LIBRARY UNIVERSITY OP CALIFORNIA SANWKJO A SURVEY OF SYMBOLIC LOGIC BY C. I. LEWIS UNIVERSITY OF CALIFORNIA PRESS BERKELEY 1918 PRESS OF THE NEW ERA PRINTING COMPANY LANCASTER, PA. TABLE OF CONTENTS PREFACE v CHAPTER I. THE DEVELOPMENT OF SYMBOLIC LOGIC. 1 SECTION I. The Scope of Symbolic Logic. Symbolic Logic and Logistic. Summary Account of their Development 1 SECTION II. Leibniz 5 SECTION III. From Leibniz to De Morgan and Boole 18 SECTION IV. De Morgan 37 SECTION V. Boole 51 SECTION VI. Jevons 72 SECTION VII. Peirce 79 SECTION VIII. Developments since Peirce 107 CHAPTER II. THE CLASSIC, OR BOOLE-SCHRODER AL- GEBRA OF LOGIC 118 SECTION I. General Character of the Algebra. The Postulates and their Interpretation 118 SECTION II. Elementary Theorems 122 SECTION III. General Properties of Functions 132 SECTION IV. Fundamental Laws of the Theory of Equations . .' . 144 SECTION V. Fundamental Laws of the Theory of Inequations. 166 SECTION VI. Note on the Inverse Operations, "Subtraction" and "Division" 173 CHAPTER III. APPLICATIONS OF THE BOOLE-SCHRODER ALGEBRA 175 SECTION I. Diagrams for the Logical Relations of Classes .... 175 SECTION II. The Application to Classes 184 SECTION III. The Application to Propositions 213 SECTION IV. The Application to Relations 219 CHAPTER IV. SYSTEMS BASED ON MATERIAL IMPLI- CATION 222 SECTION I. The Two- Valued Algebra 222 Hi IV Table of Contents SECTION II. The Calculus of Prepositional Functions. Func- tions of One Variable 232 SECTION III. Prepositional Functions of Two or More Variables . 246 SECTION IV. Derivation of the Logic of Classes from the Calcu- lus of Prepositional Functions 260 SECTION V. The Logic of Relations 269 SECTION VI. The Logic of Principia Mathematica 279 CHAPTER V. THE SYSTEM OF STRICT IMPLICATION... 291 SECTION I. Primitive Ideas, Primitive Propositions, and Im- mediate Consequences 292 SECTION II. Strict Relations and Material Relations 299 SECTION III. The Transformation {-/-} 306 SECTION IV. Extensions of Strict Implication. The Calculus of Consistencies and the Calculus of Ordinary Inference 316 SECTION V. The Meaning of "Implies" 324 CHAPTER VI. SYMBOLIC LOGIC, LOGISTIC, AND MATHE- MATICAL METHOD 340 SECTION I. General Character of the Logistic Method. The "Orthodox" View 340 SECTION II. Two Varieties of Logistic Method : Peano's Formu- laire and Principia Mathematica. The Nature of Logistic Proof 343 SECTION III. A "Heterodox" View of the Nature of Mathe- matics and of Logistic 354 SECTION IV. The Logistic Method of Kempe and Royce 362 SECTION V. Summary and Conclusion 367 APPENDIX. TWO FRAGMENTS FROM LEIBNIZ 373 BIBLIOGRAPHY 389 INDEX. . 407 PREFACE The student who has completed some elementary study of symbolic logic and wishes to pursue the subject further finds himself in a discouraging situation. He has, perhaps, mastered the contents of Venn's Symbolic Logic or Couturat's admirable little book, The Algebra of Logic, or the chapters concerning this subject in Whitehead's Universal Algebra. If he read German with sufficient ease, he may have made some excursions into Schroder's Vorlesungen iiber die Algebra der Logik. These all concern the classic, or Boole-Schroder algebra, and his knowledge of symbolic logic is probably confined to that system. His further interest leads him almost inevitably to Peano's Formulaire de Mathematiques, Principia Mathematica of Whitehead and Russell, and the increasingly numerous shorter studies of the same sort. And with only elementary knowledge of a single kind of development of a small branch of the subject, he must attack these most difficult and technical of treatises, in a new notation, developed by methods which are entirely novel to him, and bristling with logico-metaphysical difficulties. If he is bewildered and searches for some means of further preparation, he finds nothing to bridge the gap. Schroder's work would be of most assistance here, but this was written some twenty-five years ago; the most valuable studies are of later date, and radically new methods have been introduced. What such a student most needs is a comprehensive survey of the sub- ject one which will familiarize him with more than the single system which he knows, and will indicate not only the content of other branches and the alternative methods of procedure, but also the relation of these to the Boole-Schroder algebra and to one another. The present book is an attempt to meet this need, by bringing within the compass of a single volume, and reducing to a common notation (so far as possible), the most important developments of symbolic logic. If, in addition to this, some of the requirements of a "handbook" are here fulfilled, so much the better. But this survey does not pretend to be encyclopedic. A gossipy recital of results achieved, or a superficial account of methods, is of no more use in symbolic logic than in any other mathematical discipline. What is presented must be treated in sufficient detail to afford the possibility of real insight and grasp. This aim has required careful selection of material. vi Preface The historical summary in Chapter I attempts to follow the main thread of development, and no reference, or only passing mention, is given to those studies which seem not to have affected materially the methods of later researches. In the remainder of the book, the selection has been governed by the same purpose. Those topics comprehension of which seems most essential, have been treated at some length, while matters less fundamental have been set forth in outline only, or omitted altogether. My own contribution to symbolic logic, presented in Chapter V, has not earned the right to inclusion here; in this, I plead guilty to partiality. The discussion of controversial topics has been avoided whenever possible and, for the rest, limited to the simpler issues involved. Consequently, the reader must not suppose that any sufficient consideration of these questions is here given, though such statements as are made will be, I hope, accurate. Particularly in the last chapter, on "Symbolic Logic, Logistic, and Mathematical Method ", it is not possible to give anything like an adequate account of the facts. That would require a volume at least the size of this one. Rather, I have tried to set forth the most important and critical considerations somewhat arbitrarily and dogmatically, since there is not space for argument and to provide such a map of this difficult terri- tory as will aid the student in his further explorations. Proofs and solutions in Chapters II, III, and IV have been given very fully. Proof is of the essence of logistic, and it is my observation that stu- dents even those with a fair knowledge of mathematics seldom command the technique of rigorous demonstration. In any case, this explicitness can do no harm, since no one need read a proof which he already understands. I am indebted to many friends and colleagues for valuable assistance in preparing this book for publication : to Professor W. A. Merrill for emenda- tions of my translation of Leibniz, to Professor J. H. McDonald and Dr. B. A. Bernstein for important suggestions and the correction of certain errors in Chapter II, to Mr. J. C. Rowell, University Librarian, for assistance in securing a number of rare volumes, and to the officers of the University Press for their patient helpfulness in meeting the technical difficulties of printing such a book. Mr. Shirley Quimby has read the whole book in manuscript, eliminated many mistakes, and verified most of the proofs. But most of all, I am indebted to my friend and teacher, Josiah Royce, who first aroused my interest in this subject, and who never failed to give me encouragement and wise counsel. Much that is best in this book is due to him. C. I. LEWIS. BERKELEY, July 10, 1917. CHAPTER I THE DEVELOPMENT OF SYMBOLIC LOGIC I. THE SCOPE OF SYMBOLIC LOGIC. SYMBOLIC LOGIC AND LOGISTIC. SUMMARY ACCOUNT OF THEIR DEVELOPMENT The subject with which we are concerned has been variously referred to as "symbolic logic", "logistic", "algebra of logic", "calculus of logic", "mathematical logic", "algorithmic logic", and probably by other names. And none of these is satisfactory. We have chosen "symbolic logic" because it is the most commonly used in England and in this country, and because its signification is pretty well understood. Its inaccuracy is obvious: logic of whatever sort uses symbols. We are concerned only with that logic which uses symbols in certain specific ways those ways which are exhibited generally in mathematical procedures. In particular, logic to be called "symbolic" must make use of symbols for the logical relations, and must so connect various relations