U0 to H; in o t h e r words, we were l o o k i n g for the m i n i m a l value of the m i n i m a l value o f the m i n i m a l value, etc., of UO with respect to o n e of its a r g u m e n t s u~h (of the propositional f u n c t i o n ) after the o t h e r - - b y a "minimal value)' of a f u n c t i o n 4~, "with r e s p e c t to an a r g u m e n t u," we u n d e r s t a n d , now briefly speaking, a value which is actually o b t a i n e d for s o m e u, while it is c o n t a i n e d in all values which the f u n c t i o n can take on for any u - - n a m e l y we have to c o n t i n u e this until the last a r g u m e n t uih, if it exists; m o r e generally, t h e n w h e n this has b e e n d o n e with respect to each a r g u m e n t uih. T h u s Page 549 the resulting expression for the m i n i m a l value of UO has b e c o m e ind e p e n d e n t of all a r g u m e n t s u~h, in s h o r t of u; a n d t h e r e f o r e we could e l i m i n a t e the sign H in f r o n t of this c o n s t a n t by the law of tautology. Incidentally, for a = 1, b = 1', by c h a n g i n g s o m e letters, o u r first result 113) b e c o m e s 5 0 ) - - 4 h a t f o r m u l a which previously we could only find in an artful but c o n v o l u t e d way t h r o u g h an infinite n u m b e r of m u l t i p l e II. O u r result for d = 0' also includes o u r t h e o r e m 6) a n d a p p e a r s as o n e of the m o s t g e n e r a l ones of H which could so far be written in closed form. W h a t we have l e a r n e d in the last p r o b l e m can now easily be generalized "th eo re tically." If we have to know a x = II U with the "absolute extension," w h e r e U =f(u) is a given relative func"tion, we have to lookforthe general coefficient x,,k = II U,,k. We will first have no difficulties, following the stipulations (10) to (13) of w 3, r e p r e s e n t i n g the g e n e r a l factor Uhk of the last II explicitly as a "propositional function," which is built f r o m the coefficients of the u

VROM PEZRCE TO SKOLEM

375

a r g u m e n t u a n d its n e g a t i o n 6, as well as of all the possible p a r a m e t e r s a,b,c . . . . o f f ( u ) using the t h r e e species o f the p r o p o s i t i o n a l calculus and, possibly, the ~ a n d H signs in a certain way. We can "develop" this e x p r e s s i o n in the f o r m o f 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 linear a n d also h o m o g e n o u s , b u t to the h o m o g e n o u s f o r m

(~ + v)u 0 + (3 + ~,)a 0, the o n e which is n o t yet h o m o g e n o u s has to be p r e f e r r e d , as we will s o o n see. T h e p o l y n o m i a l coefficients a,/3, -y are n o t yet i n d e p e n d e n t Page 550 o f u (they carry possibly the o t h e r u-coefficients), b u t they are indep e n d e n t o f u O. To " m a k e p r o m i n e n t , " to "bring into evidencd' u 0 - - w h e r e the cases i = or r j as well as i,j = o r r h,k are to be t r e a t e d s e p a r a t e l y - - w e only have to observe: w h e n e v e r the first i n d e x 1 o f u or ~ is d o m i n a t e d by a Zt, we can multiply the g e n e r a l t e r m o f this E by l~u + 01t ( = l ) - - f o r a s e c o n d index, however, we multiply with 1~ + 0~.. If, o n the o t h e r h a n d , they are d o m i n a t e d by a Ht, we can a d d 0al' ', (=0), o r 00'1'0, respectively, to the g e n e r a l t e r m - - i n b o t h cases, b r e a k it into 0 ~th a n d 1~th a c c o r d i n g to the distributive law a(b + c) = ab + ac, o r a + bc = (a + b)(a + c), respectively. T h u s Z or H falls respectively into a s u m o r a p r o d u c t o f two o f t h e m , and, for o n e part, the s c h e m a 12) o f p a g e 121 is applicable w h e r e b y we ultimately o b t a i n u 0 or 5 o as explicit factor o r sum. In the o t h e r part, a t e r m previously effective is ineffective, e x h a u s t e d o r exc l u d e d by the factor 0~t or 0~, respectively, by the a d d e n d 1', o r 1'tj, respectively, f r o m the r e m a i n i n g Z t with r e s p e c t to H t. Now o u r Uhk actually takes o n its " m i n i m a l value" 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 n d is

w h e r e ij n o l o n g e r occurs as a suffix o f u a n d ~. If m n is any new, arbitrary suffix, we can also d e v e l o p I-

!

Vhk = ~ 'U m,, + 3 Urn,, + ~/ ,

o f which as a factor o f Xhk only the m i n i m a l value c o n t a i n e d in all Vhk for all u a n d realized for s o m e Urn,, n a m e l y w,,~ = v , , : ~ "

=

vh-,i-m.= "v' + ~'~',

in which now n e i t h e r ij n o r m n can o c c u r as an i n d e x o f u o r ~, a n d in s o m e E, H even two terms can be e x c l u d e d a n d a p p e a r e x h a u s t e d . A n d so on.

376

S C H R O D E R ' S LECTURE XI

It is now only a matter of studying the law of constructing minimal values r e p e a t e d l y m a n d until all terms have been excluded, exhausted from ~ and II referring to indices of u or 4. This is theoretically possiblemin practice, the complications can quickly astonish us. T h e completely exhausted ~ vanishes, becomes =0, the 11 is equal to 1. T h e p r o c e d u r e leaves further possibilities o p e n 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 eliminated and which consists merely of coefficients of p a r a m e t e r s a, b, c. . . . of f(u). It remains to "compress" (as I would like to say), to "condense" the propositional function, that is, to represent it as a coefficient with suffix hk i n d e p e n d e n 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 f o u n d u

Page 551

x=X. Practically, it will always succeed, at least by using sum a n d p r o d u c t 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 t e m p t misfires in p r a c t i c e ~ a n d remains o p e n for f u r t h e r research and study. Problem 26. We are looking for

x=II[a(bu;c) ; d + e{(f+ 4) ~g} ;h], u

114)

where h is not valid as a e l e m e n t l e t t e r - - s o that our 11 has eight different relatives as parameters. With the m e t h o d of exhaustion, we easily find

x O = F"keikhkjHt(buF',,,a~,,,ctmdmj+ ft + gtk),

115)

a n d 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 "

II,{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 e x c h a n g i n g letters of 113). By using l l 6 ) m a b s t r a c t i n g part of its m e t h o d - - w e transform % = [t~ = ([;/),,f---of. page 423---and ~j is transformed into {a; ([; l)d},,j. We, f u r t h e r m o r e , divide the 11t in IIt (b, + f t + gtk) = {(b + f ) J- g}~k, on the one hand, which can be r e d u c e d with the factor % to [e{(b + f ) j- g}]~k~Which may be called for the time being r~k, and, on the

FROM PEIRCE TO SKOLEM

377

o t h e r hand, 1-It{a; ([; l)d + f ; l + l; g; k}0. If r~khkg is written equivalently as (r; k" k; h)ij, 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 u n d e r the d o m i n a n c e 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 elimination or exhaustion p r o c e d u r e which contains u n s u r m o u n t a b l e 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

Page 552

We let the problem stand as it is at this point and only observe that it shows certain analogies with the mathematical p r o b l e m of "rationalizing" algebraic equations, the elimination of roots which a p p e a r in it. Even if each individual root can be eliminated by isolating it on one side of the equation and then empowering the equation on both sides with its root exponents, we do not succeed in this way to eliminate all roots. More refined methods are necessary. We u n d e r s t a n d this because when eliminating (with the m e n t i o n e d m e t h o d ) a certain root, the n u m b e r 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 o t h e r coefficients appear.

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Appendix 7: Schr6der's Lecture Xll

Introduction In his twelfth lecture, S c h r 6 d e r uses the quantifier rules of the previous lecture and relation algebra computations to construct a theory of oneto-one maps and cardinal equivalence within the calculus of relatives. T h e r e were no o t h e r 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 L 6 w e n h e i m ' s 1915 p a p e r as well. In the first part of the twelfth lecture (w 30), S c h r 6 d e r distinguishes types of relations or "mappings" [Abbildung]. Types AI t h r o u g h A4 are defined on page 561: AI are the mappings that are never u n d e f i n e d [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. S c h r 6 d e r 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). T h e n 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:~A4 (page 569), in other words, a bijective m a p p i n g where the d o m a i n and the c o d o m a i n are the same.

379

380

SCHRtDDER'S L E C T U R E X l I

Twelfth Lecture

w 31. Dedekind's Similar Mapping of One System to Another. Similar or EquipoUent Systems. Page 596 In the sections above, we have l o o k e d , so to speak, at t h e various types o f m a p p i n g s in an absolute sense, namely, as b e i n g c h a r a c t e r i z e d by c e r t a i n 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 invertible m a p p i n g [gegenseitig eindeutige Abbildung] is to be called a "substitution." F o r 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 t h e c o n c e p t s o f equipollence [Gleichmiichtigkeit], finiteness [Endlichkeit], a n d number [Anzahl]--this way o f l o o k i n g at things is n o t a d e q u a t e b e c a u s e it i m p o s e s restrictions f r o m the o u t s e t o n the m a p p i n g rule or relative x t h a t it by n o m e a n s n e e d s to fulfill, i n d e e d which it o f t e n cannot at all fulfill in p r a c t i c e in that they will conflict with essential p r e c o n d i t i o n s o f o u r i n v e s t i g a t i o n o r be i n c o m p a t i b l e with t h e m f r o m t h e outset. Thus, for example, within the domain [Gebiet] of natural numbers, through the assignment of elements standing u n d e r each other as (as x o b j e c t : )

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 [gegenseitigeindeutige 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 n u m b e r 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 w o r k ~ i n particular since the "similar mapping" is not introduced as a substitution at all! F o r r e a s o n s thus s u g g e s t e d we m u s t also study m a p p i n g s , especially t h o s e t h a t are single-valued [eindeutige] (in o n e d i r e c t i o n [einseitig], as well as) in b o t h d i r e c t i o n s [gegenseitig], in a relative sense, namely, as b e i n g

FROM PEIRCE TO SKOLEM

381

Page 597 o n e such m e r e l y with r e s p e c t to a certain system a = a j-0 = a; 1 as object a n d (possibly also) a certain system b = b~ 0 = b; 1 as (its x-)image. For the special a s s u m p t i o n a = 1 and b = 1, this relative perspective turns into the previous absolute o n e - - a n d thus what we will be d e a l i n g with is a generalization, an e x t e n s i o n of the earlier results (of w 30). P r e s e n t i n g those results earlier seems to be didactically justified b e c a u s e of their simplicity a n d i m p o r t a n c e , as well as for offering c o n s i d e r a b l y g r e a t e 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, t h e n the relative perspective, at least with r e g a r d to the image (respectively, object), turns into an "absolute" one, r e m a i n i n g 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 i n t h lecture, we set out for the highly worthy goal of b r i n g i n g the m o s t essential aspects of D e d e k i n d ' s f u n d a m e n t a l studies ( D e d e k i n d 1888) into c o n f o r m i t y with o u r t h e o r y (which did n o t prove to be a l t o g e t h e r easy), a n d in the first part of this section it is n o t the least of o u r aims to p u r s u e the f u r t h e r i n t e g r a t i o n u p to exclusively ".~64, namely, up to ",~ w 5, with the h e a d i n g "The Finite a n d 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 b e e n very largely a c c o m p l i s h e d , a n d n o t without yielding s o m e gains o n b o t h sides (in particular, a g e n e r a l i z a t i o n a n d ultimately also a simplification of chain theory). In o r d e r to carry it out completely, t h e r e r e m a i n s only to take care of a d o z e n definitions or propositions. O f the first p r o p o s i t i o n s .~1 to 63 of the oft-cited text, in fact only the d o z e n i n c l u d i n g ".~21 a n d ~ 2 5 to 35 have not already b e e n taken care of in the chain theory. W h a t r e m a i n s of o u r task, however, p r e s e n t s a series o f u n e x p e c t e d difficulties that s h o u l d not be u n d e r e s t i m a t e d , a n d o v e r c o m i n g t h e m will be m o r e than a little instructive a n d beneficial. In that I now move on to explain this, I do n o t want to suggest to s t u d e n t s that I (also) t h o u g h t I was t h e r e b y offering any simplification of D e d e k i n d ' s conclusions a n d perspectives (insofar as they are relevant Page 598 h e r e ) ! O n the contrary: we will have to go to g r e a t l e n g t h s in o r d e r to prove, by way of strict analytical d e d u c t i o n f r o m the fundamental stipulations [fundamentalen Festsetzungen] of o u r discipline, things which s e e m to be obvious to those who are familiar with the n a t u r e of invertible a s s i g n m e n t s a n d their intuition. 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 a d v a n c e b a s e d o n D e d e k i n d . We also w a n t to r e a c h the p o i n t of, so to speak, a "pasig r a p h i c " f o r m u l a t i o n , e x p r e s s e d in the n o t a t i o n a l l a n g u a g e o f o u r alg e b r a , o f those f u n d a m e n t a l c o n c e p t s , such as t h e "similarity" [Ahnlichkeit] o r "equipollence"[Gleichmiichtigkeit] o f two systems (as well as t h e "finiteness," "simple infinity," etc., o f s u c h a system a n d m i n t h e case o f t h e f i r s t m t h e " n u m b e r " [Anzahl] of its e l e m e n t s ) . D e d e k i n d ' s c o n c e p t of "similar systems" ,~32 c o i n c i d e s with G e o r g C a n t o r ' s c o n c e p t of " m a n i f o l d s " [Mannig]'altigkeiten] "having equal cardinality" [Miichtigkeit]. A c c o r d i n g to Cantor, s u c h m a n i f o l d s (see Borchardt'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 e x p r e s s i o n we p r o b a b l y have to refuse to use in o u r discipline for obvious reasons. N o w this e q u i p o l l e n c e o r similarity, for e x a m p l e , m u s t result in a r e l a t i o n b e t w e e n the systems a a n d b t h a t can be e x h a u s t i v e l y r e p r e s e n t e d by m e a n s o f t h e six species of o u r discipline. T h e first step in d e t e r m i n i n g it leads back to an e l i m i n a t i o n p r o b l e m in o u r algebra, a n d this step will also be solved explicitly for t h e lowest u n i v e r s e o f discourse. Mathematicians are aware of the great importance of the concept of equipollence. 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 f n u m 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 m e n t i o n e d m a n d even the transcendental (irrational) n u m b e r s - - o r 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

FROM PEIRCE TO SKOLEM

383

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 g r e a t deal w o u l d be g a i n e d a n d the C a r t e s i a n a n d L e i b n i z i a n i d e a o f p a s i g r a p h y w o u l d s e e m to have t a k e n a g i a n t step, p e r h a p s its m o s t significant a n d difficult step, forward, if (in this v o l u m e ) it t u r n s o u t to be possible to deliver the p r o o f t h a t the n o t a t i o n a l capital c r e a t e d with o u r s t i p u l a t i o n s (1) to (15) (an overview o f w h i c h I h a v e p u t t o g e t h e r in a h a l f m n o t very d e n s e l y p r i n t e d m p a g e in my Annalen n o t e [ 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 e x h a u s t i v e l y all definitions, propositions, a n d conclusions within the i n t e l l e c t u a l c o m p a s s o f 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 t h e 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 5 2 6 , 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 n e c e s s a r y a n d sufficient c o n d i t i o n for the "similarity" o r "equiPage 600 pollencd' o f two systems a = a~tO = a; 1,

b = b~O = b; 1

O)

is (always) t h e existence, the possibility of a "similar (or d i s t i n c t [deutlichen]) mapping' x (respectively, ; ) of the two systems i n t o e a c h o t h e r , t h a t is, an invertible assignment b e t w e e n all the e l e m e n t s o f t h e o n e system a n d t h o s e of t h e o t h e r ~ w h i c h , however, it r e m a i n s n o w to e x p l a i n in g r e a 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

384

SCHR6DER'S LECTUI~E XII

which among the elements outside of b (therefore of/~) might otherwise be ximages w h e t h e r o f a or of d. Finally, w h e t h e r t h e r e are e l e m e n t s in a that are n o t x-images o f others, as well as w h e t h e r t h e r e are e l e m e n t s in b o f which no x-image exists, is likewise to be left o p e n a n d 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 n u m b e r word "one (and only one)," and we will have to put in place of the condition with that aim a n o t h e r than the one u p o n which it is based, which in essence stipulates that other (or various) is to belong to o t h e r (various). The essential task therefore confronting 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, Xk,, =

for every element m of a, g: h, k ~ x; m must be valid, 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) @ ( n 4 ~ x; h)}. Page 601

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 i n d e p e n d e n t of k and h, the m e a n i n g of which is obvious. If, accordingly, we introduce the abbreviation y for xX, where

FROM

PEIRCE

385

TO SKOLEM

y = {(a+ ~) o~ l'}xll' 0~ ( ; +/~)} = 1~ 0~ (d + l')}xl(lq-/~) 3- ~},

2)

would be valid, then r e m a i n i n g as an expression is o u r c o n d i t i o n

I'lh(ah + ~"kbkYkh)IIk(Dk + ~"hahYkh)

= Ilkh{(a +

1 ; by)([~ + dy ; 1)}kh

m a s is a p p a r e n t if one thinks of the r u n n i n g index k h of ~k a n d ~h as having b e e n 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. T h e latter H is, now, the general coefficient of a distinguished relative, which itself is known to be equal to it. Thus is 0 j- ot/3 :t 0 itself, or if o n e prefers 1 =(=0 :t ot/3 ~ 0, where ot = a +1 ; by = a + [~ ; y, j3 = {~ + ~ty ; 1 -- [~ + y ; a, the expression of o u r condition. 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 a n d / 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) = (aj-5;b)(~:ty;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 conversion of the condition 6=(=/~;y rewritten as 1 =(=ot [and] on the left side was to be observed that a distinguished relative is equal to its converse. F o r m u l a 1) contains the formulation, and 2) and 3) t o g e t h e r contain the solution of o u r p r o b l e m . Let us express that a system a is "equipollent" or "similar" to a system Page 602 b---with G. Cantor, loc. cit., page 2 4 9 - - b y the s t a t e m e n t [Ansatz]" a ~-'' b,

which gives us the following "first version" of the definition of the similarity or equipollence of a system a with a system b: (a ,I, b) = x

[b=~-{(a+ x-) :t l'}x{l':f (s +/~)} ; a][a:~:{(/~ + x ) 3- 1'}~{1'~ ( x + d)} ; b].