that they admit of "trans- formations" and "operations", according to principles which are capable of exact statement. If we must give some definition, we shall hazard the following: Symbolic Logic is the development of the most general principles of rational pro- cedure, in ideographic symbols, and in a form which exhibits the connection of these principles one with another. Principles which belong exclusively to some one type of rational procedure e. g. to dealing with number and quantity are hereby excluded, and generality is designated as one of the marks of symbolic logic. Such general principles are likewise the subject matter of logic in any form. To be sure, traditional logic has never taken possession of more than a small portion of the field which belongs to it. The modes of Aristotle are unnecessarily restricted. As we shall have occasion to point out, the reasons for the syllogistic form are psychological, not logical : the syllogism, made up of the smallest number of propositions (three), each with the small- est number of terms (two), by which any generality of reasoning can be attained, represents the limitations of human attention, not logical necessity. To regard the syllogism as indispensable, or as reasoning par excellence, is 2 1 2 A Survey of Symbolic Logic the apotheosis of stupidity. And the procedures of symbolic logic, not being thus arbitrarily restricted, may seem to mark a difference of subject matter between it and the traditional logic. But any such difference is accidental, not essential, and the really distinguishing mark of symbolic logic is the approximation to a certain form, regarded as ideal. There are all degrees of such approximation ; hence the difficulty of drawing any hard and fast line between symbolic and other logic. But more important than the making of any such sharp distinction is the comprehension of that ideal of form upon which it is supposed to depend. The most convenient method which the human mind has so far devised for exhibiting principles of exact procedure is the one which we call, in general terms, mathematical. The important characteristics of this form are: (1) the use of ideograms instead of the phonograms of ordinary language; (2) the deductive method which may here be taken to mean simply that the greater portion of the subject matter is derived from a relatively few principles by operations which are "exact": and (3) the use of variables having a definite range of significance. Ideograms have two important advantages over phonograms. In the first place, they are more compact, + than "plus", 3 than "three", etc. This is no inconsiderable gain, since it makes possible the presentation of a formula in small enough compass so that the eye may apprehend it at a glance and the image of it (in visual or other terms) may be retained for reference with a minimum of effort. None but a very thoughtless person, or one without experience of the sciences, can fail to understand the enor- mous advantage of such brevity. In the second place, an ideographic notation is superior to any other in precision. Many ideas which are quite simply expressible in mathematical symbols can only with the greatest difficulty be rendered in ordinary language. Without ideograms, even arithmetic would be difficult, and higher branches impossible. The deductive method, by which a considerable array of facts is sum- marized in a few principles from which they can be derived, is much more than the mere application of deductive logic to the subject matter in question. It both requires and facilitates such an analysis of the whole body of facts as will most precisely exhibit their relations to one another. In fact, any other value of the deductive form is largely or wholly fictitious. The presentation of the subject matter of logic in this mathematical form constitutes what we mean by symbolic logic. Hence the essential characteristics of our subject are the following: (1) Its subject matter is The Development of Symbolic Logic 3 the subject matter of logic in any form that is, the principles of rational or reflective procedure in general, as contrasted with principles which belong exclusively to some particular branch of such procedure. (2) Its medium is an ideographic symbolism, in which each separate character represents a relatively simple and entirely explicit concept. And, ideally, all non-ideographic symbolism or language is excluded. (3) Amongst the ideograms, some will represent variables (the "terms" of the system) having a definite range of significance. Although it is non-essential, in any system so far developed the variables will represent "individuals", or classes, or relations, or propositions, or " propositional functions", or they will represent ambiguously some two or more of these. (4) Any system of symbolic logic will be developed deductively that is, the whole body of its theorems will be derived from a relatively few principles, stated in symbols, by operations which are, or at least can be, precisely formulated. We have been at some pains to make as clear as possible the nature of symbolic logic, because its distinction from "ordinary" logic, on the one hand, and, on the other, from any mathematical discipline in a sufficiently abstract form, is none too definite. It will be further valuable to comment briefly on some of the alternative designations for the subject which have been mentioned. "Logistic" would not have served our purpose, because "logistic" is commonly used to denote symbolic logic together with the application of its methods to other symbolic procedures. Logistic may be defined as the science which deals with types of order as such. It is not so much a subject as a method. Although most logistic is either founded upon or makes large use of the principles of symbolic logic, still a science of order in general does not necessarily presuppose, or begin with, symbolic logic. Since the relations of symbolic logic, logistic, and mathematics are to be the topic of the last chapter, we may postpone any further discussion of that matter here. We have mentioned it only to make clear the meaning which "logistic" is to have in the pages which follow. It comprehends symbolic logic and the application of such methods as symbolic logic exempli- fies to other exact procedures. Its subject matter is not confined to logic. "Algebra of logic" is hardly appropriate as the general name for our subject, because there are several quite distinct algebras of logic, and because symbolic logic includes systems which are not true algebras at all. "The algebra of logic" usually means that system the foundations of which were laid by Leibniz, and after him independently by Boole, and 4 A Survey of Symbolic Logic which was completed by Schroder. We shall refer to this system as the "Boole-Schroder Algebra ". "Calculus" is a more general term than "algebra". By a "calculus" will be meant, not the whole subject, but any single system of assumptions and their consequences. The program both for symbolic logic and for logistic, in anything like a clear form, was first sketched by Leibniz, though the ideal of logistic seems to have been present as far back as Plato's Republic. 1 Leibniz left frag- mentary developments of symbolic logic, and some attempts at logistic which are prophetic but otherwise without value. After Leibniz, the two interests somewhat diverge. Contributions to symbolic logic were made by Ploucquet, Lambert, Castillon and others on the continent. This type of research interested Sir William Hamilton and, though his own contribution was slight and not essentially novel, his papers were, to some extent at least, responsible for the renewal of investigations in this field which took place in England about 1845 and produced the work of De Morgan and Boole. Boole seems to have been ignorant of the work of his continental predecessors, which is probably fortunate, since his own beginning has proved so much more fruitful. Boole is, in fact, the second founder of the subject, and all later work goes back to his. The main line of this develop- ment runs through Jevons, C. S. Peirce, and MacColl to Schroder whose Vorlesungen uber die Algebra der Logik (Vol. I, 1890) marks the perfection of Boole's algebra and the logical completion of that mode of procedure. In the meantime, interest in logistic persisted on the continent and was fostered by the growing tendency to abstractness and rigor in mathe- matics and by the hope for more general methods. Hamilton's quaternions and the Ausdehnungslehre of Grassmann, which was recognized as a con- tinuation of the work begun by Leibniz, contributed to this end, as did also the precise logical analyses of the nature of number by Cantor and Dedekind. Also, the elimination from "modern geometry" of all methods of proof dependent upon "intuitions of space" or "construction" brought that subject within the scope of logistic treatment, and in 1889 Peano provided such a treatment in I Principii di Geometria. Frege's works, from the Begri/sschrift of 1879 to the Grundgesetze der Arithmetik (Vol. I, 1893; Vol. II, 1903) provide a comprehensive development of arithmetic by the logistic method. 