(4) This, however, can be p r o d u c e d in a m u c h m o r e c o m f o r t a b l e form

386

S C H R O D E R ' S L E C T U R E XII

through the introduction of y instead of x, and, indeed, in an equivalent t r a n s f o r m a t i o n ~ w e a s s e r t ~ t h e following "second version": (a,~ b) = E ( b " y ; ~ + a'~;y~-l')(a=~--f;b)(b=~--y;a), y

(5)

whereby the first b and a on the right can be replaced by /~ and d, respectively, a n d m j u s 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', 0';y=(=;+/~,

x b =(c-l ' ~ ~,

y b =~--x b,

y b =~--l ' ~ ~,

ergo

~7;~y:61',

y b ; ~ ~-- l ' .

Because, however, a and b are systems, then ~;#=/;y"

~,

yb;.~=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 statement [Ansatz]:

b" y ; ~ + a" ~;y=(=a'. This resultant--as one sees, the first factor the full one. For, if it is fulfilled by a y, then makes equation 2) true, and, indeed, in the to be seen in this way. From the converted follows that f ; y:(= d + 1',

y;)7=(=l' +/~,

62)

on the right in 5)mis now there also exists an x that form of x =y itself. This is 61) /~. y ; ~ + a'y;y=fc--l' it

y=(=)Y#(d + 1'),

y=~(lq-/~) #.~,

Page 603 and thus 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 +/~.

387

FROM PEIRCE TO SKOLEM

If, to avoid having to write the n u m b e r e d p r o p o s i t i o n s r e p e a t e d l y in p a r e n t h e s e s , we use t h e i r n u m b e r s for the n u m b e r e d p r o p o s i t i o n s , t h e n it has so far b e e n p r o v e d t h a t II{2) :(=6)} x

or

I~2) :(=6),

6) :(= Z 2);

x

therefore

6) = I; 2),

x

x

b u t in a d d i t i o n , 6) - 7), w h e r e b y 6) = 61) - 6z). Now if t h e r e exists an x such t h a t t h e r i g h t side o f e q u i v a l e n c e 4) is true, t h e n t h e r e also exists a y, n a m e l y the o n e r e p r e s e n t e d by 2), that fulfills the r i g h t side o f e q u i v a l e n c e 5), a n d conversely: if t h e r e exists a y that satisfies the last c o n d i t i o n , so is t h e r e an x satisfying t h e first, a n d , i n d e e d , at least in the f o r m x equals y. C o n d i t i o n s 4), 5) t h e r e f o r e imply e a c h o t h e r r e c i p r o c a l l y o r 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 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). x

In o r d e r to show now that the two final s u b s u m p t i o n s in 5) have t h e force o f e q u a t i o n s , as was asserted, we simply have to m a k e use o f 6) to derive the two reverse s u b s u m p t i o n s f r o m

b~--y;a

and

a:~-~;b.

and

y ; a ~-- y ; ~ ; b .

Now it follows i m m e d i a t e l y that

~;b:~-~;y;a

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" ("perpePage 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)

9)

=/~;x;5---a ;:~;b = (x;d 4 / ~ ) = ( : ~ ; b 4 a), which will reappear in a further formulation below as well as that which is given h e r e - - f o r 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 s o m e sense a l r e a d y m a d e a c o m m i t m e n t a b o u t t h e e x t e r n a l b e h a v i o r o f 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 w h i c h we m a y be s u r e that the c o m m i t m e n t can be m e t at any time, w h i c h w o u l d n o t be c o r r e c t for the n a r r o w e r c o n c e p t i o n o f the m a p p i n g r u l e as a "substitution," as we saw o n p a g e 596 in c o n t e x t . As o p p o s e d to a s t i p u l a t i o n o f the latter sort, we m u s t truly w e l c o m e s u c h restrictions o f the m a p p i n g rule with r e s p e c t to its e x t e r n a l b e h a v i o r to t h e similar systems a, b, t h e validity o f w h i c h is g u a r a n t e e d internally. A n d they c o u l d s e c u r e similar a d v a n t a g e s for us in s u c h t h i n g s as choosing 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 g e o m e t r y . O f c o u r s e , h o w e v e r (with c o m m i t m e n t s o f t h a t sort) we m u s t n o t allow a n y t h i n g to p r e j u d i c e the i n t e r n a l b e h a v i o r o f Page 605 m a p p i n g rule, t h a t is, the q u e s t i o n : b e t w e e n w h i c h e l e m e n t s o f a a n d w h i c h e l e m e n t s o f b is t h e a s s i g n m e n t to p e r t a i n ? O n l y thus will it be a s s u r e d t h a t the results of o u r investigation can be a p p l i e d in full generality. 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 assignments [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 i n d e p e n d e n t of the "versions" in which wemwith due consideration for the external behavior of the mapping rule--might formulate the (one specific) assignment.

FROM

PEIRCE

TO SKOLEM

389

In this sense, we m u s t w e l c o m e as a c o n s i d e r a b l e simplification of o u r similarity c o n d i t i o n 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~

11)

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 (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 c o n d i t i o n , if one, in the f o r m u l a t i o n of the d o u b l e r e q u i r e m e n t Xkh---committing to the e x t e r n a l b e h a v i o r of the m a p p i n g rule x in 1) in s o m e r e s p e c t - - d r o p s the restrictions m~--a a n d n=~--b (while h =~-a Page 606 a n d k ~= b r e m a i n ) , which m u s t have the s a m e effect as if o n e h a d taken a = b = 1 (in Xkh ). In o t h e r words: if, f r o m the start, we take the d i f f e r e n c e o f the x-images toward various of the objects c o m i n g into c o n s i d e r a t i o n , a n d of the objects toward various of the x-images c o m i n g into conside r a t i o n , n o t only for the e l e m e n t s of b a n d a, respectively, b u t for the e l e m e n t s of the e n t i r e universe of discourse. In that case, i n s t e a d 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) w h e r e Zkh = II,n{(m 4: h) ~=(k=~=x; place of 2), m u c h m o r e simply,

m)llI,,i(n ~ k) ~ - ( n ~ x" h)}, a n d

in

39 ~

SCHRODER'S LECTURE XII y = (~o~ l ' ) x ( l ' o~ ~),

13)

t h e r e f o r e , in place of (4) as the " f o u r t h version" of t h e similarity condition,

(a,-",b) =~{b@(~l')x(l'j-~);a}{a@(xj-X')~,(l'j-

x) ; b } ,

(14)

w h e r e , however, the x will be a different, m o r e r e s t r i c t e d relative t h a n t h e x in t h e e a r l i e r f o r m u l a s - - i n 12) to 14) this x m a y by all m e a n s be i d e n t i f i e d with o u r z. For now, we will cite the f o r m u l a s in this way ( t h o u g h t of as h a v i n g b e e n w r i t t e n for z i n s t e a d o f x), a n d w h a t m u s t be k e p t in m i n d , 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 t h e E w r i t t e n b e f o r e i t - - c o u l d be r e w r i t t e n e q u i v a l e n t l y as the r i g h t side o~f similarity c o n d i t i o n (14) o r (10), so, conversely, we c a n n o t n e g l e c t to d e r i v e the c o n s e q u e n c e s 12) (written in z) by m e a n s o f an e q u i v a l e n t transformation. 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), ILl(k

15)

~- 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 relations 16) c a n n o t , i n d e e d , by any m e a n s f r o m t h e r i g h t side of (10). It d o e s n o t n e e d to fulfill o u r t h i r d version (10). Only, if t h e r e exists any z at all it, t h e n t h e r e also exists, in the f o r m of dbz, a relative

be d e r i v e d j u s t a z t h a t satisfies t h a t d o e s satisfy s u c h that, w h e n

FROM

PEIRCE

391

TO SKOLEM

called simply z, fulfills, in a d d i t i o n to 10), the relation 16) as well. T h e r e f o r e we c o u l d take this relation 16) a l o n g 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 arriving at the "fifth version" o f d e f i n i t i o n o f similarity ,~32: (a ,I' b) = E ( z ; i + i ; z =(=l')(b ~ - z ; a)(a =~--~;b)(z ~c--gtb),

(17)

z

in which the m a p p i n g rule a p p e a r s the m o s t restricted, the m o s t narrowly conceived, by way o f the m o s t c o m p r e h e n s i v e , maximal conditions: [it appears] as one, to p u t it colloquially, that assigns the e l e m e n t s o f a a n d b, a n d only these e l e m e n t s to each o t h e r ( s o m e h o w invertible). [All conceivable instances of the assignments subject to the s a m e limitation b e t w e e n the e l e m e n t s o f the two systems m u s t p r o c e e d f r o m this a s s i g n m e n t t h r o u g h simple p e r m u t a t i o n o f the e l e m e n t s o f o n e o f 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 clear f r o m o u r n o r m a l form (17) that to the notion of similarity b e t w e e n two systems a a n d b the n o t i o n of " relativity" (in H o p p e ' s sense) as a result o f b e i n g taken with respect to a specific universe o f discourse, namely, the fundamental universe of discourse, does n o t apply. For every universe o f discourse 11 in which the similarity o f a a n d b is established in s o m e way, the e l e m e n t s o f a a n d b, at the very least, m u s t themselves b e l o n g to it, a n d t h e r e f o r e systems a a n d b as well as their 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) o u r z is c o m p l e t e l y i n c l u d e d in the r e c t a n g u l a r quadrilateral-relative [Augenquaderrelativ] rib, then, if t h e r e exists a relative z that fulfills the c o n d i t i o n s o f (17) in any arbitrary universe o f discourse 12, so also in every such universe o f discourse 12. Page 608

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 corollary to (17), respectively, (10), we have the p r o p o s i t i o n a,~" a

18)

that is, every system is similar to itself. In o t h e r words, the r e l a t i o n s h i p o f similarity b e l o n g s 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 =

392

SCHRODER'S LECTURE XlI

dl';a=al';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 r e q u i r e m e n t ~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 h e 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 i m i l a r to b, so m u s t b b e s i m i l a r to a, a n d conversely. It is this c i r c u m s t a n c e t h a t first gives us t h e r i g h t to say t h a t t h e s y s t e m a a n d s y s t e m b a r e " s i m i l a r to e a c h o t h e r . " For proof, it is e n o u g h to see that the right side of (10) or (17), and i n d e e d directly the propositional product, which forms the general term of the sum over z likewise, is simply m a p p e d to itself if a is e x c h a n g e d with b and at the same time z with L With this latter exchange, the first propositional factor characterizing the m a p p i n g rule z only maps to itself. T h e second and third propositional factors change places. For (17) the fourth (adventive) propositional factor that comes up characterizing z maps to the reciprocally converted and thus equivalent one. Admittedly, however, the z u n d e r the I2 turns back to ~?u n d e r 12. Nevertheless, this must be irrelevant. For, if there exists one value, one relative, that fulfills any r e q u i r e m e n t stipulated for z, there must also existwin the form of its conv e r s e m a relative that fulfills precisely this r e q u i r e m e n t conceived for L [Indeed, every condition for z can also be r e g a r d e d m b y taking z = rS, ff = w---as a condition (and therefore for w) for L] .~33. P r o p o s i t i o n . Similarity is also a transitive relationship; in other words, if a a n d b are similar systems, then every system that is similar to b is also similar to a:

(a ,I, b)(b ,-,', c) ~--(a ,.", c).

20)

T h e s i m i l a r o f similars is similar. Page 609

Proof.

T h a t m i n reference to ( 1 7 ) m i t must be that

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 u n d e r the left-hand side conditions

z=y'x,

s

certainly fulfill requirements on the right, and indeed the first propositional factor, as has already been shown implicitly in w 30, but can be shown again here, however, with

FROM PEIRCE TO SKOLEM y; x;:~;~7+ ~ ; ~ ; y ; x ~ - y ;

393 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 equations; 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 statement of the three adventive requirements, could be saved (if desired) by appeal to (10) instead of (17). m If, however, we attempt 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 subsumptions 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 condition which we should expect to be characteristic for the composed mapping rule and which will take on the role " o f the r e q u i r e m e n t A,~A 4 along with the adventive r e q u i r e m e n t 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 treatment and complete clarification in future research. The proof given here for 20) distinguishes itself, in a not altogether 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 proposition , ~ 3 1 m w h e r e he sets his a r g u m e n t forth regarding the elements. We will return to this below on page 621, once we have gained a bit more information. We m a y now, given t h a t t h e p r e m i s e s o f 20) h a v e b e e n fulfilled, also call t h e 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 t h e c o n c l u s i o n s , w h i c h we a r e f a m i l i a r with r e g a r d i n g e q u a l i t i e s , f o r o u r s t a t e m e n t s o r a s s e r t i o n s o f similarity, we easily arrive at a n e x t e n s i o n o f Page 610 this c o n c e p t o f "similar to e a c h o t h e r " f r o m t h r e e to a n y a r b i t r a r y sets, i n c l u d i n g e v e n t u a l l y an i n f i n i t e system o f " s y s t e m s " ~ a s o f a c o n c e p t t h a t is i n d e p e n d e n t o f t h e o r d e r in w h i c h t h e y a r e e n u m e r a t e d o r named. ".~34. D e f i n i t i o n . It is t h e r e f o r e p o s s i b l e to s e p a r a t e all systems i n t o 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 o n l y systems (a), b, c, ... t h a t a r e s i m i l a r 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 t h e class; a c c o r d i n g to t h e p r e v i o u s p r o p o s i t i o n ~ 3 3 , t h e class d o e s n o t c h a n g e if a n y 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 representative.

394

SCHRODER'S LECTURE XII

~ 3 5 . P r o p o s i t i o n . If a a n d b are similar systems, t h e n every s u b s y s t e m o f a is also similar to a s u b s y s t e m o f b, every p r o p e r s u b s y s t e m o f a is similar to a p r o p e r s u b s y s t e m o f 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, since a, b, c, d are to be t h o u g h t o f as systems, we w a n t to c o n s i d e r [ t h e m ] in the f o r m u l a s , t h e n all we n e e d to d o is r e p l a c e t h e i r n a m e s - - w i t h t h e e x c e p t i o n o f t h e d u n d e r t h e Z - - b y , respectively, a;1, b;1, c;1, d;1. Proof of the first proposition. Among the assumptions is (a,-', b) =~ (z;s

=~.;b)(b=z;a)(z =(ectb),

s

z

whereby t h e m u n d e r l i n e d m a d v e n t i v e condition can be disregarded for the moment. If, furthermore, c 4= a, then we can say

z;c - d and it follows that z; c =(e=z;a; therefore,

d =geb,

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 subsumption, 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,

FROM PEIRCE TO SKOLEM

d=~-b

and

395

(z;i+s

i.e.,

c,-',d,

q.e.d. T h e adventive r e q u i r e m e n t foreseen for c and d in the n o r m a l form similarity condition 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 n e e d merely to i n t r o d u c e [dz as a new z. T h a t for a and b e q u i p o l l e n t 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 particular as a corollary of the proposition. For the s e c o n d proposition, 22), the assumption c :/: a is simply a d d e d to the hypothesis of 21) as a propositional factor, and likewise the assertion d ~ b to the thesis after the s u m m a t i o n sign. This can be easily proved apagogically [Translator's note: by contradiction]. Namely, if d = b, then with z; c = d according to the above, we would also have c = ~; b = a; therefore, c - a, contradicting the assumption. T h e above proofs also differ essentially from D e d e k i n d ' s for ".~35mwhich we will also discuss in m o r e detail below on page 622. In what we have covered thus far, however, the most i m p o r t a n t propositions of this a u t h o r about similar mappings and similar s y s t e m s - - t h o s e for which the others served merely as p r e p a r a t i o n m h a v e been taken over into o u r discipline from the parts of his text delimited at the outset and justified, legitimated in spirit! T h e r e remains nonetheless a whole series of studies of the material to be p r o d u c e d . In o r d e r to f a c i l i t a t e "argumentation regarding the elements" f o r a n y similar m a p p i n g o r to l e g i t i m a t e it f o r o u r d i s c i p l i n e , t h e f o l l o w i n g o b s e r 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 b e o f i n t e r e s t . A c c o r d i n g t h e a s s u m p t i o n s laid d o w n in ( 1 7 ) , II,,k

{(k:~--z;h):~--(h:~--a)(k:~--b)(z;h:~--k)(~;k:~--h)} (h ~ i; k)

a r e valid, w h e r e t h e s u b s u m p t i o n c a l l e d to m i n d a b o v e it.

Page 612

24)

in t h e s e c o n d l i n e s h o u l d m e r e l y b e

as a n 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

immediately

First, we want to prove this (in a fashion) for the s u b s u m p t i o n sign. If h, k are e l e m e n t s and if k=g~--z;h,then, first of all, h=(c--amust 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 contradiction 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), inclusions s u c h as

(k~z;h)=(z;h=k),

(h~;k)=(~;k=h)

m u s t have the force of e q u a t i o n s ; e q u a t i o n s , s u c h as the two in (z;h=k)

=(~;k=h)

[are] e q u i v a l e n 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 c a n m a n d 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 e s t a b l i s h e d f u r t h e r is the p r o p o s i t i o n

IIhk { (Z ; h ~-- k)(h ~-- a) ~_ (k ~-- b)(k ~-- z ; h) } x {(~.;k~-h)(k ~ b) ~-(h~--a)(h@~.;k)}

25)

- - w h e r e 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 i n d e p e n d e n t 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 t h e a s s u m p t i o n s h ~ - a a n d k =~ b, or at least by s o m e o n e o f t h e m ,

FROM PEIRCE TO SKOLEM

Page 613

397

an equivalence must exist between two of the total of six p r o p o s i t i o n s the following thesis:

of

II,, k (h ~ a)(k ~-- b) ~-

= ( s ; k ~ h) = (h ~ s ; k) = ( s ; k = h)

"

T h i s - - 2 4 ) to 26)---completely f o r m u l a t e d yields a series of distinguished relatives that are recognized as valid (i.e., = 1) . . . . In his definition .~26 of the similar m a p p i n g of a (to b), D e d e k i n d focused on its nature, that which makes it different from the singlePage 615 valued m a p p i n g : that different images should always c o r r e s p o n d to different elements 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 proposition: 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 =0J-(d+a

=(= l'hk]

+i;s

=

YIhk{dhk "~ (lkh "~-

l')j-0'-a

=(a;1;d~--i'i+z;z+

j-(s

( Z ; Z'Jff z ; Z)hk -"

l't,k}

l')j-d

1').

It appears, therefore, that 0~d ~= s i + z ; z

27~,)

is the most concise expression in o u r notational system for the requirem e n t p r e s e n t e d above. Only with z ; h =(=z ; k as the s e c o n d premise, even 27) is already valid---cf. X) of w 3 0 - - a s 0~d~=s ~.

27~)

This relation can t h e n - - a s will be seen below r e g a r d i n g 31 ) - - a l s o 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 m a d e a s o m e w h a t different version of the definition of similarity f u n d a m e n t a l [to this work], so we must t h e r e f o r e prove p r o p o s i t i o n 27~) from o u r own [definition] with which the two previous ones are given a fortiori. This can be d o n e very easily as follows. F r o m the identity 0~6=(=0~z it follows, by substituting a = s 1 = s (~ + z) from 25~) on the right side, 'The rest of page 613 and page 614, where Schr6der re-proves propositions 24)-26), are omitted here.

398

SCHRODER'S LECTURE XII

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 h e o n e r e m a i n i n g o n t h e r i g h t t h e f a c t o r 0' c a n o b v i o u s l y b e s u p p r e s s e d - - q . e . d . O n t h e basis o f 27), h o w e v e r , we a r e n o w also a b l e to t a k e c a r e o f p r o p o s i t i o n s , ~ 2 7 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 . ~ 2 7 . ( c + d:~--a)(z; c:~-z; d) :~--(c:~--d),

(c + d:~--b)(~.; c:/~-~.; d) :~c--(c~--d), 28)

' ~ 2 8 . (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 t h e two p r o p o s i t i o n s o f D e d e k i n d

Page 616

named

above.

Proof of 28). For if h is an e l e m e n t of c and therefore also of a, then z; h is an e l e m e n t of z;c and therefore also of z;d; consequently = z ; k , where k is an e l e m e n 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 e m e n t h of c is also an e l e m e n t of d, as was to be proved. C o n s i d e r i n g definition (1) of equality, the next proposition ~ 2 8 is thus a very close corollary of the previous one. T h e assumptions c + d =(= a, etc., may not be suppressed h e r e because, since the z-images vanish for all elements outside of a (because z; fi =0), there is in fact no n e e d at all f o r a p a r t o f c t h a t d o e s n o t b e l o n g to a to b e c o n t a i n e d in d. T h e next proposition 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 r e 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 a n y case. To prove them as reverse [subsumptions], we can follow D e d e k i n d ' s considerations: Every e l e m e n t of z;c" z ; d " ... is in every case c o n t a i n e d in z;a; therefore the image k -- z; h of an e l e m e n t h is c o n t a i n e d in a. Since, however, z; h is a c o m m o n e l e m e n t of z;c and z ; d "'", so, according to 28), must h be a c o m m o n e l e m e n t of c and d ' " . Consequently, every e l e m e n t k of z; c" z; d" ... is the z-image of an e l e m e n t h of cd"" and therefore an e l e m e n t of z;c" z ; d " . . . - - q . e . d . No d o u b t it would be possible to produce, for the last three propositions with II prefixed to c, d ..... o t h e r analytical proofs besides these, which would o p e r a t e

FROM PEIRCE TO SKOLEM

399

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 m e r e l y establishes that the i d e n t i c a l m a p p i n g o f a system a is also a similar m a p p i n g of i t - - a n d for us is n o less selfevident. W i t h i n b, however, the s a m e is possible only i n s o f a r 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 "explicit" o n e ) version o f the d e f i n i t i o n o f similarity: It is also possible to f o r m u l a t e i n d i v i d u a l l y the f o u r c o n d i t i o n s o f w h i c h the r e q u i r e m e n t o f the similar m a p p i n g o f a to b consists ( a n a l o g o u s l y to the A~ t o A 4 o f w 30) a n d t h e n s u m m a r i z e it in r e t r o s p e c t . It w o u l d mean: 3'1 = For every e l e m e n t h o f a t h e r e exists at least o n e e l e m e n t k of b such that k =(=x; h, 3'2 = For every e l e m e n t h o f a t h e r e exists at most o n e e l e m e n t k o f b such that k ~ - ; h , 3"3 = For every e l e m e n t k o f b t h e r e exists at least o n e e l e m e n t h o f a such that k =~- x; h, Page 618 3'4 = For every e l e m e n t k o f b t h e r e exists at most o n e e l e m e n t h o f a such that k =(=x; h. T h u s

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)},

32)

3"4 = 1-Ihk,,,{(k:(c--b)(h @a)(k=(ex ; h)(m~-a)(m r h) ~--(k ~- x; m)}, We t h e n find (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';bx =0)

= a a~ {(x +/~1 a~ 1' + x } a~/~= (~/~;0' 9~/~a = 01,

33)

3"~ = ~ j- x; a = (b=~--x; a),

3"4 =/~ ~ {(J~'~-a ) ~" 1' + ~}j-d = ( x d ; 0 ' 9x~b = 0). Yet j u s t as quickly it is possible to give ")'2 a n d 3'4 in the s i m p l e r f o r m s of expression: A list of formulas that Schr6der gives for the student's convenience (pp. 616-617) are omitted here.

/

400

SCHR()DER'S LECTURE XII

T z = 1 ;{1'~- (/~+ x - ) l ~ - d = l d ( O ~ . O ' ; b x )

=0}

= a ~- { (x +/~) 0~ 1'} ; 1 = {(;/~; 0' 0~ 0 ) a = 0},

34)

"y4 -/~ o ~ { ( ; + a )~-1'} ; 1 = { ( x d ; 0 ' o ~ 0 l b =0},

o r also

{

T2=a0~lxo~(l'+/~)l;l=l(;;0'bo~0)a=0}, 3'4 =/~ ~ - I ; # (1' + d)} ;1 = { ( x ; 0 ~ e 0 ) b

= 0}.

35)

Justification. These [values] must be given i n d e p e n d e n t l y only for 3'1 and 3'2 because, a m o n g the expressive forms required for these two conditions and propositions, those for 7:~ and 3'4 are p r o d u c e d by e x c h a n g i n g 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 propositions 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 r o o f of the statements in 33). In o r d e r now to get two of these for forms 34), 35), we state the lemma: ( a ' 0 " a ~ b " 1) : ( a ' 0 ' ~ 0 : ~ - b" 1),

(b'l : ( = a o ~

(a'0'

(l'b=(=a+

9

a=~-l'b)

~

(0~

0 /

;a=(=l'b)

,

+a)

Ib'l:(=(aj-1)'l}

1' j-a) = {l'b=(=l 9(1 ' j-a)} . 36)

What is to be presented here are a, an arbitrary relative, and b; 1, just 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 t h e r hand, a;0'o~ 0 = 11100, according to which the two subjects in 36) differ: this proposition must be valid, as can be seen by c o m p u t i n g the rows [zeilenrechnerisch]. It can be proved m o r e elegantly as L -- R thusly. Because a; 0' 9a =(= 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 PEIRCE TO SKOLEM

401

from L, which, because ( a ; 0 ' ) a ; 1 :(= a ; 0 ' j - 0 , cf. 30) o f w 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' 9x~i =(=/~(=/~; 1), is easily transformed to x6; 0',t- 0 :(=/~, that is, into 3 4 ) h a n d c o r r e s p o n d i n g 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 propositions about systems. It is v e r y m u c h w o r t h r e m a r k i n g that, while, 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 h e systems a a n d b, t h e r e l a t i o n s h i p s ~ a n d 3':t a r e transitive, s u c h is by no means t h e case with "Y2 a n d "g4. T h a t c a n b e i m m e d i a t e l y p r o v e d a n a l y t i c a l l y with t h e c o n d i t i o n s a fortiori valid:

(a~;b)(b~;c)

~(a=~;);;c)

=(a=(=i;c)

( b : ~ - x ; a ) ( c @ y ; b ) @ ( c : ~ - y ; x ; a ) = (c:~c--z;a)

for z=y;x, for 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 that, if f o r e v e r y a ( a c c o r d i n g to m a p p i n g r u 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 f o r e v e r y 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 r u l e y) at least one c b e l o n g s , t h e n , for e v e r y a ( a c c o r d i n g to b o t h m a p p i n g r u l e s 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, t h r o u g h o u t , we replace the words "at least" here with "at most," it may be that for some the proposition will be every bit as illuminating. Nevertheless, the rhetorical evidence would simply have led to error (namely, to such an a to w h i c h - - a s "at most o n e " r a n 0 b belongs, any n u m b e r of c could b e l o n g much m o r e 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 neither with z =y; x nor with any o t h e r z, in that the conclusion in question would indeed have to have been a resultant of the elimination of b from the premisses. No such [elimination] is at hand, however, because the premisses have always proved satisfied for b--0. T h e conclusion, accordingly, must signify n o t h i n g [nichtssagend] as a relation; in o t h e r 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. T h e r e 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 c o m p u t a t i o n 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 m a p p i n g rule, once

402

SCHRI~)DER'S LECTURE Xll

again there belongs to every e l e m e n t of the universe of discourse at most one e l e m e n t from this entire universe of discourse, likewise according to a second m a p p i n g rule, then also according to the composite m a p p i n g rules. W i t h r e s u l t s 32) to 35), t h e f o l l o w i n g n e w v e r s i o n o f t h e s i m i l a r i t y r e q u i r e m e n t f o r 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, systems a a n d b will b e c a l l e d s i m i l a r if a n d o n l y if, if t[aere exists a m a p p i n g r u l e x, w h i c h 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 t h e "sixth v e r s i o n " :

( 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 condition in principle as follows: for every e l e m e n t h of a there should be within b (at least) one e l e m e n t k, which is "uniquely an x-image of it and of it alone" (and conversely--using "s instead of x-image), then it is already clear to c o m m o n sense that there c a n n o t also be a second e l e m e n t 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 t h e r elements of a. And one feels, or believes to feel, that the earlier version must essentially coincide with the one formulated now, although the latter poses the relevant requirement, not as a d e p e n d e n t o n e - - i n a relative p r o p o s i t i o n - - b u t independently. Only, we must not c o n t e n t ourselves with such an intuition 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)a0 also that (s163 Thus, if we take

xdb = z, the last condition 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 consequently it must be valid that

Page 621

z;0" z+ s

s

which is easily transformed 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 requirements 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). T h e r e f o r e (10) follows from (39), that is, if there exists an x that fulfills r e q u i r e m e n t (39) with respect to given systems

FROM PEIRCE TO SKOLEM

4o3

a a n d b, t h e n t h e r e will also exist a z (= glbx) that, with r e s p e c t to the same systems, fulfills r e q u i r e m e n t (10) a n d even (17). Conversely: if any z satisfies the c o n d i t i o n (10), t h e n x = z must also satisfy the c o n d i t i o n (39), which is obvious with respect to the first two r e q u i r e m e n t s , but can be shown with respect to the last ones as follows. We have z; 0 ' 9z = 0, and t h e r e f o r e z; 0 ~ 9z~-/~, as well. A c c o r d i n g to o u r proposition 36), this is equivalent to

z;0'0~0=(:/~,

a n d from t h e r e it follows a f o r t i o r i :

z;0~j-0:~-/~,

i.e., ( z ; 0 ~ 0 ) b = 0 .

A n d just like that, it is shown that (i;0'b0~0)a =0, q.e.d. According condition

to this, t h e m a p p i n g

r u l e z t h a t satisfies t h e normal s i m i l a r i t y

(17) a l s o satisfies, in a n y case, t h e c o n d i t i o n s

any one of the other of our versions of the mapping not conversely.m W e a r e n o w in a p o s i t i o n to t a k e u p D e d e k i n d ' s

expressed

in

r u l e (x o r y ) m b u t p r o o f s o f , ~ 3 1 , 33,

35 w i t h t h e i r argumentation regarding the elements.