1 See the criticisms of contemporary mathematics and the program for the dialectic or philosophic development of mathematics in Bk. vi, Step. 510-11 and Philebus, Step. 56-57. The Development of Symbolic Logic 5 In 1894, Peano and his collaborators began the publication of the Formulaire de Mathematiques, in which all branches of mathematics were to be presented in the universal language of logistic. In this work, symbolic logic and logistic are once more brought together, since the logic presented in the early sections provides, in a way, the method by which the other branches of mathematics are developed. The Formulaire is a monumental production. But its mathematical interests are as much encyclopedic as logistic, and not all the possibilities of the method are utilized or made clear. It remained for Whitehead and Russell, in Principia Mathematica, to exhibit the perfect union of symbolic logic and the logistic method in mathematics. The publication of this work undoubtedly marks an epoch in the history of the subject. The tendencies marked in the development of the algebra of logic from Boole to Schroder, in the development of the algebra of relatives from De Morgan to Schroder, and in the foundations for number theory of Cantor and Dedekind and Frege, are all brought together here. 2 Further researches will most likely be based upon the formulations of Principia Mathematica. We must now turn back and trace in more detail the development of symbolic logic. 3 A history of the subject will not be attempted, if by history is meant the report of facts for their own sake. Rather, we are interested in the cumulative process by which those results which most interest us today have come to be. Many researches of intrinsic value, but lying outside the main line of that development, will of necessity be neglected. Reference to these, so far as we are acquainted with them, will be found in the bibliography. 4 II. LEIBNIZ The history of symbolic logic and logistic properly begins with Leibniz. 5 In the New Essays on the Human Understanding, Philalethes is made to say : 6 "I begin to form for myself a wholly different idea of logic from that which I formerly had. I regarded it as a scholar's diversion, but I now see that, in the way you understand it, it is like a universal mathe- 2 Perhaps we should add "and the modern development of abstract geometry, as by Hilbert, Fieri, and others", but the volume of Principia which is to treat of geometry has not yet appeared. 3 The remainder of this chapter is not essential to an understanding of the rest of the book. But after Chapter i, historical notes and references are generally omitted. 4 Pp. 389-406. 6 Leibniz regards Raymond Lully, Athanasius Kircher, John Wilkins, and George Dalgarno (see Bibliography) as his predecessors in this field. But their writings contain little which is directly to the point. 8 Bk. iv, Chap, xvn, 9. 6 A Survey of Symbolic Logic matics." As this passage suggests, Leibniz correctly foresaw the general character which logistic was to have and the problems it would set itself to solve. But though he caught the large outlines of the subject and actually delimited the field of work, he failed of any clear understanding of the difficulties to be met, and he contributed comparatively little to the successful working out of details. Perhaps this is characteristic of the man. But another explanation, or partial explanation, is possible. Leibniz expected that the whole of science would shortly be reformed by the appli- cation of this method. This was a task clearly beyond the powers of any one man, who could, at most, offer only the initial stimulus and general plan. And so, throughout his life, he besought the assistance of learned societies and titled patrons, to the end that this epoch-making reform might be instituted, and never addressed himself very seriously to the more limited tasks which he might have accomplished unaided. 7 Hence his studies in this field are scattered through the manuscripts, many of them still unedited, and out of five hundred or more pages, the systematic results attained might be presented in one-tenth the space. 8 Leibniz's conception of the task to be accomplished altered somewhat during his life, but two features characterize all the projects which he entertained: (1) a universal medium ("universal language" or "rational language" or "universal characteristic") for the expression of science; and (2) a calculus of reasoning (or "universal calculus") designed to display the most universal relations of scientific concepts and to afford some sys- tematic abridgment of the labor of rational investigation in all fields, much as mathematical formulae abridge the labor of dealing with quantity and number. "The true method should furnish us with an Ariadne's thread, that is to say, with a certain sensible and palpable medium, which will guide the mind as do the lines drawn in geometry and the formulae for operations which are laid down for the learner in arithmetic." 9 This universal medium is to be an ideographic language, each single character of which will represent a simple concept. It will differ from existing ideographic languages, such as Chinese, through using a combina- 7 The editor's introduction to "Scientia Generalis. Characteristica" in Gerhardt's Philosophischen Schriften von Leibniz (Berlin, 1890), vn, gives an excellent account of Leibniz's correspondence upon this topic, together with other material of historic interest. (Work hereafter cited as G. Phil.) 8 See Gerhardt, op. dt. especially iv and vn. But Couturat, La logique de Leibniz (1901), gives a survey which will prove more profitable to the general reader than any study of the sources. 9 Letter to Galois, 1677, G. Phil, vn, 21. The Development of Symbolic Logic 7 tion of symbols, or some similar device, for a compound idea, instead of having a multiplicity of characters corresponding to the variety of things. So that while Chinese can hardly be learned in a lifetime, the universal characteristic may be mastered in a few weeks. 10 The fundamental char- acters of the universal language will be few in number, and will represent the "alphabet of human thought": "The fruit of many analyses will be the catalogue of ideas which are simple or not far from simple." n With this catalogue of primitive ideas this alphabet of human thought the whole of science is to be reconstructed in such wise that its real logical organiza- tion will be reflected in its symbolism. In spite of fantastic expression and some hyperbole, we recognize here the program of logistic. If the reconstruction of all science is a project too ambitious, still we should maintain the ideal possibility and the desirability of such a reconstruction of exact science in general. And the ideographic language finds its realization in Peano's Formulaire, in Principia Mathe- matica, and in all successful applications of the logistic method. Leibniz stresses the importance of such a language for the more rapid and orderly progress of science and of human thought in general. The least effect of it "... will be the universality and communication of different nations. Its true use will be to paint not the word . . . but the thought, and to speak to the understanding rather than to the eyes. . . . Lacking such guides, the mind can make no long journey without losing its way . . . : with such a medium, we could reason in metaphysics and in ethics very much as we do in geometry and in analytics, because the characters would fix our ideas, which are otherwise too vague and fleeting in such matters in which the imagination cannot help us unless it be by the aid of characters." 12 The lack of such a universal medium prevents cooperation. "The human race, considered in its relation to the sciences which serve our welfare, seems to me comparable to a troop which marches in confusion in the darkness, without a leader, without order, without any word or other signs for the regulation of their march and the recognition of one another. Instead of joining hands to guide ourselves and make sure of the road, we run hither and yon and interfere with one another." 13 The "alphabet of human thought" is more visionary. The possibility of constructing the whole of a complex science from a few primitive con- 10 Letter to the Duke of Hanover, 1679 (?), G. Phil, vii, 24-25. 11 G. Phil, vii, 84. 12 G. Phil., vii, 21. 18 G. Phil, vii, 157. 8 A Survey of Symbolic Logic cepts is, indeed, real vide the few primitives of Principia Mathematica. But we should today recognize a certain arbitrariness in the selection of these, though an arbitrariness limited by the nature of the subject. The secret of Leibniz's faith that these primitive concepts are fixed in the nature of things will be found in his conception .of knowledge and of proof. He believes that all predicates are contained in the (intension of the) subject and may be discovered by analysis. Similarly, all truths which are not absolutely primitive and self-evident admit of reduction by analysis into such absolutely first truths. And finally, only one real definition of a thing "real" as opposed to "nominal" is possible; 14 that is, the result of the correct analysis of any concept is unambiguously predetermined in the concept itself. The construction, from such primitives, of the complex concepts of the various sciences, Leibniz speaks of as "synthesis" or "invention", and he is concerned about the "art of invention". But while the result of analysis is always determined, and only one analysis is finally correct, synthesis, like inverse processes generally, has no such predetermined character. In spite of the frequent mention of the subject, the only im- portant suggestions for this art have to do with the provision of a suitable medium and of a calculus of reasoning. To be sure there are such obvious counsels as to proceed from the simple to the complex, and in the early essay, De Arte Combinatoria, there are studies of the possible permutations and combinations or "syntheses" of fundamental concepts, but the author later regarded this study as of little value. And in Initia et Specimina Scientice nova Generalis, he says that the utmost which we can hope to accomplish at present, toward the general art of invention, is a perfectly orderly and finished reconstruction of existing science in terms of the absolute primitives which analysis reveals. 15 After two hundred years, we are still without any general method by which logistic may be used in fields as yet unexplored, and we have no confidence in any absolute primi- tives for such investigation. The calculus of reasoning, or universal calculus, is to be the instrument for the development and manipulation of systems in the universal language, and it is to get its complete generality from the fact that all science will be expressed in the ideographic symbols of that universal medium. The calculus will consist of the general principles of operating with such ideo- 14 See G. Phil, vn, 194, footnote. 15 G. Phil, vn, 84. The Development of Symbolic Logic 9 graphic symbols: "All our reasoning is nothing but the relating and sub- stituting of characters, whether these characters be words or marks or images. " 16 Thus while the characteristica universalis is the project of the logistic treatment of science in general, the universal calculus is the pre- cursor of symbolic logic. The plan for this universal calculus changed considerably with the development of Leibniz's thought, but he speaks of it always as a mathe- matical procedure, and always as more general than existing mathematical methods. 17 The earliest form suggested for it is one in which the simple concepts are to be represented by numbers, and the operations are to be merely those of arithmetical multiplication, division, and factoring. When, later, he abandons this plan of procedure, he speaks of a general calculus which will be concerned with what we should nowadays describe as "types of order" w T ith combinations which are absolute or relative, symmetrical or unsymmetrical, and so on. 1 * His latest studies toward such a calculus form the earliest presentation of what we now call the "algebra of logic". But it is doubtful if Leibniz ever thought of the universal calculus as restricted to our algebra of logic: we can only say that it was intended to be the science of mathematical and deductive form in general (it is doubtful whether induction was included), and such as to make possible the appli- cation of the analytic method of mathematics to all subjects of which scientific knowledge is possible. Of the various studies to this end our chief interest will be in the early essay, De Arte Combinatorial and in the fragments which attempt to develop an algebra of logic. 20 Leibniz wrote De Arte Combinatoria when he was, in his own words, mx egressus ex Ephebis, and before he had any considerable knowledge of mathematics. It was published, he tells us, without his knowledge or consent. The intention of the work, as indicated by its title, is to serve the general art of rational invention, as the author conceived it. As has been mentioned, it seems that this end is to be accomplished by a complete analysis of concepts of the topic under investigation and a general survey of the possibilities of their combination. A large portion of the essay is concerned with the calculation of the possible forms of this and that type 16 G. Phil, vn, 31. 17 See New Essays on the Human Understanding, Bk. iv, Chap, xvn, 9-13. 18 See G. Phil, vn, 31, IQSff., and 204. 19 G. Phil., iv, 35-104. Also Gerhardt, Leibnizens malhematische Schriften (1859), v, 1-79. 20 Sdentia Generalis. Characteristica, xv-xx, G. Phil., vn. 10 A Survey of Symbolic Logic of logical construct: the various dyadic, triadic, etc., complexes which can be formed with a given number of elements; of the moods and figures of the syllogism; of the possible predicates of a given subject (the com- plexity of the subject as a concept being itself the key to the predicates which can be analyzed out of it); of the number of propositions from a given number of subjects, given number of predicate relations, and given number of quaestiones ; 21 of the variations of order with a given number of terms, and so on. In fact so much space is occupied with the computation of permutations and combinations that some of his contemporaries failed to discover any more important meaning of the essay, and it is most fre- quently referred to simply as a contribution to combinatorial analysis. 22 Beyond this the significance of the essay lies in the attempt to devise a symbolism which will preserve the relation of analyzable concepts to their primitive constituents. The particular device selected for this purpose representation of concepts by numbers is unfortunate, but the attempt itself is of interest. Leibniz makes application of this method to geometry and suggests it for other sciences. 23 In the geometrical illustration, the concepts are divided into classes. Class 1 consists of concepts or terms regarded as elementary and not further analyzable, each of which is given a number. Thereafter, the number is the symbol of that concept. Class 2 consists of concepts analyzable into (definable in terms of) those of Class 1. By the use of a fractional notation, both the class to which a concept belongs and its place in that class can be indicated at once. The denomi- nator indicates the number of the class and the numerator is the number of the concept in that class. Thus the concept numbered 7 in Class 2 is represented by 7/2. Class 3 consists of concepts definable in terms of those in Class 1 and Class 2, and so on. By this method, the complete analysis of any concept is supposed to be indicated by its numerical symbol. 24 21 Leibniz tells us that he takes this problem from the Ars Magna of Raymond Lully. See G. Phil., v, 62. 22 See letter to Tschirnhaus, 1678, Gerhardt, Math., iv, 451-63. Cf. Cantor, Geschichle d. Math., in, 39 ff. 23 See the Synopsis, G. Phil., iv, 30-31. 24 See Couturat, op. cit., appended Note vi, p. 554.$'. The concepts are arranged as follows (G. Phil., iv, 70-72): "Classis I; 1. Punctum, 2. Spatium, 3. intervallum, 4. adsitum seu contiguum, 5. dis- situm seu distans, 6. Terminus seu quae distant, 7. Insitum, 8. inclusum (v.g. centrum est insitum circulo, inclusum peripheriae), 9. Pars, 10. Totum, 11. idem, 12. diversum, 13. unum, 14. Numerus, etc. etc. [There are twenty-seven numbered concepts in this class.] "Classis II; 1. Quantitas est 14 T&V 9 (15). [Numbers enclosed in parentheses have their usual arithmetical significance, except that (15) signifies 'an indefinite number'.] 2. Includens est 6.10. III. 1. Intervallum est 2.3.10. 