Proposition ~ 3 1 , which c o n f o r m s m o r e closely to the s t a n d a r d t e r m i n o l o g y of o u r discipline, says: If x is a similar m a p p i n g of a to b a n d y is a similar m a p p i n g of b to c, t h e n 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 e m e n t s h a n d h' of a c o r r e s p o n d as e l e m e n t s of b to d i f f e r e n t images k = x; h, k ' = x; h ~ a n d these again as e l e m e n t s of c to different images l - y ; k = y ; x ; h , l ~=y;k ~=y;x;h ~, t h e r e f o r e z-- y;xis 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, a n d this by ~ to an e l e m e n t ~;~; l = ~; k = h of a, so that ~= ~;37 is s i m u l t a n e o u s l y a similar m a p p i n g of c to a. Various e l e m e n 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 l e m e n t s of b, or, as the case may be, of a, for otherwise the reverse conclusions would necessarily lead to c o n t r a d i c t i o n . T h e conclusiveness of this p r o o f rests essentially, as we see, on the justification: to express the relationship b e t w e e n image e l e m e n t s a n d object e l e m e n t s by m e a n s of a m a p p i n g rule in the form of an equationmwhich we m a d e sure of

Page 622

analytically u n d e r 26). Now the transitivity of the similarity relationship b e t w e e n the systems a, b, a n d c stated in ,~33 is also obvious. For, if t h e r e exists a similar m a p p i n g x b e t w e e n the systems a a n d b, and o n e y of b to c, t h e n a c c o r d i n g to ~ 3 1 t h e r e also exists, in the form y;x, a similar m a p p i n g z of a to c, q.e.d. S o m e w h a 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 m i g h t well recall the definition of similarity in the version f r o m 1) a n d 4):

404

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 somewhat m o r e conveniently, as in 1), take the expression given in 12) for Z^k, and w h e r e replacing the inclusion sign and its n e g a t i o n by the equality sign and its n e g a t i o n from 26) already appears justified for the theses (or propositional s u b s u m p t i o n predicates, such as k=(= z; h). If now, a c c o r d i n g to the right side of 40), which has b e e n raised to the level of a p r e c o n d i t i o n , z maps system a similarly to system b, therefore, b - - z ; a , ~; b - a is valid, and c = c; 1 :(= a r e p r e s e n t s a subsystem of a, t h e n it also follows that z; c =(= z; a, and, if z; c = d is taken, d =(=b. W h a t is to be shown now* is that this system d - z;c must be similar to c, and c o n s e q u e n t l y 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 s u b s t a t e m e n t s of o u r thesis, i n c l u d i n g Zkh, are i m m e d i a t e l y c o n t a i n e d in 40), with the e x c e p t i o n of these two: that k =(= d would be on the left, a n d h ~(= c on the right. [Already i n t r o d u c e d into l-I^ and II k by various p r e c o n d i t i o n s are h a n d k, and they t h e r e f o r e have different m e a n i n g s a priori and n o t h i n g to d o with each other. If, now, for the first h, which :(= c=g~--a, it is i m m e d i a t e l y established that t h e r e exists a k=(~-b such that z; h = k, then we have still not achieved o u r 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 b e e n proved to be a m e m b e r of system b:

( k =g~--b d - d ) + ( k -~-- b d )

Page 623

a n d for the h b e l o n g i n g to a: (h =~-ac = c) + (h=~-ac-), w h e r e b y the two possibilities are mutually exclusive. If, now, k=~-d, where d - z; c, were on the left, t h e n it would follow that k=~2?a-g ergo s k=(=g, a n d given z;h = k, ergo k=gc-z; h as well as h ~--~.; k, a fortiori: h ~ g, in contradiction to the a s s u m p t i o n 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.

4o5

FROM PEIRCE TO SKOLEM

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 6 1 0 - - a much better way to arrive at our goal, if we want to stay with an a r g u m e n t 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 t h e similarity 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 p u r e l y logical r e l a t i o n b e t w e e n t h e two, for t h e a d e q u a t e e x p r e s s i o n o f w h i c h o u r d i s c i p l i n e has t h e m e a n s . It a p p e a r s as t h e r e s u l t a n t o f t h e e l i m i n a t i o n o f t h e m a p p i n g r u l e x, o r y o r z, as t h e case m a y be, f r o m t h e 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 similarity in a n y o n e o f t h e versions. As l o n g as t h e e l i m i n a t i o n is n o t actually a c c o m p l i s h e d , t h e ~ o r l-I, i n d e e d , with t h e h e l p o f w h i c h t h e r e s u l t a n t s m a y b e r e p r e s e n t e d f o r g e n e r a l p r o p o s i t i o n s , a r e n o t e v a l u a t e d - - i n s h o r t : as l o n g as t h e n a m e o f t h e m a p p i n g r u l e c o n t i n u e s to f i g u r e in t h e e x p r e s s i o n o f t h e r e s u l t a n t as a n i n d e f i n i t e b i n a r y r e l a t i v e - - w e m i g h t still call t h e d e f i n i t i o n o f similarity an i m p l i c i t o n e . 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, u n d e r a n u m b e r 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. T h e 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

The following, however, may be added: a.~?;b+/J.z;a+(a+[,)z, as well as others. T h e n

O'. z;~+O'. ~?;z

41)

406

SCHRIDDER'S LECTURE XII

( a v , b) =E{f(z) = 0 } = [ l = E { 0 0 z

~f(z)

Or (av, b)--(L=0), where L = I I U a n d conversion of the third term, yields

~O}]=[II1;f(z);1-O]. z

42)

U=l;f(u);1, which, by 41), with the

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 similarity c o n d i t i o n s , to b e g i n with, certain (two) partial o r s u b r e s u l t a n t s can be listed. M o r e o v e r , we are in a position, for the lowest universe o f d i s c o u r s e m i n d e e d , theoretically, for every finite universe o f discourse, to p r e s e n t the c o n d i t i o n o f similarity "explicitly," in a truly closed f o r m . To this e n d , all we n e e d , 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 coefficients as II0[If(z)}ij = 0]

or

E0lf(z)} 0 = 0

e l i m i n a t i n g , by the m e t h o d s established, f r o m this c o m b i n e d null e q u a tion, in which the s u m e x t e n d s over all o f the suffixes ij o f o u r (finite) u n i v e r s e o f discourse, the coefficients Zhk o f the m a p p i n g rule o r elim i n a n t s as a w h o l e (if necessary, successively). I w a n t n e x t to offer this explicit similarity c o n d i t i o n as such for the Page 625 f o u r lowest universes of discourse. It is: (a,l, b) is e q u i v a l e n t to

43)

407

FROM PEIRCE TO SKOLEM under

11

under

1 ~2 (1 ; a -

under

11 ( 1 ; a = l ; b ) ( O # O " , a = O # O " , b ) ( O # a = O # b )

(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)

or under

1 ;b)(O#a=O#b)

(1;a=l;b){1;(l'#a) 114 ( 1 ; a = l ; b ) ( O # {1;(l'#a)

=l;(l'#b)}(O#a=O#b)

O' ; a = O # O

' ;b)

=l;(l'#b)}(0#a=0fb)

But I do not dare say "etc.," because how it will go on r e m a i n s s h r o u d e d in darkness. 1.~ would join to the four previous r e q u i r e m e n t s as yet a fifth o n e in the middle, and discovering it promises to be very instructive. A l t h o u g h a possible way to do it has already been indicated, it remains a question worthy of being posed by an academy for a prize; for, lacking truly exceptional skill, no o n e is likely to discover the trick for eliminating 25 (double suffixed) u n k n o w n s Zll, Zl2. . . . . Z55 that is r e q u i r e d here---or, as the case may be, with 20 of t h e m , insofar as the coefficients of the individual self-relatives of z (here five in n u m b e r ) would also be generally subject to elimination in the case of unqualified [voraussetzungslos] universes of discourse. It is possible to avoid the calculations to this point (up to and i n c l u d i n g 1~4), which quickly build to a terrible c u m b e r s o m e crescendo, by c o n s i d e r i n g a colu m n s c h e m a t i c - - a l t h o u g h , admittedly, not without a p p e a l i n g to a c o m m o n und e r s t a n d i n g "that can c o u n t to four (or at least to m o r e than one)." I want to give the c o m p l e t e i n t r o d u c t i o n for the final case here. As systems c o n t a i n i n g the "same n u m b e r " of elements, a and b consist of either 0, 1, 2, 3 or of all 4 elements, which, however--with the exception of the two e x t r e m e cases---can be completely different for b than for a. At the same time, 5 and /~, respectively, must contain (all) 4, 3, 2, 1 elements. T h e five possibilities that could occur for a can be r e p r e s e n t e d by the equations [Ansdtze]: a = 0 , a=i, a = i + j = h + k, a = i + j + h=fq a = i + j + h + k = l , with the u n d e r s t a n d i n g that the e l e m e n t letters i,j, h, k s h o u l d 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 o n e of these five possibilities with respect to system ais characterized by a different value system of the four distinguished relatives 0#0';a, 0 0 1 1 1

1;(l'#a), 0, 0, 0, 1, 1,

a=klc~/3~,0 we have

1;a=llll0,

a =0

by a=i by a--2i by a=3i by a--4i by because

for

1;a, 0, 1, 1, 1, 1,

0#a, 0 0 0 0 1,

n a m e l y with

0#0';a=lll00,

1;(l'#a)=

1 1 0 0 0 , 0 # a -- 1 0 0 0 0 . Page 626

Now if a consists of one full row, so does every c o l u m n of a, as an occupied one, a m e m b e r of c o l u m n category % and it will be cast off in the relative 0 0 ' ; a , which is why it then vanishes. But if a consists of m o r e than o n e full row,

408

SCHRODER'S LECTURE XII

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 114, 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 t a b l e is t h e r e f o r e j u s t i f i e d . Conversely, on the basis of 43) u n d e r 14, the value systems of the above table, as one also valid for b, again characterize for its part the composition or m o d e of formation of b as 0, lj, 2j, 3j, respectively, 4)'. And it is thus clear that in fact the four conditions taken together make up the necessary and sufficient expression of the equinumerability [Gleichzahligkeit] of the elements in a and b u n d e r 1'4" T h e first 1 ; a - 1 ; b o f t h e s e 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 h a t is, it always r e q u i r e s s i m u l t a n e o u s v a n i s h i n g o r n o n v a n i s h i n g in t h e s i m i l a 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, e x p r e s s e s , solely f o r itself, t h a 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, respectively, n o t m o r e t h a n o n e e l e m e n t ( i n s o f a r as t h e y a r e s u p p o s e d to b e s i m i l a r ) . In c o m b i n a t i o n with t h e first c o n d i t i o n , h o w e v e r , it is also t h e sufficient e x p r e s s i o n for t h e r e q u i r e m e n t that, as s o o n as system a consists o f exactly o n e e l e m e n t ( a c c o r d i n g l y , is itself an e l e m e n t ) , t h e n b m u s t also consist o f exactly o n e e l e m e n t (likewise b e i n g s o m e elem e n t ) , as well as conversely. N o w t h e s e two c o n d i t i o n s a r e (obviously) valid for e v e r y u n i v e r s e o f d i s c o u r s e , i n c l u d i n g t h e u n q u a l i f i e d u n i v e r s e o f d i s c o u r s e , a n d l a t e r we will also h a v e to d e r i v e it analytically f r o m o u r d e f i n i t i o n o f similarity Page 627

(|7). It is o t h e r w i s e with t h e last two c o n d i t i o n s u n d e r 14I in 43).

FROM PEIRCE TO SKOLEM

409

T h e 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 l e m e n t s of the universe of discourse, the same must be true of the other, and if not there, then not here either. In c o m b i n a t i o n with this last one, the second to the last c o n d i t i o n expresses 1 ; (l'a- a) = 1 ; (l'a-b) or 0a-0'; d =0a,-0';/~, that, f u r t h e r m o r e , if a contains all of the e l e m e n t s of the universe of discourse except for one, then b must also contain all e l e m e n t s with the exception of some one of them, and otherwise notmlikewise conversely. In fact, however, these two conditions 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 m a p p e d to p r o p e r parts of itself (page 596). Just as little, therefore, will these two "last" conditions be able to claim validity for the unqualified universe of discourse; they could not at all, for example, be necessary conclusions from our definition of similarity and they c a n n o t in fact be proved as such. In general, a cut has been put roughly in the middle of the s t a t e m e n t 43) to be m a d e about 14, with the dots "..." mentally inserted into the opening, which can be t h o u g h t of in the higher universes of discourse as being replaced by an always increasing n u m b e r of further interpolations and previously u n k n o w n conditions, while the first two conditions (as partial resultants) must exist and go on existing for every universe of discourse. L e a p i n g past the points ... now, the one before last and the last of the four conditions p r e s e n t e d u n d e r 14~ must retain their validity in every finite universe of discourse and c o n c l u d e 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 beingfollowed here, the explicit definition of similarity can ever be e x t e n d e d (to a full resultant), which it still lacks, by a d d i n g further and further conditions (as partial resultants) to the first two [conditions] u n d e r 141 in 43), then the two "last" conditions will "never come" from t h e r e . m The

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.

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 l i n e s u n d e r t h e r u l e in 31), as we said: Similar systems

45) t h e first two

can only vanish Page 628 simultaneously and must otherwise be altogether different from O. It c a n b e i n f e r r e d easily a n d in a v a r i e t y o f ways f r o m o u r d e f i n i t i o n o f similarity. First, we have, for the two main conditions of (17) a =(c-z; b, b 4= z; a, the single resultants (the elimination of z): a=(=l ;b, b=~=l ;a. Since, however, a c c o r d i n g to

410

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 combination of these two yield precisely the equation 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 a n o t h e r way of writing the same resultant. However, the two main conditions 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 t h e s e 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 c o n c l u s i o n - - i n s t e a d 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 condition (14) appears to be completely eliminated t o g e t h e r with ~b(x)=y. However, we already had the o p p o r t u n i t y in a footnote on page 287 to stress what we here find confirmed: that when x appears merely as an element, an a r g u m e n t of an expression ~b(x), the resultant of the elimination of q~(x) does not represent by a long shot the resultant of x, but merely a subresultant of it. For the condition d; a =/~; b is far from sufficient to g u a r a n t e e the similarity of a and b. Rather, it is merely equivalent to 44).] In fact, d; a = 1 ; a a - 1 ; a, and consequently the statement 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 p r e s e n t e d in 31) by multiplying the last [resultant] by the converted [resultant] on both sides, and which has yet to be the same also by shifting it to the form d ; l ' ; a - b ; l ' ; b , justified. As t h e second c o n d i t i o n a n d partial resultant was (a~b)~-(0'0

Page 629

~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 h a v e a l r e a d y b e e n l i s t e d t o w a r d t h e e n d o f 31). W h a t it s t i p u l a t e s we a l r e a d y d i s c u s s e d in t h a t c o n t e x t . To p r o v e it f r o m 1 7 ) n w h i c h is w h a t we a r e o b l i g e d to d o n o w n w a s n o t all t h a t easy. I will give three proofs, and also make mention of a failed a t t e m p t that was instructive insofar as it discloses n u m e r o u s new expressive forms and relations. T h e first thing to note is, according to the most well-known propositions about systems, that c~;0';a = 1 ;a(O';a) - - 0 ' d - 0 ' ; a m t h e latter according to 30), page 916. T h e 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, furthermore, 0~z, as well as a" 0'" a (= aO'" a), is a system, the distinguished

FROM

PEIRCE

TO

411

SKOLEM

relative O' 0'- O' 9a 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 further the distinguished relative 1 ;a(O';a) will not be 1 when a(O';a) =0, which is why the same ones [expressions] must also be equivalent with (a" O ' ; a = O ) = (b" O';b=O). With this, the forms p r e s e n t e d u n d e r 31), insofar as z does not occur in them, a p p e a r to be etc., traced back to each other. Because b = z ; a , a=ff;b, O';z =I(=s and s we can conclude: Oj- O' "b=Oj- O' " z" a=(c-O j- s a = O j- i" ~" b=(c-O a- O' " b, Oj-O"a=O0,-

0 / "Y.;b=(c-Oj-z;b=O,t-z;~.;a=gc-Oj-O

t "a.