2. Aequale A rf/s 11. J. 3. Continuum est A ad B, si TOV A r) 9 est 4 et 7 TU B.; etc. etc." The Development of Symbolic Logic 1 1 In point of fact, the analysis (apart from any merely geometrical defects) falls far short of being complete. Leibniz uses not only the inflected Greek article to indicate various relations of concepts but also modal inflections indicated by et, si, quod, quam faciunt, etc. In later years Leibniz never mentions this work without apologizing for it, yet he always insists that its main intention is sound. This method of assuming primitive ideas which are arbitrarily symbolized, of introducing other concepts by definition in terms of these primitives and, at the same time, substituting a single symbol for the complex of defining symbols this is, in fact, the method of logistic in general. Modern logistic differs from this attempt of Leibniz most notably in two respects: (1) modern logistic would insist that the relations whereby two or more concepts are united in a definition should be analyzed precisely as the substantives are analyzed; (2) while Leibniz regards his set of primitive concepts as the necessary result of any proper analysis, modern logistic would look upon them as arbitrarily chosen. Leibniz's later work looks toward the elimina- tion of this first difference, but the second represents a conviction from which he never departed. At a much later date come various studies (not in Gerhardt), which attempt a more systematic use of number. and of mathematical operations in logic. 25 Simple and primitive concepts, Leibniz now proposes, should be symbolized by prime numbers, and the combination of two concepts (the qualification of one term by another) is to be represented by their product. Thus if 3 represent "rational" and 7 "animal", "man" will be 21. No prime number will enter more than once into a given combination a rational rational animal, or a rational animal animal, is simply a rational animal. Thus logical synthesis is represented by arithmetical multipli- cation: logical analysis by resolution into prime factors. The analysis of "man", 21, would be accomplished by finding its prime factors, "rational", 3, and "animal", 7. In accordance with Leibniz's conviction that all knowledge is analytic and all valid predicates are contained in the subject, the proposition "All S is P" will be true if the number which represents the. concept S is divisible by that which represents P. Accordingly the 25 Dated April, 1679. Couturat (op. tit., p. 326, footnote) gives the titles of these as follows: "Elemenla Characteristicae Universalis (Collected manuscripts of Leibniz in the Hanover Library, PHIL., v, 8 b); Calculi universalis Elemenla (PHIL., v, 8 c); Calculi universalis investigaliones (PHIL., v, 8 d); Modus examinandi consequentias per numeros (PHIL., v, 8 e); Regulae ex quibus de bonilate consequentiarum formisque et modis syllogis- morum categoricum judicari potest per numeros (PHIL., v, 8f)." These fragments, with many others, are contained in Couturat's Opuscules et fragments inedits de Leibniz. 12 A Survey of Symbolic Logic universal affirmative proposition may be symbolized by S/P = y or S = Py (where y is a whole number). By the plan of this notation, Py will represent some species whose "difference", within the genus P, is y. Similarly Sx will represent a species of S. Hence the particular affirmative, "Some S is P," may be symbolized by Sx = Py, or S/P = y/x. Thus the uni- versal is a special case of the particular, and the particular will always be true when the universal is true. There are several objections to this scheme. In the first place, it presumes that any part of a class is a species within the class as genus. This is far-fetched, but perhaps theoretically defensible on the ground that any part which can be specified by the use of language may be treated as a logical species. A worse defect lies in the fact that Sx = Py will always be true. For a given S and P, we can always find x and y which will satisfy the equation Sx = Py. If no other choice avails, let x = P, or some multiple of P, and y = S, or some multiple of S. " Angel-man" = "man-angel" although no men are angels. "Spineless man" = "ra- tional invertebrate", but it is false that some men are invertebrates. A third difficulty arises because of the existential import of the particular a difficulty which later drew Leibniz's attention. If the particular affirma- tive is true, then for some x and y-, Sx = Py. The universal negative should, then, be Sx =f Py. And since the universal affirmative is S = Py, the particular negative should be S =t= Py- But this symbolism would be practically unworkable because the inequations would have to be verified for all values of x and y. Also, as we have noted, the equality Sx = Py will always hold and Sx ={= Py, where x and y are arbitrary, will never be true. Such difficulties led Leibniz to complicate his symbolism still further, introducing negative numbers and finally using a pair of numbers, one positive and one negative, for each concept. But this scheme also breaks down, and the attempt to represent concepts by numbers is thereafter abandoned. Of more importance to symbolic logic are the later fragments included in the plans for an encyclopedia which should collect and arrange all known science as the proper foundation for future work. 26 Leibniz cherished the 26 G. Phil., vn, xvi-xx. Of these, xvi, without title, states rules for inference in terms of inclusion and exclusion; Difficultates quaedam logicae treats of subalternation and conversion and of the symbolic expression for various types of propositions; xvin, Specimen Calculi universalis with its addenda and marginal notes, gives the general prin- ciples of procedure for the universal calculus; xix, with the title Non inelegans specimen The Development of Symbolic Logic 13 notion that this should be developed in terms of the universal characteristic. In these fragments, the relations of equivalence, inclusion, and qualification of one concept by another, or combination, are defined and used. These relations are always considered in intension when it is a question of apply- ing the calculus to formal logic. "Equivalence" is the equivalence of concepts, not simply of two classes which have the same members; "for A to include B or B to be included in A is to affirm the predicate B universally of the subject A". 27 However, Leibniz evidently considers the calculus to have many applications, and he thinks out the relations and illustrates them frequently in terms of extensional diagrams, in which A, B, etc., are represented by segments of a right line. Although he preferred to treat logical relations in intension, he frequently states that relations of intension are easily transformed into relations of extension. If A is included in B in intension, B is included in A in extension; and a calculus may be inter- preted indifferently as representing relations of concepts in intension or relations of individuals and classes in extension. Also, the inclusion rela- tion may be interpreted as the relation of an antecedent proposition to a consequent proposition. The hypothesis A includes its consequence B, just as the subject A includes the predicate B. 28 This accords with his frequently expressed conviction that all demonstration is analysis. Thus these studies are by no means to be confined to the logic of intension. As one title suggests, they are studies demonstrandi in abstractis. demonstrandi in abstractis struck out, and xx, without title, are deductive developments of theorems of symbolic logic, entirely comparable with later treatises. The place of symbolic logic in Leibniz's plans for the Encyclopedia is sufficiently indicated by the various outlines which he has left. In one of these (G. Phil., vu, 49), divisions 1-6 are of an introductory nature, after which come: "7. De scientiarum instauratione, ubi de Systematibus et Repertoriis, et de Encyclo- paedia demonstrativa codenda. "8. Elementa veritatis aeternae, et de arte demonstrandi in omnibus disciplinis ut in Mathesi. "9. De novo quodam Calculo generali, cujus ope tollantur omnes disputationes inter eos qui in ipsum consenserit; est Cabala sapient um. "10. De Arte Inveniendi. "11. De Synthesi seu Arte combinatoria. "12. DeAnalysi. " 13. De Combinatoria speciali, seu scientia formarum, sive qualitatum in genere (de Characterismis) sive de simili et dissimili. "14. De Analysi speciali seu scientia quantitatum in genere seu de magno et parvo. " 15. De Mathesi generali ex duabus praecedentibus composita." Then various branches of mathematics, astronomy, physics, biological science, medi- cine, psychology, political science, economics, military science, jurisprudence, and natural theology, in the order named. 27 G. Phil, vii, 208. 28 "Generales Inquisitiones" (1686): see Couturat, Opuscuks etc., pp. 356-99. 14 A Survey of Symbolic Logic It is a frequent remark upon Leibniz's contributions to logic that he failed to accomplish this or that, or erred in some respect, because he chose the point of view of intension instead of that of extension. The facts are these: Leibniz too hastily presumed a complete, or very close, analogy between the various logical relations. It is a part of his sig- nificance for us that he sought such high generalizations and believed in their validity. He preferred the point of view of intension, or connotation, partly from habit and partly from rationalistic inclination. As a conse- quence, wherever there is a discrepancy between the intensional and ex- tensional points of view, he is likely to overlook it, and to follow the former. This led him into some difficulties which he might have avoided by an opposite inclination and choice of example, but it also led him to make some distinctions the importance of which has since been overlooked and to avoid certain difficulties into which his commentators have fallen. 29 In Difficulties quaedam logicae, Leibniz shows that at last he recognizes the difficulty in connecting the universal and the corresponding particular. He sees also that this difficulty is connected with the disparity between the intensional point of view and the existential import of particular proposi- tions. In the course of this essay he formulates the symbolism for the four propositions in two different ways. The first formulation is: 30 Univ. aff.; All A is B: AB = A, or A non-5 does not exist. Part, neg.; Some A is not B; AB 4= A, or A non-5 exists. Univ. neg.; No A is B; AB does not exist. Part, aff.; Some A is B; AB exists. AB = A and AB 4= A may be interpreted as relations of intension or of extension indifferently. If all men are mortal, the intension of "mortal man" is the same as the intension of "man", and likewise the class of mortal men is identical in extent with the class of men. The statements concerning existence are obviously to be understood in extension only. The interpretation here put upon the propositions is identically that of contemporary symbolic logic. With these expressions, Leibniz infers the subaltern and the converse of the subaltern, from a given universal, by 29 For example, it led him to distinguish the merely non-existent from the absurd, or impossible, and the necessarily true from the contingent. See G. Phil., vn, 231, foot- note; and "Specimen certitudinis seu de conditionibus," Dutens, Leibnitii Opera, iv, Part in, pp. 92 ff., also Couturat, La Logique de Leibniz, p. 348, footnote, and p. 353, footnote. 30 G. Phil., viz, 212. The Development of Symbolic Logic 15 means of the hypothesis that the subject, A, exists. Later in the essay, he gives another set of expressions for the four propositions : 31 All ,4 is B: AB = A. Some A is not B: AB =(= A. No A is B: AB does not exist, or AB 4= AB Ens. Some A is B : AB exists, or AB = AB Ens. In the last two of these, AB before the sign of equality represents the possible AB's or the AB "in the region of ideas"; "AB Ens" represents existing AB's, or actual members of the class AB. (Read AB Ens, " AB which exists".) AB = AB Ens thus represents the fact that the class AB has members; AB 4= AB Ens, that the class AB has no members. A logical species of the genus A, "some A", may be represented by YA; YA Ens will represent existing members of that species, or "some exist- ing A". Leibniz correctly reasons that if AB = A (All A is J5), YAB = YA (Some A is B); but if AB 4= A, it does not follow that YAB 4= YA, for if Y = B, YAB = YA. Again, if AB 4= AB Ens (No A is B), YAB 4= YAB Ens (It is false that some A is 5); but if AB = AB Ens (Some A is B), YAB = YAB Ens does not follow, because Y could assume values incompatible with A and B. For example, some men are wise, but it does not follow that foolish men are foolish wise persons, because "foolish" is incompatible with "wise". 32 The distinction here between AB, a logical division of A or of B, and AB Ens, existing AB's, is ingenious. This is our author's most successful treatment of the relations of extension and intension, and of the particular to the universal. In Specimen calculi universalis, the "principles of the calculus" are announced as follows : 33 1) "Whatever is concluded in terms of certain variable letters may be concluded in terms of any other letters which satisfy the same conditions; for example, since it is true that [all] ab is a, it will also be true that [all] be is b and that [all] bed is be. . . . 2) "Transposing letters in terms changes nothing; for example ab coincides with ba, 'animal rational' with 'rational animal'. 3) "Repetition of a letter in the same term is useless. . . . 4) "One proposition can be made from any number by joining all the subjects in one subject and all the predicates in one predicate : Thus, a is 6 and c is d and e is /, become ace is bdf. . . . 31 G. Phil., vii, 213-14. 32 G. Phil., vn, 215: the illustration is mine. 33 G. Phil, vn, 224-25. 16 A Survey of Symbolic Logic 5) "From any proposition whose predicate is composed of more than one term, more than one proposition can be made; each derived proposition having the subject the same as the given proposition but in place of the given predicate some part of the given predicate. If [all] a is bed, then [all] a is b and [all] a is c and [all] a is d. " 24 If we add to the number of these, two principles which are announced under the head of "self-evident propositions" (1) a is included in a; and (2) ab is included in a we have here the most important of the funda- mental principles of symbolic logic. Principle 1 is usually qualified by some doctrine of the "universe of discourse" or of "range of significance", but some form of it is indispensable to algorithms in general. The law numbered 2 above is what we now call the "principle of permutation"; 3, the "principle of tautology"; 4, the "principle of composition"; 5, the "principle of division". And the two "self-evident propositions" are often included in sets of postulates for the algebra of logic. There remain for consideration the two fragments which are given in translation in our Appendix, XIX and XX of Scientia Generalis: Char- acteristica. The first of these, with the title Non ineleqans specimen demon- strandi in abstractis, stricken out in the manuscript, is rather the more inter- esting. Here the relation previously symbolized by AB or ab is represented by A+B. And A+B = L signifies that A is contained or included in (est in) B. A scholium attached to the definition of this inclusion relation distinguishes it from the part-whole relation. Comparison of this and other passages shows that Leibniz uses the inclusion relation to cover (1) the relation of a member of the class to the class itself; (2) the relation of a species, or subclass, to its genus a relation in extension; (3) the rela- tion of a genus to one of its species a relation of intension. The first of these is our e-relation; (2) is the inclusion relation of the algebra of logic; and (3) is the analogous relation of intension. Throughout both these fragments, it is clear that Leibniz thinks out his theorems in terms of extensional diagrams, in which classes or concepts are represented by segments of a line, and only incidently in terms of the intension of concepts. The different interpretations of the symbols must be carefully dis- tinguished. If A is "rational" and B is "animal", and A and B are taken in intension, then A+B will represent "rational animal". But if A and B are classes taken in extension, then A + B is the class made up of those things which are either A or B (or both). Thus the inclusion relation, 34 4. and 5. are stated without qualification because this study is confined to the proper- ties of universal affirmative propositions. 