47)

Because the initial subject and final predicate coincide here, all i n t e r m e d i a t e termini must be equal to them and a m o n g themselves, whereby, when we also take into consideration that z; a = z; 1 and ~; b = ~; 1, it yields a long list of o t h e r forms of expression p r e s e n t e d in 31) in addition, as soon as o u r assertion 46) is proved in some o t h e r way. It c a n n o t itself be gained in this way, unless we should m a n a g e first to d e m o n s t r a t e equality between some o n e expression of the o n e and the o t h e r line of 4 7 ) m s u c h as, for example, the e q u a t i o n 0ji; 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 p e t i t i o p n ' n c i p i i . Meanwhile, with 48) a n d proposition 46), the rest of the expressive forms in 31) are s e c u r e d m w i t h the exception 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). C o n c e r n i n g failed attempts, it will be n o t e d that it is i n d e e d possible to show (l:(=0';a) =(z ~-0';a),

Page 630

(l:(=s

=(a=(=0';a),

49)

but with that appears the real difficulty of bringing the two s t a t e m e n t s closer together. T h e 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. T h e second thus:

SCHRODER'S LECTURE XII

4x2 (1 :~-s

={1 :(= (s

a=~-s

a}=(=(1 =(=s

a)

= (27' 1 =(=0" a). O n 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 t h e r e would i n d e e d exist a ~, respectively z of that sort, as m a p p i n g a c o m p l e t e l y identically to itself, it would be a different o n e m Z - - t h a n the z that maps o u r a to b (and possibly even to a). T h e r e f o r e , this is not the way that the goal is to be reached.] To first prove proposition 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 r d e r for Et01tal = a a + alj + "'" (without a;) to be equal to 1, every l t h e r e m u s t be more than one I for which a t equals 1. For if only one a~ma~ s a y - - w e r e equal to 1, then for this i the sum would vanish, and if t h e r e were no a t equal to 1 at all, then it would vanish for every i. T h e r e exist t h e r e f o r e 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 t h e r words, both h=(~-a as [well as] m=(~--a with h ~: m. A c c o r d i n g to the original version of the c o n c e p t of similar m a p p i n g z t h e r e exist now a k ~=b and an n ~=b such that k -- z" h and n = z" m, and i n d e e d , because h ~: m, it must be that k ~ n as well. T h e r e are t h e r e f o r e for l at least two bt, namely, b~ and b,, that are equal to 1, and thus Et01tb ~is also equal to 1 for every i, q.e.d. A n d conversely. This proof, a l t h o u g h binding, is not satisfying m e t h o d o l o g i c a l l y because it is based on r e a s o n i n g with regard to the e l e m e n t s and the distinction between the singular and plural a m o n g them. Analytically, a s e c o n d p r o o f of o u r 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" a d ; ~ ? = 0 " z" ( l ~ d + 0~a') "~?=0" z" l ~ d ' ~ ? : ( = 0 " z" l " i = 0" z'i~0' 91'= 0, t h e r e f o r e 0'b/~=0 as well, q.e.d. T h e same conversely. A third p r o o f 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 9z" 0 " 27; 1, b" 0 " b = 1 9~; 0 " z" 1, c o n s e q u e n t l y showPage 631

ing the equality of the two right sides, or to also transform the first of the four expressions into the last one. We also have: d ' 0 " a - 1 "0~d" 1 = 1" 0'(~?; 1 "z)" 1. Breaking down the m i d d l e 1 h e r e 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', t h e r e f o r e 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. T h e relation used h e r e a l o n g the way z . 0 ' . ~ + i; 0' 9z=(=0'

50)

FROM

PEIRCE

413

TO SKOLEM

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 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 I

~hkZhiOhkZki = ( 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 o f the f u n d a m e n t a l c o n d i t i o n s o f similarity or cardinality, incidentally, is simply w h a t o c c u r r e d to m e o n the first a p p r o a c h to t h e p r o b l e m , a n d it is n o t yet n e c e s s a r y to give u p h o p e t h a t with a d e e p e r t r e a t m e n t of t h e e l i m i n a t i o n p r o b l e m as t h a t c h a r a c t e r i z e d o n p a g e 624---following t h e m e t h o d s at the e n d of w 29 w h i c h s h o u l d 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 o b t a i n a concise e x p r e s s i o n for t h e explicit c o n d i t i o n in g e n e r a l . This would, of course, be a great triumph for our discipline: we can, propaedeutically, 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. N o w that, in the f o r e g o i n g , we have also d e a l t f r o m t h e s t a n d p o i n t o f o u r t h e o r y with e v e r y t h i n g t h a 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 invertible m a p p i n g s , we w a n t to t u r n to t h e p r o p o s i t i o n s in the text t h a t refer to the s i n g l e - v a l u e d m a p p i n g in possibly o n e d i r e c t i o n 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. ~ 2 1 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 elements h by a relative x.

SCHRODER'S LECTURE XII

414 To be quoted from the outset, mutatis mutandis:.

By a single-valued mapping x of a system a = a;1 we understand 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 l e m e n t i of system c, which i possesses as an element of a; at the same time the system, which consists of all images x; i, should be called the ximage 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 w a n t n o w to f o r m a l i z e t h a t w h i c h is c a l l e d for in t h e f o r e g o i n g c o n c e r n i n g t h e s i n g l e - v a l u e d m a p p i n g o f a system a in a way t h a t is p a r a l l e l to w h a t has b e e n said a b o u t s i m i l a r m a p p i n g s - - - o f several types a n d o n c e a g a i n relative with r e s p e c t to a d e t e r m i n a t e s e c o n d system b as t h e r e c i p i e n t , r e c e i v e r o f t h e x-images o f a. H e r e [it is i m p o r t a n t ] n o t to c o n f u s e a ( s i n g l e - v a l u e d ) 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." F o r t h e f o r m e r e a c h e l e m e n t o f b ( a n d also c o n v e r s e l y ) w o u l d have to be an x - i m a g e o f e l e m e n t s o f a. F o r t h e l a t t e r it is n e c e s s a r y o n l y for every e l e m e n t o f a also to h a v e an x - i m a g e inside b, a n d it is this latter, as t h e less n a r r o w c o n d i t i o n , t h a t is o f i n t e r e s t to us next. As t h e 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 s u c h t h a t x singlevaluedly maps system a to system b a p p e a r s t h e following: F o 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 t h a t is its x-image, while at t h e s a m e t i m e t h e c o n d i t i o n Xkh is fulfilled t h a t e v e r y e l e m e n t n o f b t h a t differs f r o m 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 i n s i d e o f b t h e r e s h o u l d be o n e a n d o n l y o n e e l e m e n t k w h i c h is :(= x; h. T h i s by itself will in n o way p r e j u d i c e t h e e x t e r n a l b e h a v i o r o f x t o w a r d a and b.F r o m t h e above, we have n o w IIh{ (h :(=a) :~--F~k(k:~--b)(k:~--x; h)Xkh}, where

Xkh=II,,{(n:~-b)(n :/: k):~--(n4~x;h)};

thus

51)

IIh(a-h + ZkbkXkhXkh),

415

FROM PEIRCE TO SKOLEM

w h e r e Xkh = II,,(D,, + lk,, + Y,,h = {1'0~ (/~ + X)}kh, a n d we let xX = y, thus yielding:

I-Ih(dh + Ekbkykh) = 0 ~ (a + 1 ; by) j- 0 : [J; y j- d = (a d"y; b) = (a ~--~; b). O u r result is thus: a=~-lxo ~(l'+/~)}~;b

or

a0 ~{x0~ ( l ' + / ~ ) } ~ ; b

or

a=(=)7;b,

52)

w h e r e y = x{l'0 ~ (/) + x-)} = x{l' +/~) o~ ~)}. 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 x, a a(=l ;b m u s t be valid; that is, b may not vanish without a. We want to stipulate f r o m h e r e o n that this is fulfilled. It will b e c o m e e v i d e n t that it was the full r e s u l t a n t , that is, that it will always be the case that for every [ s o m e h o w ] given a a n d b, t h e r e will exist a m a p p i n g x that satisfies t h e c o n d i t i o n : Every system

a can be mapped single-valuedly to a nonvanishing system b. F r o m the e q u a t i o n for y, it is now possible to e l i m i n a t e x as follows: y=(=l'j-(/~+ ;), therefore, by'O';y=O

y~--x,

O;y~(=/~+ ;,

[or also b x ~ l ' # ~ ,

by :~--l' j- ~,

by ; y:~c--l',

b" O';y=(=;=(=)7;

t h e r e f o r e afortion]: b;y;y~(=l',

a n d this r e s u l t a n t that c o m e s in two f o r m s is the full o n e , for, if it is fulfilled, t h e n 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 e p l a c e d e q u i v a l e n t l y by

(b" y ; ~ @ l ' ) ( a @ ~ ; b )

or

(by. O';y=O)(a=(=y;b).

52,~)

this, however, it follows with y; a@b, namely: y; a ~ - y ; f ; b = (y;y')D;b:~--l';b= b, b e c a u s e c o n v e r s i o n also yields b'y;~:~c--l'.

From

We w o u l d thus like to n o t e o u r result in the "full" f o r m as well: {(b+/~) " y ; ~ - - l ' } ( a ~ - f ;

b)(y; a@b).

Page 634

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

416

S C H R O D E R ' S LECTURE XII

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 afortiori the earlier conditions 53), 54). T h a t 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 condition 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).

Page 635

56)

This approach is also an expression for the fact that system a is m a p p e d 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 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 condition 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 element 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 t h e r

FROM

PEIRCE

417

TO SKOLEM

e l e m e n t s k' o f bb, w h e t h e r as x - i m a g e s o f h, o r w h e t h e r also as m e r e l y in t h e r a n g e x; h as s u c h , 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 h a v e to follow. T h e n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n f o r a to b e s i n g l e - v a l u e d l y 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 b e

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 u s e t h e s c h e m a 0) o r ~j) o f w 30 f o r t h e e x p r e s s i o n o f t h e 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 f o r 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 h a v e a l o o k at b o t h o f t h e m . W i t h t h e first o n e , we g e t / ~ ; x ( l ' ~ ~) ~ d = act ( x ~ l ' ) ~ ; b = { a @ ( x ~ 1 ' ) ~ ; b} = (a:~--~;b){a:~--(x~ 1 ' ) ; b } ,

58)

a n d with t h e s e c o n d , in c o n 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 altogether 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, positioned as a predicate to 1, also maps to the third, according to the first inversion theorem. T h e equivalence of this to the last subsumption and fourth form, however, rests u p o n a general proposition: (ci,j- l')a" b = (dd- 1') 9b" 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 proposition 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, considering that all terms in which l ~ h must drop out of the latter d o u b l e sum because if k ~: /, then aih becomes an effective factor of the rl, t o g e t h e r with a~h; and if we now let l--h, then Rq coincides completely with L O, q.e.d. Page 636 This proposition obviously combines with certain propositions given in w page 525ff. For 59) we have, first,

SCHR()DER'S LECTURE XlI

418

IIh{d h + E~,bj. f~'x'h. (/~j-s h)} =IIh{6 h + F.qfi;bx'h. (]~j-~" h)}, because, namely, bk = b,h--k" b" h as well as bl,xk^- (bx)kh, etc. A c c o r d i n g to this, o u r next step is the auxiliary task of d o i n g the s u m m a t i o n in the last term, that is, conceived s o m e w h a t m o r e generally, to carry o u t in closed form a sum of the following form z = X j ; a" ( i # b) for any a,b. For purposes of solving it, we form its g e n e r a 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,, w h e r e b y z has b e e n found. T h u s it is worth n o t i n g the proposition:

{

~ , i ; a " (i'j-b) =1 ; a ( l ' j - b ) ,

II;(i;a+~'b)

Zia'i

IIi(a'i+b'z)=b'O'+a)j-O.

(b~i)=a(b~l'),

=0o,-(a+0"b),

61)

A c c o r d i n g to the first of these schemata, o u r ~, is equal to

1 "(bx" h)(l' j- s h) =/~" bx" (l'j- x')" h =/~; (x~ 1')" xb" h --cf. 9) of w p. 444 and 27) of w p. 419. [ A c c o r d i n g to this, in particular for b = 1, we may note the schema: Ek(x" h = k) =/~" (xj- 1') "x" h = {(xj- 1') 9x}^^.]

62)

With this, o u r c o n d i t i o n b e c o m e s IIh{(d + (xj- 1')"xb}hh=a j- l'{(xj- 1')"xb}" 1. For if, for a m o m e n t , we call the c o n t e n t s of the braces c, a n d the s e c o n d t e r m in t h e m d, then, first, Hhchh = Hhk(l~; 1)^j, = O,r 1~" 1, because the j-0 at the e n d can be suppressed. Moreover, 1~:' 1 = 1'5" 1 + breaks down and, since h e r e 6 is a system, we have 1'6; 1 = 6" 1'; 1 = 6, which arises 0 j- ~ + 1~/; 1) =a j- 1~/; 1, which is what was to be shown at the O u r c o n d i t i o n thus c o m e s out as a=~l~/; 1. Now, it is worth n o t i n g that a c o n d i t i o n for any d, if a is a system, m u s t be equivalent to the s i m p l e r d, because, t h e n (l~=(=d) --(a" 1 : ( = d + 0') : {a=(=(0' + d) j - 0 : 1~/" 1}. Accordingly, it is possible to note the proposition"

1~/" 1 from start. such 1~=(=

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 f o u n d last, the second Page 637

form 59) of o u r condition, q.e.d. In o r d e r to derive the two forms 58) and 59) directly from each other, we can first go with the distingished relative and second go with the s u b s u m p t i o n of o u r condition. In regard to the first, it is easy to prove for every a, bb, c from the coefficient evidence of the proposition 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 r o d u c t is a system because b -- b" 1, and if we call that system e, then 1~" 1 = e, thereby t r a n s f o r m i n g the distingished relative 59) into 58) without f u r t h e r ado, q.e.d. In regard to the latter, the proposition (l~z; 1 :(= l'b; 1) : (l~z~(=l'b) = (1 ; al' :(:: 1 ; bl'), (1~- 1 = l'b- 1) = (1~ - l'b) - ( l ' a l ' -

l'bl')

65)

is to be established as valid for any relatives a, bb Of these equivalences, only the first o n e needs to be p r o v e d ~ p r o v e d , indeed, as a forward s u b s u m p t i o n , since it is self-evident as a reverse one. To be proved h e r e is

L=(l'a,~-l'b;1j-O) = (l~=(=l'b) as =(=(1 ;a=(=l'bl) = R . This can be d o n e with L = L(l~z=(=l') = (l~@-l'b; 1 91 ' - l'b) = R. In o r d e r now to obtain from the first s u b s u m p t i o n 5 8 ) i n l e t us call it L - - t h e last o n e 59)--call it R m a n d conversely, we conclude by using the above abbreviations d = (xd- 1');xb and e = ( x d - l ' ) ~ ; b as follows: L=(a~(~-e)~(l'a;1 :(~1~; 1), where now, according to 64), 1~= 1~/ must be valid, t h e r e f o r e 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, either 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 s u b s u m p t i o n s in 58), 59) is a l s o - - f o r a = b = l m t h e reference back of the two e x t r e m e forms of the characteristic 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 h o u g h t by which I f o u n d 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 initial e q u a t i o n [Ansatz] 58) o r 59) is likewise a n e x p r e s s i o n f o r t h e c o n d i t i o n t h a t a s y s t e m a c a n b e s i n g l e - v a l u e d l y m a p p e d to b by x. T h i s x o b v i o u s l y n e e d n o t e v e n b e a m a p p i n g in t h e s e n s e o f w 30, f o r

420

SCHRODER'S LECTURE XII

t h e c o n d i t i o n c o i n c i d e s with t h e c h a r a c t e r i s t i c o f n o t o n e o f o u r fifteen types. However, this c o n d i t i o n is essentially d i f f e r e n t for all p r e v i o u s 51), 52), 53), 55), which can be u n d e r s t o o d in t h a t it o n c e a g a i n allows Page 638 a n o t h e r e x t e r n a l r e l a t i o n o f x with r e s p e c t to a a n d b. If, however, we let x ( l ' 0 ~ ;) =y, t h e n we g e t a=(=37; b, w h e r e y = x ( l ' ~ ;) o r y ;~=(=1',

66)

a n d c a n g e t f r o m h e r e back to o u r n o r m a l f o r m 55) by a p p l y i n g t h e e q u a t i o n [Ansatz] dby = z. It is also possible to s h o w with f o r m 59) that, if an x satisfies t h e c o n d i t i o n , t h e n it m u s t also be satisfied by z = dbx, a n d c o n v e r s e l y ( w h e r e t h e c o n v e r s e is i m m e d i a t e l y o b v i o u s for x - z). To be s h o w n , t h e r e f o r e , o n t h e basis o f 59), is t h a 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. C o n s e q u e n t l y , t h e assertion b r e a k s d o w n i n t o l~z=(=d, w h i c h is o b v i o u s bec a u s e l~z = 1'6, a n d l~z~=ld ; bx + {(/~ + x) 0~ 1'} ; bx, w h i c h , b e c a u s e o f the c o n t a i n m e n t [Einordnung] o f l~z in t h e t h e und e r l i n e d p a r t o f t h e r i g h t - h a n d side, is a l r e a d y valid a fortiori by virtue o f 59). N o w 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 f o r similar [ m a p p i n g s ] c o r r e s p o n d s f o r t h e m e r e l y single-valued m a p p i n g w h i c h was n o t explicitly s t a t e d by D e d e k i n d as m u c h as t a k e n u p i n c i d e n t a l l y in ,~21: P r o p o s i t i o n . If a system a m a p s single-valuedly to b by x, or, respectively y o r z, t h e n by t h e s a m e m e a n s will every s u b s y s t e m c o f a also m a p single-valuedly to b. Proof. This follows (for c = c ; 1 ) with c=(~--afrom 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 - a c , ~= (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

FROM PEIRCE TO SKOLEM

4 2 1

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 " D e f i n i t i o n a n d P r o p o s i t i o n " ~ 2 5 has to d o with t h e "Zusammensetzung," c o m p o s i t i o n o f two single-valued m a p p i n g s i n t o a third, as well as t h e associative law g o v e r n i n g s u c h 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. T h u s , for us, t h e a s s e r t i o n o f t h e transitivity o f t h e s i n g l e - v a l u e d m a p p i n g r e m a i n s as t h e c o r e o f t h e 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 similar 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 singlevalued mapping is understood in merely "relative" terms: as such from one welldetermined system to another. To be established here is: P r o p o s i t i o n . If a system a is m a p p e d by a relative x single-valuedly to a n o t h e r system b a n d t h e latter is m a p p e d single-valuedly by y to a system c, t h e n system a will be single-valuedly m a p p e d to c by t h e relative c o m p o s e d o f t h e two t o g e t h e r (z =)y; x. This is valid in fact for t h e 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" single-valued m a p p i n g s a c c o r d i n g to v e r s i o n 55), w h e r e for z - - y ; x we have t h e 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 (conspicuously) 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 t a k i n g b = 1 in 55), we can also c o n c e i v e o f this d e f i n i t i o n o f single-

422

S C H R O D E R ' S LECTURE XII

v a l u e d m a p p i n g as o n e t h a t is m e r e l y relative with r e s p e c t to t h e o b j e c t o f t h e s a m e [not, however, also, like 55), with r e s p e c t to t h e i m a g e o r t h e r e c i p i e n t o f the latter], a n d i n d e e d in t h e f o r m : (z; i:(=l')(a:~-~?; 1)(z:(=a') = (z ; i=(=l')(z:(=6 = 1 ;z)

67)

w h i c h is t h e n o r m a l f o r m for 54). With this version we can also easily p r o v e the p r o p o s i t i o n : If system a Page 640 is m a p p e d single-valuedly by x, w h o s e i m a g e 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 t h e last p r o p o s i t i o n can also be d e l i v e r e d 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 t h e e l e m e n t s in p r e c i s e conn e c t i o n to D e d e k i n d ' s r e a s o n i n g , in t h a t we p u t v e r s i o n 57) for b = 1 at t h e 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 t h e u n d e r l i n e d e x p r e s s i o n s are s u p p o s e d m e r e l y to be r e m a r k s t h a t c o u l d also be s u p p r e s s e d ; w h e n p r e s e n t , however, t h e y m a k e it possible to r e c o g n i z e t h a t t h r o u g h t h e thesis o f t h e first p r e m i s e ( b e f o r e Ilk), t h e h y p o t h e s i s o f t h e s e c o n d o n e (after t h e Ilk) is s i m u l t a n e o u s l y c e r t a i n to be fulfilled. To s p e a k generally: If for every element h of system a there exists one (and only* one) element 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 element 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.

423

FROM PEIRCE TO SKOLEM

l~z=(=~; (l'j- x-) and 1" x;a=r

(l'j-37)

the conclusion l~=~-~;f; (l'j- 3~0,-~). T h a t task is hereby r e c o m m e n d e d to researchers, and a similar task could also be c o n n e c t e d to version 58) for b = 1. In regard to the latter, o n e 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=(=~; ~; x m w h e r e b y the conclusion now follows afortiori. T h e o t h e 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 i n d e e d must be possible only if, on the left, we retain the premises from the previous assertion. Page 641

If a s y s t e m a s i n g l e - v a l u e d l y m a p s to a s y s t e m b, t h e n t h e r e m u s t e x i s t 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 i v e n by t h e e l i m i n a t i o n o f x o r z, r e s p e c t i v e l y , f r o m t h e v e r s i o n o f o u r r e l a t i v e m a p p i n g d e f i n i t i o n . T h i s is

1 ; a=(=l ;b

68)

a n d says solely t h a t b m a y n o t v a n i s h 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 o p us f r o m u s i n g it as t h e i m a g e o f e v e r y 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 fulfilled, it is also p o s s i b l e to o b t a i n t h e m o s t g e n e r a l r e l a t i v e x t h a t s i n g l e - v a l u e d l y m a p s a to b, a n d b e s t s u i t e d f o r t h a t is v e r s i o n 59) in t h a t in t h e 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 h a v e t a k e n t h e r i g h t side 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 o n l y t h e c o e f f i c i e n 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 completely u n d e t e r m i n e d . T h e system converse 1 ;dl' therefore receives an arbitrary definition in x, or 1 ; 41' 9u 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 o n e term bh,x^fiafiB,... (without ~^;) 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~ = bh = 1, while for o t h e r [values of] h it could also be that bh; = bh =0. T h e n we n e e d merely to keep an bullt out in this c o l u m n i" for x, on any one of the places where it is

424

SCHRI~DER'S

LECTURE

XII

intersected by the full rows of system b, while all o t h e r 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 representing every root of proposition 59). Only for the case b = 1 this seems to be possible without considerable additional effort. T h e situation in this case is, namely, that the columns of the system converse 1 9al' at x can only be simply occupied; that is, we have x=l'dl"

u+ l'al"

f

if f represents the most general "function " that is, a relative with all occupied columns. T h e (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 "dl'=a,

Page 642

l "a l ' =

gL,

in that, e.g., 1 " a l ' = 1 "51'= 1 "(1 "d)l'= 1 91" 1 "d-- 1 9d, and thus x=au+ -(au+

a[u(l'a- ti) + {O,t- ( a + 0'" u)}l']

69)

1' d- zi)u + al'{O o'- (*i + O' u)}

is the general root of the condition l~z=(=)~. (1' cl-:~) or a=(=l . x 1. (1' .,J- :?) . for a

=

a 1.

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 t h e c o n c e p t o f t h e " s i n g l e - v a l u e d " m a p p i n g , it is also p o s s i b l e to c o n c e i v e o f t h e "identicalential r e c i p i e n t o f t h e 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 " a b s o l u t e , " as it has b e e n u p to this p o i n t as l ' - - - n a m e l y , g i v e n 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 s y s t e m a as t h a t o f t h e s u b s t r a t e ( o f t h e o b j e c t as well as t h e i m a g e ) o f t h e m a p p i n g in t h e u n i v e r s e o f d i s c o u r s e . 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 r e l a t i v e x will be c a l l e d a n i d e n t i c a l m a p p i n g with respect to a s y s t e m a = a ; 1 - - n o t o n l y in t h e e n t i r e x ; a = a, b u t r a t h e r i f - - t o e v e r y e l e m e n t h o f a, h is its o w n x - i m a g e - - w h e r e t h e e l e m e n t s o f d c a n b e m a p p e d a n y w h e r e , if at all. This condition 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. T h e last two II split into I I ^ ( d + x ) h h - O d - ( d + 1%'1) and II^(d+ l'd-

FROM

PEIRCE

425

TO SKOLEM

= 0 j - (1' + x-) j- d, which is x-),h = 0,t-{d + l'(l',t- s 1)}, respectively, II,(d h + [z3- s obtained 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 altogether: l'd~(=x~(= 1' + a,

73)

from which, according to the rules of the identity calculus, we can calculate x=l'~+u(l'+a) x = 1'ci + u(~ Page 643

or: 74)

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 ( ~ 3 0 ) 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 m a p p i n g (for b = a ) m a s 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 n o t h i n g at all to prejudice the external behavior of m a p p i n g rule x. With o t h e r versions, in contrast--- such as (10), for e x a m p l e - - w h e r e there is some d e g r e e 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 t h e " n o r m a l f o r m " o f t h e r e l a t i v e l y i d e n t i c a l m a p p i n g o f a, we a d d t h e c o n d i t i o n x; d = 0 (as a n a d v e n t i v e c o n d i t i o n ) to t h e c o n d i t i o n s we h a d t h u s far, t h e n it m u s t b e t h a t u a ~ - g , t h e r e f o r e u a ~= ~ a = 0, a n d

x=al'

75)

r e m a i n s as a n e x p r e s s i o n for t h e fully determined m a p p i n g t h a t m e r e l y m a p s a identically. T h e s a m e w o u l d also m a p e v e r y 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 i d e n t i c a l l y " ) . O b v i o u s l y it also satisfies t h e s i m i l a r i t y 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 " v e r s i o n (17). I I n s o f a r as p a r t o f t h e g o a l we h a d in m i n d was t h e e m b o d i m e n t in o u r d i s c i p l i n e o f t h e d e f i n i t i o n s , 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 Ded e k i n d ' s t e x t u p to t h e p o i n t i n d i c a t e d o n 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 h e p r o p o s i t i o n s

426

SCHRODER'S LECTURE XII

21,

25,

26, 27, 28

o n page 638, 639ff, 615,

29,30

31,

32,

33,

34, 35

616,

621, 598607 622, 610(622),

are taken up, r e p r e s e n t e d , a n d dealt with, with the m o d i f i c a t i o n s necessary f r o m the s t a n d p o i n t o f o u r discipline. To convey an idea, however, of the multiplicity o f the c o n d i t i o n s that i m p o s e themselves, at the same time p r o v i d i n g the s t u d e n t with m o r e c o m p r e h e n s i v e practice m a t e r i a l - - w h e t h e r in w o r d i n g c o n d i t i o n s in the f o r m o f affirmative or also n e g a t e d s u b s u m p t i o n s as well as o f disting u i s h e d relatives; w h e t h e r in the i n t e r p r e t a t i o n o f the l a t t e r - - w h a t we w a n t to d o n e x t is p r o c e e d t h r o u g h a n u m b e r o f the m o s t n o t a b l e c o n d i t i o n s , p r e s e n t i n g t h e m f o r m u l a t e d in the symbolic l a n g u a g e o f Page 644 o u r discipline. Since it is always the case that 0 =(= x; a, it is m e a n i n g l e s s to r e q u i r e that "one" x-image o f 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 f o r m x ; a = 0 o f "the" x-image of a. Now, if a = a ; 1 a n d b = b;1 are systems, a n d if in e n c o d i n g the conditions we m a k e use o f the d e s i g n a t i n g m a p p i n g rules given in the c o n t e x t o f p a g e 597, t h e n what follows i m m e d i a t e l y is an overview o f obvious possibilities for the c o n d i t i o n s a n d their f o r m u l a t i o n . 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 e m e n t s of a whose x-image does n o 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 l e m e n t s of a vanishes: c~,, = IIh{(h=~--a) =~--(x; h =/= 0)}= 1 ; x j - d = (a:~--~,;1).

T h e r e exist e l e m e n t s of a whose x-image does vanish: &2 = F , h ( h ~ a ) ( x ; h = O ) } -

(0 j - ~ ) ; a =

(a~;

1).

To be n o t e d is that the universal j u d g m e n t s o~ a n d og2 are also valid for a = 0, that is, w h e n t h e r e exists n o e l e m e n t o f a at all. For, in the spirit o f o u r a l g e b r a of logic of "every" e l e m e n t o f a is valid, as we know, for e v e r y t h i n g conceivable: b o t h that it vanishes as well as that it does n o t vanish. In that case, or, a n d og2 b e c o m e consistent ( m e a n i n g : that the xi m a g e o f each e l e m e n t of a does n o t vanish) . . . . :~

The rest of the conditions (pp. 644-648), closely analogous to his first example, are omitted here.

427

FROM PEIRCE TO SKOLEM

Page 649

In o r d e r once m o r e 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 n o r m a l form (17) of the similarity condition to a system b that is similar to a and is conceived of as =(=a. It would be p o s s i b l e m u n d e r s t a n d i n g u(= u ; 1 ) as an u n d e t e r m i n e d s y s t e m ~ s i m p l y to enter b = ua in that formula, having only to associate with the previously written E a n o t h e r E. Better, however, if we add to this the condition b ~ a and do comp~'etely away with the n a m e b by replacing it wherever it appears with the equivalent z ; a . T h e n 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 u n d e r l i n e d factor expresses the adventive condition and as such could also be suppressed. The same, however, could be replaced yet m o r e 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 n e e d e d as such. Formula 76) now forms the point of d e p a r t u r e for further i m p o r t a n t 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 n u m b e r theoretical as well as arithmetical c o n c e p t s ~ i n c l u d i n g "being ordered," "being discrete," "being dense," and "constancy," e t c . m o f an amount, and how the goals of reasoning and p r o o f can be advanced by such representation.

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Appendix 8" Norbert Wiener's Thesis

Introduction O n e of the brightest lights in American mathematics of the first half of the twentieth century was N o r b e r t Wiener, f o u n d e r of cybernetics, the m o d e r n theory of Brownian motion, and of the Paley-Wiener Theory, a m o n g many other accomplishments. Wiener started his career with a thesis at Harvard (at the age of 18) in mathematical logic, a n d subsequently traveled to Cambridge, England, to study u n d e r Russell. Wiener's thesis turns out to be a careful examination of S c h r 6 d e r ' s algebra of relatives vis-fi-vis Russell's t r e a t m e n t of relatives in Principia Mathematica. Wiener quotes Russell's disparaging remarks a b o u t Schr6der, and then with meticulous detail makes the case that the o r d e r of propositions and proofs regarding relatives in S c h r 6 d e r was essentially copied into Principia Mathematica with an adjustment of basis from classes to propositions, a distinction without m u c h essential difference. In o t h e r words, he is saying that his m e n t o r Russell did n o t give credit where credit was due. He also makes a case, as L 6 w e n h e i m 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 m a d e before. This is natural if we think of the Boolean algebra o r d e r i n g of the subsets of a power set u n d e r inclusion as the basic datum. T h e accusation of confusion was m a d e by Alessandro Padoa and implied by Russell, a n d Wiener refutes it. However, Wiener also notes that S c h r 6 d e r misread Peano so far as the e relation goes. H e r e we give the introduction and last c h a p t e r 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 a u t h o r to begin a serious and detailed study of the Algebra of Relatives, but his work was so h a m p e r e d by an inordinately complex and awkward symbolism that it thereby lost m u c h of the value which it might have had to other workers in this field. Schroed e r took up Peirce's work, and developed it in a form at once m o r e m a n a g e a b l e and m o r e sequential. After S c h r o e d e r 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 d o n e on this subject. Since Russell claims that his work is d o n e on a foundation different from that of Schroeder, and since their symbolisms are almost totally dissimilar, the question of their connection and the translatability of the formulae of the o n e into the terms of the other becomes of the first importance. Russell says 2 of Schroeder, "Peirce and S c h r o e d e r have realized the great i m p o r t a n c e 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 o u g h t to be m a d e are practically not feasible. In addition to the defects of the old symbolic logic, their m e t h o d 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 s u m m a t i o n 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 r e a t m e n t of relations. This treatment, w h e t h e r m o r e philosophically correct or not, is certainly far m o r e convenient and far more powerful as an engine of discovery in actual mathematics." Without any desire to belittle in any degree the magnificent work of Russell, I would like to raise the question w h e t h e r the advances which he has m a d e 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.

FROM PEIRCE TO SKOLEM

431

a n d m a r k such a radical d e p a r t u r e f r o m the d i r e c t i o n o f work p o i n t e d o u t by S c h r o e d e r as he t h e n s e e m e d to think. It is p e r f e c t l y t r u e that m u c h of S c h r o e d e r ' s work is c u m b r o u s a n d difficult, :~ a n d that, o n the whole, Russell's work in the Principia Mathematica is c o m p a c t a n d simple, b u t even h e r e it is an o p e n q u e s t i o n to m e w h e t h e r , in g e n e r a l , w h e n S c h r o e d e r a n d Russell treat 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 S c h r o e d e r ' s r e g a r d i n g a r e l a t i o n 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 true that S c h r o e d e r r e g a r d s a relative as a sum o f relatives which are of such a n a t u r e that they h o l d b e t w e e n the two terms o f a u n i t c o u p l e , w h e r e a s Russell r e g a r d s a relative as a class whose members are u n i t couples, b u t to m e n o vast d i f f e r e n c e in t h e c o m p l e x i t y o f ~(xRy) a n d EoR!~(i" j) is a p p a r e n t . In fact, as I shall show, n o t only are these two e x p r e s s i o n s equivalent, b u t any o p e r a t i o n o n the o n e can be c a r r i e d o u t o n the o t h e r with exactly the s a m e ease in an exactly parallel manner.4 As to S c h r o e d e r ' s t e n d e n c y to r e d u c e relative p r o p o s i t i o n s to class p r o p o s i t i o n s , he is perfectly willing o n occasion to r e d u c e class p r o p ositions to relative p r o p o s i t i o n s , as in his w h o l e t r e a t m e n t of systems, 5 and, as a m a t t e r of fact, it seems to be m e r e l y for the sake o f c o n v e n i e n c e , entirely a p a r t f r o m any metaphysical c o n s i d e r a t i o n s , that S c h r o e d e r develops his t r e a t m e n t of relatives f r o m that of classes a n d p r o p o s i t i o n s . S c h r o e d e r tells us why he starts with s u b j e c t - p r e d i c a t e p r o p o s i t i o n s . H e says: '~ " B e g i n n e n wir s o n a c h damit, die Urteile in's A u g e zu fassen, wie sie die W o r t s p r a c h e als Saetze formulirt! Es muss sich uns h i e r b e i e m p f e h l e n , u n t e r Beiseitelassung d e r z u s a m m e n g e s e t z t e r e n , z u n a e c h s t uns an die e i n f a c h s t e n A r t e n d e r Urteile zu halten. 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 d a r s t e l l e n in F o r m eines Satzes, d e r mit e i n e m "Subject" ein "Praedikat" v e r k n u e p f t . " In o t h e r words, t h e r e is n o m e t a p h y s i c a l q u e s t i o n involved h e r e at all. As S c h r o e d e r finds predicative p r o p o s i t i o n s easier to deal with t h a n th'e o t h e r types of p r o p o s i t i o n which he considers, h e p r o c e e d s in an entirely justifiable m a n n e r to analyze all o t h e r p r o p o s i t i o n s in t e r m s o f t h e m . It may well be that S c h r o e d e r was m i s t a k e n as to the simplicity a n d the c o n v e n i e n c e of this m e t h o d of p r o c e d u r e , a n d that it is actually an easier a n d a c l e a r e r c o u r s e to prove the p r o p e r t i e s o f classes a n d of relatives f r o m those of p r o p o s i t i o n s , b u t t h e r e is n o " p h i l o s o p h i c a l e r r o r " h e r e , for t h e r e is n o m e t a p h y s i c a l p u r p o s e . T h e r e m i g h t s e e m to be s o m e tacit m e t a p h y s i c a l 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