4. is true also for universal negatives. The Development of Symbolic Logic 17 A + B = L, may be interpreted either in intension or in extension as "A is in L ". This is a little confusing to us, because we should nowadays invert the inclusion relation when we pass from intension to extension; instead of this, Leibniz changes the meaning of A + B from "both A and B" (in intension) to "either A or B" (in extension). If A is "rational", B "ani- mal ", and L "man", then A + B = L is true in intension, "rational animal" = "man" or "rational" is contained in "man". If A, B, and L are classes of points, or segments of a line, then A + B = L will mean that L is the class of points comprising the points in A and the points in B (any points common to A and B counted only once), or the segment made up of segments A and B. The relation A+B does not require that A and B should be mutually exclusive. If L is a line, A and B may be overlapping segments; and, in intension, A and B may be overlapping concepts, such as "triangle" and "equilateral", each of which contains the component "figure". Leibniz also introduces the relation L A, which he calls detractio. L A N signifies that L contains A and that if A be taken from L the remainder is N. The relations [+] and [ ] are not true inverses: if A+B = L, it does not follow that L A = B, because A and B may be overlapping (in Leibniz's terms, communicantia) . If L A = N, A and N must be mutually exclusive (incommunicantia) . Hence if A+B = L and A and B have a common part, M, L A = B M. (If the reader will take a line, L, in which A and B are overlapping segments, this will be clear.) This makes the relation of detractio somewhat confusing. In extension, L A may be interpreted " L which is not A ". In intension, it is more difficult. Leibniz offers the example: "man" - "rational" = "brute", and calls our attention to the fact that "man" "rational" is not "non-rational man" or "man"+ "non-rational". 35 In intension, the relation seems to indicate an abstraction, not a negative qualification. But there are difficulties, due to the overlapping of concepts. Say that " man" + "woodworking" = "carpenter" and " man" + "white-skinned" 36 G. Phil., vii, 231, footnote. Couturat in commenting on this (op. tit., pp. 377-78) says: "Ailleurs Leibniz essaie de pr6ciser cette opposition en disant: 'A A est Nihilum. Sed A non-A est Absurdum.' "Mais il oublie que le n6ant (non-Ens) n'est pas autre chose que ce qu'il appelle 1'absurde ou 1'impossible, c'est-a-dire le contradictoire. " It may be that Couturat, not Leibniz, is confused on this point. Non-existence may be contingent, as opposed to the necessary non-existence of the absurd. And the result of abstracting A from the concept A seems to leave merely non-Ens, not absurdity. 3 18 A Survey of Symbolic Logic = "Caucasian". Then " Caucasian "+" carpenter " = "man" + "white- skinned " + " woodworking ". Hence (" Caucasian " + " carpenter ") " car- penter" = "white-skinned", because the common constituent "man" has been abstracted in abstracting "carpenter". That is, the abstraction of "carpenter" from "Caucasian carpenter" leaves, not "Caucasian" but only that part of the concept "Caucasian" which is wholly absent in "carpenter". We cannot here say "white-skinned man" because "man" is abstracted, nor "white-skinned animal" because "animal" is contained in "man": we can only say "white-skinned" as a pure abstraction. Such abstraction is difficult to carry out and of very little use as an instrument of logical analysis. Leibniz's illustration is scribbled in the margin of the manuscript, and it seems clear that at this point he was not thinking out his theorems in terms of intensions. Fragment XX differs from XIX in that it lacks the relation symbolized by [ ]. This is a gain rather than a loss, both because of the difficulty of interpretation and because [+ ] and [ ] are not true inverses. Also XX is more carefully developed : more of the simple theorems are proved, and more illustrations are given. Otherwise the definitions, relations, and methods of proof are the same. In both fragments the fundamental operation by which theorems are proved is the substitution of equivalent expressions. If the successors of Leibniz had retained the breadth of view which characterizes his studies and aimed to symbolize relations of a like generality, these fragments might well have proved sufficient foundation for a satis- factory calculus of logic. III. FROM LEIBNIZ TO DE MORGAN AND BOOLE After Leibniz, various attempts were made to develop a calculus of logic. Segner, Jacques Bernoulli, Ploucquet, Tonnies, Lambert, Holland, Castillon, and others, all made studies toward this end. Of these, the most important are those of Ploucquet, Lambert and Castillon, while one of Holland's is of particular interest because it intends to be a calculus in extension. But this attempt was not quite a success, and the net result of the others is to illustrate the fact that a consistent calculus of logical relations in intension is either most difficult or quite impossible. Of Segner's work and Ploucquet's we can give no account, since no copies of these writings are available. 36 Venn makes it clear that Plouc- 36 There seem to be no copies of Ploucquet's books in this country, and attempts to secure them from the continent have so far failed. The Development of Symbolic Logic 19 quet's calculus was a calculus of intension and that it involved the quanti- fication of the predicate. Lambert 37 wrote voluminously on the subject of logic, but his most important contribution to symbolic procedure is contained in the Seeks Versuche einer Zeichenkunst in der Vernunftlehre. These essays are not separate studies, made from different beginnings; later essays presuppose those which precede and refer to their theorems; and yet the development is not entirely continuous. Material given briefly in one will be found set forth more at length in another. And discussion of more general prob- lems of the theory of knowledge and of scientific method are sometimes introduced. But the important results can be presented as a continuous development which follows in general the order of the essays. Lambert gives the following list of his symbols: The symbol of equality (Gleichgiiltigkeit) = addition (Zusetzung) + abstraction (Absonderung) opposition (des Gegentheils} X universality > particularity < copula given concepts (Begri/e) a, b, c, d, etc. undetermined concepts n, m, I, etc. unknowns x, y, z. the genus 7 the difference d The calculus is developed entirely from the point of view of intension: the letters represent concepts, not classes, [ + ] indicates the union of two concepts to form a third, [ ] represents the withdrawal or abstraction of some part of the connotation of a concept, while the product of a and b represents the common part of the two concepts. 7 and 5 qualify any term "multiplied" into them. Thus ay represents the genus of a, ad the difference of a. Much use is made of the well-known law of formal logic that the concept (of a given species) equals the genus plus the difference. (1) ay + ad = a(y +5) = a 37 Johann Heinrich Lambert (1728-77), German physicist, mathematician, and astrono- mer. He is remembered chiefly for his development of the equation x n +px = q in an infinite series, and his proof, in 1761, of the irrationality of v. 38 In Logische und philosophische Abhandlungen; ed. Joh. Bernoulli (Berlin, 1782), vol. i. 20 A Survey of Symbolic Logic ay + ad is the definition or explanation (Erklarung) of a. As immediate consequences of (1), we have also (2) ay = a ad (3) a8 = a ay Lambert takes it for granted that [+ ] and [ ] are strictly inverse opera- tions. We have already noted the difficulties of Leibniz on this point. If two concepts, a and b, have any part of their connotation in common, then (a + &) b will not be a but only that part of a which does not belong also to b. If "European" and "carpenter" have the common part "man", then ("European"* "carpenter") minus "carpenter" is not "European" but "European" minus "man". And [+ ] and [ ] will not here be true inverses. But this difficulty may be supposed to disappear where the terms of the sum are the genus and difference of some concept, since genus ajid difference may be supposed to be mutually exclusive. We shall return to this topic later. More complex laws of genus and difference may be elicited from the fact that the genus of any given a is also a concept and can be "explained," as can also the difference of a. (4) a = a(y + 5) 2 = ay z + ayd + ady + ad 2 Proof: ay = ayy + ayd and ad = ady + add But a = ay + ad. Hence Q.E.D. That is to say: if one w y ish to define or explain a, one need not stop at giving its genus and difference, but may define the genus in terms of its genus and difference, and define the difference similarly. Thus a is equiva- lent to the genus of the genus of a plus the difference of the genus of a plus the genus of the difference of a plus the difference of the difference of a. This may be called a "higher" definition or "explanation" of a. Obviously, this process of higher and higher "explanation" may be carried to any length; the result is what Lambert calls his "Newtonian formula". We shall best understand this if we take one more preliminary step. Suppose the explanation carried one degree further and the resulting terms arranged as follows: a = a(y 3 + yyd + ydd + 5 3 ) + ydy + dyd + dyy + ddy The three