d e r ' s Der Operationskreis des Logikkalkuls, for t h e r e 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 i m s e l f a d m i t t e d in the Principles of Mathematics, the case is different. H e a t t e m p t e d to c o n s t r u c t a sort o f universal g r a m m a r o f thinking, a definitive set of postulates for all m a t h e m a t i c s , never to be s u p e r s e d e d . H e said: ~ "I h o l d - - a n d it is an i m p o r t a n t part of my p u r p o s e to p r o v e - - t h a t all p u r e m a t h e m a t i c s ( i n c l u d i n g G e o m e t r y a n d even rational Dynamics) c o n t a i n s only o n e set o f indefinables, n a m e l y the f u n d a m e n t a l logical c o n c e p t s discussed in p a r t I. W h e n the various logical constants have b e e n e n u m e r a t e d , it is s o m e w h a t arbitrary which of t h e m we r e g a r d as i n d e f i n a b l e , t h o u g h t h e r e are a p p a r e n t l y s o m e which m u s t be i n d e f i n a b l e o n any theory." This last clause indicates the fatal weakness o f Russell's e n t i r e position. It is, as a m a t t e r of fact, entirely arbitrary which p r o p o s i t i o n a m o n g those true in a given system we take as a postulate for that system, p r o v i d e d we m a k e a right selection of o t h e r postulates to go with it. In any system, we may e i t h e r d e d u c e all the p r o p o s i t i o n s true within that system f r o m a given set o f postulates for the system, or that set o f postulates f r o m the r e m a i n i n g p r o p o s i t i o n s true within the system. If it can be s h o w n that o n e can select f r o m S c h r o e d e r ' s t r e a t m e n t a set of postulates ade q u a t e for the Algebra o f Relatives, then, h o w e v e r m u c h we may call his t r e a t m e n t of the subject c r u d e or c u m b e r s o m e or i n c o m p l e t e , we c a n n o t , as Russell does, brush it aside as based on a false p h i l o s o p h i c a l theory. I a m glad to be able to say, however, that Russell's t r e a t m e n t o f the n a t u r e 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 a n d m o r e closely to a c o r r e c t position. H e says:" " S o m e p r o p o s i t i o n s m u s t be a s s u m e d w i t h o u t proof, since all inf e r e n c e p r o c e e d s from p r o p o s i t i o n s previously asserted ... these, like the primitive ideas, are to s o m e e x t e n t a m a t t e r o f arbitrary choice; t h o u g h , as in the previous case, a logical system grows in i m p o r t a n c e a c c o r d i n g as the primitive p r o p o s i t i o n s are few a n d simple ... T h e p r o o f o f a logical system is its a d e q u a c y a n d its c o h e r e n c e . " This is surely a d e p a r t u r e f r o m the position that m a t h e m a t i c a l postulates are metaphysical ultimates. In a c o m p a r i s o n o f two systems o f m a t h e m a t i c s o r m a t h e m a t i c a l logic, the natural way to begin is by a c o m p a r i s o n o f their postulates. Now, as the initial postulates o f S c h r o e d e r ' s algebra o f relatives are those which d e f i n e the n a t u r e o f classes, w h e r e a s Russell begins with p r o p o s i t i o n s , a n d since it is their t r e a t m e n t s 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 d e v e l o p 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.

FROM

PEIRCE

TO SKOLEM

433

of relatives. T h e r e f o r e we must be c o n t e n t with making, as it were, crosssections of their m e t h o d s of procedure, and c o m p a r i n g these. In o t h e r words, we shall c o m p a r e Schroeder's postulates for classes, not with Russell's postulates for propositions, but with a set of t h e o r e m s concerning classes which Russell claims are sufficient to define all their formal properties. Similarly, we shall c o m p a r e Russell's postulates for propositions with certain formulas of S c h r o e d e r from which we may d e d u c e all the laws to which propositions are subject. As both S c h r o e d e r 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 n d e p e n d e n t set of postulates for the Algebra of Relatives from a m o n g the formulas of either. However, it is an exceedingly i m p o r t a n t fact that the Algebra of Relatives possesses every formal law of the Algebra of Classes, and only differs from the latter in having o t h e r formal laws also. I intend to devote the first c h a p t e r of my thesis to a p r o o f that the sets of postulates for classes and propositions given by S c h r o e d e r 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. After I have proved these things, I propose to devote the second c h a p t e r to a comparative study of the m e t h o d s by which S c h r o e d e r and Russell make the transition from the calculi of classes a n d propositions to the calculus of relatives. I shall first discuss the two f o r m u l a e which S c h r o e d e r and Russell respectively make the basis of their t r e a t m e n t of the subject, a = ~oao(i'j)

and R = 5c~(xRy),

and show that Schroeder's formula is equivalent to a proposition true in the system of Russell (although I c a n n o t prove that Russell's f o r m u l a is true in the system of Schroeder, because, as I shall show, it c a n n o t be expressed in his language). Afterwards I shall take up Schroeder's definitions of the various operations which can be p e r f o r m e d a m o n g relatives a n d the relations of subsumption, equality, and inequality which hold between them. To do this, I shall d e m o n s t r a t e that the a + b, ab, a j-b, and a ; b of S c h r o e d e r are equivalent to the a ~ b , a ~ b , - ( - a ] - b ) , and alb of Russell; that 6 and d c o r r e s p o n d to - a and ~ (or cnv 'a); that :(= c o r r e s p o n d s to C a n d = to =.; that 1 --V, 0 = A, 1'= I, and 0' =J. I shall also show that the relative i of S c h r o e d e r 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. O n the basis

434

NORBERT WIENER'S THESIS

of these facts I shall establish a general m e t h o d 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 propositions formed by the combination 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 a n o t h e r 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 p r o b l e m of the connections between classes and relatives. It will be shown that, although 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 m e n t i o n e d 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 r e n d e r i n g 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 "Augenquaderrelativ", 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 rendering 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 Schroeder's terminology. Although there can be rmthing in Schroeder's system c o r r e s p o n d i n g 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 S c h r o e d e r 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 propositions 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

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

"a is similar to b" S c h r o e d e r writes a ~ a ~' b is d e f i n e d as

435

b, and Russell writes a s m b.

( z ; ~ + ~; z :(= l')(b = z; a ) ( a = ~.;b)(z=~-[tb), z

whereas an e q u a t i o n sufficient to define c~ sm 13 is k : c~sm/3.-=: ( 3 R ) . R e 1 - - * I . a = D ' R . ~ = C I ' R .

.73.1

I shall show that these two notions of similarity are equivalent. I shall also show that R e a sm b, or "R is a relation which puts a a n d b in oneto-one c o r r e s p o n d e n c e " , can be translated by ( R ; R + k; R=~--l')(a = R;1)(b = R;1). F r o m these e x a m p l e s I shall be able to show that I can translate into S c h r o e d e r ' s t e r m i n o l o g y without any alteration of its truth-value, every p r o p o s i t i o n of Russell which deals solely with relatives of the same "type", a n d classes of the type of their d o m a i n s a n d c o n v e r s e - d o m a i n s (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 e d u c e d to the form 1 ; a ; 1. T h e s e have only two possible v a l u e s - - l , if a ~ : 0, a n d 0, if a = 0 . Now, since a - l = a = a " i and a 90 = 0-- a" 0, b(1 ; a ; 1) is equal to b (a ~: 0), or b(Eoao). This makes it possible in an expression which involves the p r o d u c t of a p r o p o s i t i o n by a class or relative to replace the proposition by an "ausgezeichnetes Relativ". As the a o in Zoao(i:j) is a proposition, a n d is multiplied by the relative (i: j), we may replace it by an "ausgezeichnetes Relativ." S c h r o e d e r gives us the relative i; a;j as this "ausgezeichnetes Relativ." T h a t it is an "ausgezeichnetes Relativ" is shown by the f o r m u l a (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 m i g h t have p r o c e e d e d f r o m the calculi of classes a n d propositions to the calculus of relatives. T h a t this is a valid way I shall show. To c o n c l u d e the chapter, I shall try to show that 1 ; a ; 1 ; a ; 1, a; 1, a n d a f o r m a regular s e q u e n c e , a n d that the step f r o m class or system to relative is closely analogous to that from p r o p o s i t i o n or "ausgezeic h n e t e s Relativ" to class. In general, the steps from binary to ternary, from ternary to quaternary, from q u a t e r n a r y to quinary, and, in general, f r o m n-ary to (n + 1)-ary relatives, are of the same nature. F r o m this I shall argue that the Algebra of Logic may be treated m o r e symmetrically by starting with propositions r a t h e r than with classes, since propositions are 0-ary relatives, a n d h e n c e form the lower e n d of the scale of relatives.

436

NORBERT WIENER'S THESIS

This does not tend in the least, however, to disprove that a perfectly cogent, rigorous, and altogether logically satisfactory treatment 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 r e a t m e n t 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 S c h r o e d e r 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 m e t h o d of p r o c e d u r e can be dealt with by him with perfect rigor and correctness. I shall also show that Schroeder's symbolism involves the treatment of none of the notions which the e 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. 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 misunderstood the e-relation of Peano in a most peculiar way. I believe that I have been able to prove the following theorems, t h o u g h I have not included them in this thesis ... [A list of theorems has been omitted 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, although 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 introduction I should like to call attention 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 d e v e l o p m e n t of the whole of mathematics from a few simple logical postulates. Accordingly, the t h e o r e m s are arranged in a clear-cut, def-

FROM PEIRCE

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437

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 m a n n e r 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 d e p a r t m e n t 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 assimilated 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 treatment 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.

438

NORBERT WIENER'S THESIS

relatives, at any rate, S c h r o e d e r is not only able to treat of everything with which Russell deals, but to treat of it in a way which is in m a n y cases at least as simple, if not simpler. I shall leave it for the r e a d e r to j u d g e how m u c h S c h r o e d e r ' s "obsession with the forms of o r d i n a r y 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 b e e n repeatedly asserted in the course of this thesis that S c h r 6 d e r does n o t treat of individuals, in Russell's sense, xRy c a n n o t be translated into the language of S c h r 6 d e r unless x a n d y are both a p p a r e n t variables, or else unless we know some proposition such as i = t ' x . j = t'y. T h a t is, we can translate (~ 'i)R(t'~/'), but not merely xRy. Similarly, we saw, we may find S c h r 6 d e r ' s equivalent for (~'i) ~ o~, but n o t merely for x c~. T h e z-relation is untranslatable into the language of S c h r 6 d e r simply because it has no c o u n t e r p a r t in the actual system of Schr6der. T h e q u o t a t i o n from the Algebra tier Logik which we gave in the b e g i n n i n g of c h a p t e r II showed us that the ' I n d i v i d u e n ' of S c h r 6 d e r are g e n u i n e unitclasses, quite equivalent to the unit-classes of Russell, a n d are c o n n e c t e d with o t h e r classes by the relation of s u b s u m p t i o n r a t h e r 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 t h e r symbols ~2 is a symbol of anything but a proposition, a class, or a relative; a n d S c h r 6 d e r explicitly refuses to deal with classes involving the existence of o t h e r classes, ~:~ as such, and, as a c o n s e q u e n c e , with classes of relatives or of propositions, for he treats of relatives as a part of classes, a n d all his propositions c o n c e r n themselves with classes or with relatives. H e says, TM " U n d d a m i t auch in d e r ursprfinglichen Mannigfaltigkeit dis S u b s u m t i o n (2+) a u f r e c h t ero halten w e r d e n k6nne, ist von v o r n h e r e i n e r f o r d e r l i c h ( u n d hinreic h e n d ) , dass unter ihren als "Individuen" gegegebenen Elementen sich keine

Klassen befinden, welche ihrerseits Elemente derselben Mannigfaltigkeit als Individuen unter sich begreifen." This clearly excludes any f o r m u l a e in which classes of classes or relatives occur t o g e t h e r with the classes or relatives which they classify, c o n n e c t e d with t h e m by any of the o p e r a t i o n s a n d copulae of the calculi o f classes a n d of relatives. As for classes of propositions, it should be r e m e m b e r e d that there are, from the s t a n d p o i n t of the calculus of J~ i.e., all symbols not representing operations or copulae. Is T h e r e is an important exception to this in I, w 23, p. 481-3. I shall treat of this later. ~4A. der L., I, p. 248. All of w 9, 'Pure Manifolds', in which this passage occurs, is invaluable for a p r o p e r understanding 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 conditioned by another.

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439

propositions, only two propositions, i a n d (). Treating i a n d () as unitclasses, then, there are only four classes of propositions: 0, 1, 0, 1, w h e r e 1 contains both 1 and 0. Since 0 =(= 1, we m i g h t write these as 0, (0 =(= 1), (1 :~-0), 1. Since, in o u r system, (0=(= 1)(1 : ~ 0 ) , 1 = 0 . This makes o u r system dwindle down to 0 alone, a n d shows its absurdity. ~ Such a class would not be a p u r e manifold, a n d could not be r e p r e s e n t e d by 1, for its e l e m e n t s c a n n o t coexist. To quote S c h r 6 d e r ' s words, ~6 "Als eine erste A n f o r d e r u n g h a b e n wir schon in w 7 u n t e r Postulat ((1+)) die n a m h a f t g e m a c h t : dass die E l e m e n t e d e r Mannigfaltigkeit sS.mtlich vereinbar, 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 S c h r 6 d e r ' s treating of classes of propositions as such, a n d c o m p l e t e s the p r o o f that he c a n n o t treat of a class t o g e t h e r with its m e m b e r s , as he virtually admits. S c h r 6 d e r actually never does unreservedly treat of a class t o g e t h e r with its m e m b e r s in the same formula. As the e - r e l a t i o n is the relation between a term a n d the class to which it belongs, he m a k e s no g e n e r a l use of it n o r of any equivalent symbol, n o r n e e d he do so. It is an utterly inexcusable mistake for Padoa to say ~7 "I1 est d o n c 5. p r o p o s de r a p p e l e r ici que M. S c h r 6 d e r - - q u i s'etait mis fi l'oeuvre avant M. P e a n o et qui en 1877 avait d6jfi publi6 son Operationskreis des Logikkalkuls---n'a pas r6ussi/l nous laisser u n e i d e o g r a p h i e logique satisfaisante; et cela, princ i p a l e m e n t , parce qu'il n ' a pas distinqu6 les appartenances des inclusions et par suite il les r e p r 6 s e n t a 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 rec o n n a i t r e la n6cessit6 de cette distinction. "Maisje crois que d e u x e x e m p l e s suffriront/t vous 6claircir la diff~rente signification des d e u x symbols " e " et "D",ls que d'ailleurs les logiciens scholastiques d i s t i n q u a i e n t en "sensus compositf' et "sensus divisi," touffois sans d o n n e r ~ cette distinction l ' i m p o r t a n c e que j u s t e m e n t lui d o n n a M. Peano. "Voici les d e u x exemples: d ' u n c6t6 vous avez les inclusions genevois D suisse

suisee D e u r o p 6 e n

desquelles on tire genevois D e u r o p 6 e n , et d ' u n autre c6t6 vous avec les appartenances Pierre ~ ap6tres

ap6tres e d o u z a i n e

15Cf. A. der L., I, w 9, p. 245. '" A. cLerL., I, w 9, p. 247. 17Rev. de Met. et de Mor., 19, 1911, pp. 852, 853. i~ Padoa, following Peano, uses D for both implication and subsumption.

44 ~

NORBERT WIENER'S THESIS

( q u ' o n p e u t lire "Pierre fut un des ap6tres" et "les ap6tres 6taient u n e d o u z a i n e " ) desquelles o n ne p e u t p a r tirer Pierre ~ d o u z a i n e . " I can say definitely that I k n o w o f n o case w h e r e the =(= o f S c h r 6 d e r definitively r e p r e s e n t s a n y t h i n g but the D of Peano, o r its equivalents, D, C , a n d C in the system o f Russell. W h e n i:~--a r e p r e s e n t s x ~ a, we have, as we have shown repeatedly, i - L'x, n o t i = x. It is perfectly true that S c h r 6 d e r would translate "Peter ~ apostles" into his own termin o l o g y as "Peter =(= apostles", b u t S c h r 6 d e r ' s 'Peter' w o u l d be the class o f m e n o f which P a d o a ' s ' P e t e r ' is the sole m e m b e r . If, however, S c h r 6 d e r s h o u l d translate "Peter ~ apostles" in this m a n n e r , he w o u l d n o t r e n d e r "apostles ~ d o z e n " as "apostles :(= dozen," for this w o u l d involve a f o r m u l a d e a l i n g with a class t o g e t h e r with a n o t h e r class o f which the first is a m e m b e r , which, by the principles laid d o w n in A. der L., I, w 9, he is u n a b l e to discuss. H e would translate "Apostles d o z e n " as " n u m . 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 o f 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 e m e n t which may be m a d e entirely in the l a n g u a g e o f the calculus o f classes. I:' As n u m . a = 12, however, would be quite a l e n g t h y e x p r e s s i o n to e x p a n d , let us translate into the l a n g u a g e o f S c h r 6 d e r a n o t h e r pair of expressions quite a n a l o g o u s to those o f Padoa, but having ' c o u p l e ' in place o f 'dozen'. Let us suppose, then, that we have the two z - p r o p o s i t i o n s , '[John is o n e o f the J o n e s twins", a n d "the J o n e s twins are a couple", f r o m which we c a n n o t infer that J o h n is a couple. Let x stand for J o h n , ol for the class consisting o f the J o n e s twins, a n d 2 for the class o f all couples. Russell would write the two p r o p o s i t i o n s as x ~ ol a n d c~ ~ 2, respectively, while S c h r 6 d e r w o u l d write t h e m as i=(=ol a n d n u m . ot - 2, i s t a n d i n g for the t'x o f Russell. Now, n u m . ol = 2 is d e f i n e d as TM

x, y

J"Jy (x ~

y)(a = 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 c o n c e r n i n g number, series, etc., which, as far as I know, have never been followed out further by Schr6der. ~~ der I.., II, 1, p. 349.

441

FROM PEIRCE TO SKOLEM

( N u m . 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 f o r m of s t a t e m e n t , a n d any o b j e c t i o n against S c h r 6 d e r ' s position h e r e w o u l d tell equally against Russell o r P e a n o . Russell s e e m s to be guilty of the same e r r o r which we have j u s t exh i b i t e d in the case of Padoa, a l t h o u g h he is n o t q u i t e so explicit as to the object of his attacks. H e says, 21 "Before P e a n o a n d Frege, the relation of m e m b e r s h i p ( e ) was r e g a r d e d as m e r e l y a p a r t i c u l a r case of the relation o f inclusion ( C ) . For this reason, the t r a d i t i o n a l f o r m a l logic t r e a t e d such p r o p o s i t i o n s as "Socrates is a m a n " as instances of the universal affirmative A, "All S is P," which is what we e x p r e s s by "c~ C 13." This involved a c o n f u s i o n of f u n d a m e n t a l l y d i f f e r e n t kinds of p r o p ositions, which greatly h i n d e r e d the d e v e l o p m e n t a n d usefulness of symbolic logic." Since S c h r 6 d e r does n o t deal with the e - r e l a t i o n , all his classes are of o n e 'type', for Russell's types are r e l a t e d to o n e a n o t h e r by the erelation, or s o m e relation derived f r o m the e - r e l a t i o n . 22 T h e s a m e holds true of relatives. 2:~ All the e x p r e s s i o n s of Russell which involve classes a n d relatives of d i f f e r e n t types are w i t h o u t d i r e c t e q u i v a l e n t s in the system of S c h r 6 d e r , for if an e x p r e s s i o n involves classes a n d relatives of d i f f e r e n t types, it ultimately involves a given class, a n d a class of classes of which the given class is a m e m b e r . S c h r 6 d e r , t h e n , is u n a b l e to express objects of d i f f e r e n t types by his f o r m u l a e . W h e n he wishes to m a k e a s t a t e m e n t , for e x a m p l e , c o n c e r n i n g the fact that a certain class is a c o u p l e , he must, as we have seen, express this as the a f f i r m a t i o n of a certain formal p r o p e r t y of this class e n t i r e l y w i t h i n t h e c a l c u l u s o f classes. Yet, for all this, t h e r e is a c e r t a i n i m p o r t a n t a n a l o g y b e t w e e n the p r o c e d u r e s of S c h r 6 d e r a n d of Russell in, for exa m p l e , t h e i r discussions of the n o t i o n of a c o u p l e , o r in g e n e r a l of a n u m b e r . It will be r e m e m b e r e d that Russell's oL e 2 was s h o w n to be e q u i v a l e n t to S c h r 6 d e r ' s n u m . 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 e x c e p t in a p r o p o s i t i o n such as o~ e 2, which stands as an a b b r e v i a t i o n for a c e r t a i n ~ Cf..63.02-.051, 103.5.51.52. ~2Cf..63.02-.051, 103.5.51.52. u:~Cf..64.01-041.3-34. ~1 Principia Mathematica, pp. 75-84, inclusive.

NORBERT WIENER'S THESIS

44 2

p r o p o s i t i o n a l function of ~. Similarly, the 2 in n u m . ot = 2 is an incomplete symbol, and has no m e a n i n g e x c e p t in s o m e e x p r e s s i o n such as n u m . c~ = 2. Now, j u s t as Russell defines u , r C , 2~ etc., by m e a n s of c o m b i n a t i o n s of the p r o p o s i t i o n a l functions d e t e r m i n i n g the classes which they c o n n e c t , in terms of the o p e r a t i o n s a n d c o p u l a e of the p r o p o s i t i o n a l calculus, so S c h r 6 d e r defines +, x , =,26 etc., b e t w e e n n u m b e r s by m e a n s of a p r o p o s i t i o n a l f u n c t i o n of classes of the given n u m b e r of terms, expressed by the o p e r a t i o n s and c o p u l a e of the calculi of classes, propositions, a n d relatives. T h e r e is a certain parallelism between Def.

(.22.01)

a n d S c h r 6 d e r ' s definition: 27 ( n u m . a = n u m . b)

= ( ~ ; a =/~" b)((~; 0'" a = b'0'" b){~" 0'(0'" ,

,

,

,

,

aO')'a =/~" 0'(0' ,

,

,

9b0') "b} ,

9

In o t h e r words, t h e r e is an analogy, which seems to m e m o r e than trivial, b e t w e e n S c h r 6 d e r ' s step f r o m a to n u m . a, a n d Russell's step f r o m individual to class, or, in general, from type to type. A p o i n t which seems to m e worthy of at least passing notice is that Russell can treat =, C , a n d C as relatives, while their equivalents, = a n d =(=, c a n n o t be relatives for Schr6der. T h e reason for this is c l e a r m t h e y are of a type h i g h e r than the classes or relatives which they relate. S u p p o s e that R C S. T h e n , by .64.201, t'R = t'S, or R a n d S are of the s a m e type. By .55.3, R C S may be read as (R ~ S ) C ( C ) , w h e r e the first C is o n e type h i g h e r than the second. By .64.31 a n d . 6 3 . 1 6 , t'(R $ S) = t'(t'R $ t'S) = t'(t'R "~ t'R). By this we get, by m e a n s of . 6 4 . 1 1 , 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 s e c o n d C is that of R ~ S, so that we get, t ' ( C ) = t"'R, w h e r e t " R stands for a certain type above that of P~ T h e r e f o r e , :(= c a n n o t be t r e a t e d as a relative by Schr6der. Similarly, he c a n n o t treat = as a relative. (i =j) = 1'0 seems to be an e x c e p t i o n to this, but it really is not, for 1,)does not c o r r e s p o n d to the iIj of Russell, but to the xIy, w h e r e i = i'x a n d j = i'y, a l t h o u g h it is true in this case that F x I y . - , iIj, by .51.23, .50.1. T h e g e n e r a l equality relation, a = b, w h e r e a a n d b are not ' I n d i v i d u e n ' , is not e q u i v a l e n t to any relative coefficient. We have now show that, in general, S c h r 6 d e r ' s symbolism has no place for objects of different types, a n d have e x h i b i t e d the c o n s e q u e n c e s !

2~, 22.01-03. '~" On Pasigraphy, Monist, IX, 1898-9, p. 54. ~7Ibid.

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which follow this perfectly justifiable limitation. It would be utterly unjust, however, to claim that S c h r 6 d e r was u n a w a r e of the possibility of f o r m i n g a system involving a h i e r a r c h y of types. H e discusses at s o m e l e n g t h the h i e r a r c h y of classes, zs classes of classes, etc., a n d is very careful to k e e p these various grades of classes sharply s e p a r a t e f r o m o n e another, d i s t i n g u i s h i n g t h e m as the " u n s p r / i n g l i c h e Mannigfaltigkeit," the "erste a b g e l e i t e t e Mannigfaltigkeit," the "zweite a b g e l e i t e t e Mannigfaltigkeit,"etc. T h e s e notions, t h o u g h they have an i m p o r t a n t place in his t h o u g h t , have no place, in general, in his symbolism. T h e r e is, as I have said in a note, o n e highly significant passage in the Algebra der Logik, w h e r e S c h r 6 d e r breaks his g e n e r a l rule, a n d tries to r e p r e s e n t objects of d i f f e r e n t types t h r o u g h his symbolism. ~~ H e wishes to express the fact that the ' q u o t i e n t ' of two classes may have m a n y values by the formula, x=~--a :: b, w h e r e x is a p a r t i c u l a r value of the ' q u o t i e n t ' , a n d a :: b is the class of all values of the ' q u o t i e n t ' . To give his own words, "Auch diese S u b s u m t i o n s z e i c h e n [the =(= in x4= a + b :~0or x =(=a] :: b w/iren a b e r als solche d e r abgeleiteten M a n n i g f a l t i g k e i t zu i n t e r p r e t i e r e n , u n d nicht als solche d e r u r s p r f i n g l i c h e n . Die Subs u m t i o n besagte hier nicht, das G e b i e t x sei als Teil e n t h a l t e n in e i n e m rechts a n g e f ~ h r t e n Gebiete, s o n d e r n nur, es sei als Individuum e n t h a l t e n in d e r rechts s t e h e n d e n Klasse von G e b i e t e n . " G e r a d e in j e n e n Grenzf/illen aber, wo die Klasse a + b rechts selbst n u r ein G e b i e t umfasst, mfisste das S u b s u m t i o n s z e i c h e n Missverstandnisse n a h e 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, n u r G l e i c h h e i t gelten kann. Zur V e r m e i d e n s o l c h e r ( u n d / i h n l i c h e r schon in w 9 u n t e r ~b) c h a r a c t e r i z i r t e r MisstS.nde mfisste m a n eigenlich zweierlei Subsumtionszeichen v e r w e n d e n f/ir die urspriingliche u n d die abgeleitete Mannigfaltigkeit." In o t h e r words, S c h r 6 d e r wishes to c o n s t r u c t a symbol to signify that x is a "Individuum," n o t a part, of the class a + b. Now, since S c h r 6 d e r has no s e p a r a t e symbol for x as an o r d i n a r y class a n d x as a unit-class (the c~ a n d t'o~ of Russell, respectively) the word, " I n d i v i d u u m , " in this passage m u s t have the s a m e 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 n o t i o n could hardly have b e e n b o r r o w e d f r o m any of the early works of Frege or P e a n o , for S c h r 6 d e r makes no r e f e r e n c e to this p o i n t in his discussion of F r e g e ' s work, :~ whereas he says that he b e c a m e a c q u a i n t e d with the writings of P e a n o too late to m a k e use of t h e m 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. :~lA. der L., I., pp. 703, 704. ~ A. der L., I, pp. 709, 710.

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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 attention of the reader to an extremely curious passage in the Algebra der Logik. In the p o s t h u m o u s second part of the second volume of this work, which Schr6der never finally revised, the following s t a t e m e n t occurs: :~:~ "Statt unseres Subsumtionszeichen, wendet H e r r Peano---was sicher kein Vorzug i s t B m e i s t zweierlei Zeichen an, n~mlich zwischen Klassen ein ~ als den Anfangsbuchstaben von 'eorL, "ist," zwischen Aussagen dagagen ein u m g e k e h r t e s C, also D, was e r i n n e r n soll an concluditur (6 contenutao). Ich glaube unwiderleglich d a r g e t h a n zu haben, dass ein besonderes Zeichen ffir die letztere Beziehung entbehrlich, m.a.W, dass die Kopula der kategorischen Urteile auch ffir die hypothestischen verwendbar ist ... B A l l e r d i n g s bin ich auch fiberzeugt, u n d habe es schon gelegentlich (Bd. 1, S. 482) ausgesprochen, dass m a n u n t e r U m s t a n d e n noch einer b e s o n d e r e n zweiten Art von Subsumtionszeichen bedarf, nfimlichen n e b e n einem solchen ffir die ursprfingliche noch eines a n d e r e n ffir die abgeleitete Mannigfaltigkeit. So kann, wenn wie gew6hnlich, a~--b die E i n o r d n u n g eines Gebeites a in ein Gebiet b ausdrfickt, der Satz dass da Gebiet a zur Klasse J d e r Individuen geh6re, o d e r 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 t h e r words, Schr6der mistakes the z-relation of Peano for the general relation of inclusion between classes, and misunderstands 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~ O f 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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Wissenschaftlicher Br~vechsel, ed. G. Gabriel et al. (Hamburg: Meiner), pp. 157-159. ~ . 1977, Leopold L6wenheim: life, work, and early influence, in Logic Colloquium 76, Proceedings of a Conference at Oxford in July 1976, ed. R. Gandy and J. M. E. Hyland (Amsterdam: North Holland), pp. 235-252. ~ . 1984, Leopold L6wenheim, in Enzyklopiidie Philosophie und Wissenschaftstheorie, ed. J. Mittelstrass (Amsterdam: North Holland), pp. 715-716. ~ . 1990a, Leopold L6wenheim, in Dictionary of Scientific Biography, vol. 18, ed. E L. Holmes (New York: Scribner), pp. 571-572. ~ . 1990b, Ernst Schr6der and the distribution of quantifiers, Modern Logic, 1, 160-173. ~ . 1994, Zur Rezeption des Satzes von L6wenheim, preprint Vaught, R. L. 1974, Model theory before 1945, in Proceedings of Symposia in Pure Mathematics 25: Proceedings of the Tarski Symposium, ed. L. Henkin et al. (Providence: American Mathematical Society), pp. 153-172. Wang, H. 1967, The axiomatization of arithmetic, Journal of Symbolic Logic, 22, 145-158. ~ . 1970, A Survey of Skolem's work in logic, in Selected Works in Logic, ed. J. Fenstad (Oslo: Universitetforlaget, 1970), pp. 17-52. Wiener, E, and Young, E H., eds. 1952, Studies in the Philosophy of Charles Sanders Peirce, ed. P. and (Cambridge: Harvard University Press). Wigner, E. E 1960, The unreasonable effectiveness of mathematics in the natural sciences, in Communications on Pure and Applied Mathematics, 13, 1-14. Zeman, J. 1963, The graphical logic of C. S. Peirce, Ph.D. thesis, University of Chicago.

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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; Boolean, 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: 461

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 exposition 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, 16-17, 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; Mitchell's quantification for, 77-94; in

INDEX

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 Dedekind'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 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

FROM PEIRCE TO SKOLEM

463

Extensionality, 26-28, 31, 32

Heine, Eduard, 157 Herbrand, Jacques, 1, 4, 163, 204 Herbrand universe, 204 Hilbert, David, In, 2, 7, 13, 133, 195

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 Logik, 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

~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; Dedekind'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" 1,6wenheim'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; arithmetical, 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, o f a 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,

FROM PEIRCE TO SKOLEM 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, Exponentiation, 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, 16-17, 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, Benjamin 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 reviewed 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 calculus of logic," 14-17 "On the algebra oflogic," 10, 51-74 "On tile algebra of logic: A contribution 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 treatment of, 143-144, 151-152, 207; elimination of, 163-165; in firstand 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, 1 3 1 ; 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

467

FROM PEIRCE TO SKOLEM

1), 207-221; Schr6der's Lecture II (Appendix 2), 223-249; Schr6der's Lecture III (Appendix 3), 251-256: Schr6der's Lecture V (Appendix 4), 257-294; Schr6der's Lecture IX (Appendix 5), 295-338; Schr6der's Lecture XI (Appendix 6), 339-377; Schr6der's Lecture XII (Appendix 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 Ackermann's statement of the L6wenheim-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,

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 predicate logic, 109-110, 163 Tarski, Alfred, 5, 7, 37-38, 48, 95, 121, 163, 174, 251, 258; and Lindenbaum algebras, 121; and relation algebras, 142; Schr6der's influence on, 5, 12-13, 159; semantics ofquantifiers 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 (-