possible arrangements of two y's and one d might be summarized The Development of Symbolic Logic 21 by 37 2 5; the three arrangements of two 6's and one 7 by 375 2 . With this convention, the formula for an explanation carried to any degree, n, is: n(n-l) n(n-l}(n-2) (5) a = a(y n + ny n ~ l b + - - - 7*- 2 5 2 + - - y*~ 3 5 3 + . . . etc. . O 1 This "Newtonian formula" is a rather pleasant mathematical conceit. Two further interesting laws are given: (6) a = ad + ay8 + ay 2 8 + ay 3 8 + . . . etc. Proof : a = ay + a8 But ay = ay 2 + ay 8 and a7 2 = ay 3 + ay 2 8 ay 3 = ay 4 + ay 3 8, etc. etc. (7) a = ay + ady + a8 2 y + a8 3 y + . . . etc. Proof: a = ay + a8 But a8 = a8y + a5 2 and ad 2 = a8 2 y + ad 3 a8 3 = a8 3 y + a6 4 , etc. etc. Just as the genus of a is represented by ay, the genus of the genus of a by a7 2 , etc., so a species of which a is genus may be represented by ay~ l , and a species of which a is genus of the genus by 7~ 2 , etc. In general, as ay n represents a genus above a, so a species below a may be represented by a ay~ n or y n Similarly ainy concept of which a is difference of the difference of the differ- ence . . . etc., may be represented by a aS-' or -~ Also, just as a = a(y + 8) n , where a is a concept and a(y + 8} n its "explana- tion", so- ^ = a, where ^' ls the concept and a the "explanation" of it. Certain cautions in the transformation of expressions, both with respect to "multiplication" and with respect to "division," need to be observed. 40 39 Seeks Versuche, p. 5. 40 Ibid., pp. 9-10. 22 A Survey of Symbolic Logic The concept ay 2 + ady is very different from the concept (ay + ad~)y, because (8) (ay + a8)y = 0(7+5)7 = ay(y+d) = ay while O7 2 + a8y is the genus of the genus of a plus the genus of the difference of a. Also 7 must be distinguished from . - 7 is the genus of any 7 77 species x of which a is the genus, i. e., (9) - 7 = a 7 But 07/7 is any species of which the genus of a is the genus, i. e., any species x such that a and x belong to the same genus. We turn now to consideration of the relation of concepts which have a common part. Similarity is identity of properties. Two concepts are similar if, and in so far as, they comprehend identical properties. In respect to the remaining properties, they are different. 41 ab represents the common properties of a and b. a ab represents the peculiar properties of a. a + b ab ab represents the peculiar properties of a together with the peculiar properties of b. It is evident from this last that Lambert does not wish to recognize in his system the law a + a = a; else he need only have written a + b ab. If x and a are of the same genus, then xy = ay and ax = ay = xy If now we symbolize by a \ b that part of a which is different from 6, 42 then (10) a\b + b\a+ ab + ab = a+b Also x x a = ay, or x = ay + x \ a ax = a8 a ax = ad a = ax + a8 ax a aS = ay = xy 41 Ibid., p. 10. 42 Lambert sometimes uses a \ b for this, sometimes a : b. The Development of Symbolic Logic 23 And since ay X "~~ 7 ax + a x = a ax + x\a = x ax = a a x = x x\a a\x a ax x a = x ax The fact that y is a property comprehended in x may be expressed by y = xy or by y + x y = .T. The manner in which Lambert deduces the second of these expressions from the first is interesting. 43 If y is a property of x, then y x is null. But by (10), 2xy + x y + y x = x + y Hence in this case, 2xy + x\y = x + y And since y = xy, 2y + x\y = x + y Hence y + x \ y = x He has subtracted y from both sides, in the last step, and we observe that ty y = y- This is rather characteristic of his procedure; it follows, throughout, arithmetical analogies which are quite invalid for logic. With the complications of this calculus, the reader will probably be little concerned. There is no general type of procedure for elimination or solution. Formulae of solution for different types of equation are given. They are highly ingenious, often complicated, and of dubious application. It is difficult to judge of possible applications because in the whole course of the development, so far as outlined, there is not a single illustration of a solution which represents logical reasoning, and very few illustrations of any kind. The shortcomings of this calculus are fairly obvious. There is too much reliance upon the analogy between the logical relations symbolized and their arithmetical analogues. Some of the operations are logically uninterpretable, as for example the use of numerical coefficients other than and 1. These have a meaning in the " Newtonian formula", but 2y either has no meaning or requires a conventional treatment which is not given. And in any case, to subtract y from both sides of 2y = x + y and get y = x represents no valid logical operation. Any adequate study of the properties of the relations employed is lacking, x = a + b is transformed into a = x b, regardless of the fact that a and b may have a common part and that 43 Seeks Versuche, p. 12. 24 A Survey of Symbolic Logic x b represents the abstraction of the whole of b from x. Suppose, for example, man = rational + animal. Then, by Lambert's procedure, we should have also rational = man animal. Since Leibniz had pointed out this difficulty, that addition and subtraction (with exactly these meanings) are not true inverses, it is the more inexcusable that Lambert should err in this. There is a still deeper difficulty here. As Lambert himself remarks, 44 no two concepts are so completely dissimilar that they do not have a common part. One might say that the concept "thing" (Lambert's word) or "be- ing" is common to every pair of concepts. This being the case, [ + ] and [ ] are never really inverse operations. Hence the difficulty will not really disappear even in the case of ay and a8; and a ay = a8, a ad = ay will not be strictly valid. In fact this consideration vitiates altogether the use of "subtraction" in a calculus based on intension. For the meaning of a b becomes wholly doubtful unless [ ] be treated as a wholly con- ventional inverse of f + ], and in that case it becomes wholly useless. The method by which Lambert treats the traditional syllogism is only remotely connected with what precedes, and its value does not entirely depend upon the general validity of his calculus. He reconstructs the whole of Aristotelian logic by the quantification of the predicate. 45 The proposition "All A is B" has two cases: (1) A = B, the case in which it has a universal converse, the concept A is identical with the concept B. (2) A > B, the case in which the converse is particular, the concept B comprehended in the concept A . The particular affirmative similarly has two cases : (1) A < B, thejcase in which the converse is a universal, the subject A comprehended within the predicate B. (2) The case in which the converse is particular. In this case the subject A is comprehended within a subsumed species of the predicate and the predicate within a subsumed species of the subject. Lambert says this may be expressed by the pair: mA > B and A < nB Those who are more accustomed to logical relations in extension must not make the mistake here of supposing that A > mA, and mA < A. mA is a species of A, and in intension the genus is contained in the species, Ibid., p. 12. 76td., pp. 93 jf. The Development of Symbolic Logic 25 not vice versa. Hence mA > B does not give A > B, as one might expect at first glance. We see that Lambert here translates "Some A" by mA, a species comprehended in A, making the same assumption which occurs in Leibniz, that any subdivision or portion of a class is capable of being treated as some species comprehended under that class as its genus. In a universal negative proposition Lambert says the subject and predicate each have peculiar properties by virtue of whose comprehension neither is contained in the other. But if the peculiar properties of the subject be taken away, then what remains is contained in the predicate; and if the peculiar properties of the predicate be taken away, then what remains is contained in the subject. Thus the universal negative is repre- sented by the pair m and A> B - n The particular negative has two cases : (1) When it has a universal affirmative converse, i. e., when some A is not B but all B is A. This is expressed by A < B (2) When it has not a universal affirmative converse. In this case a subsumed species of the subject is contained in the predicate, and a sub- sumed species of the predicate in the subject. mA > B and A < nB Either of the signs, < and >, may be reversed by transposing the terms. And if P < Q, Q > P, then for some I, P = IQ. Also, "multi- plication" and "division" are strict inverses. Hence we can transform these expressions as follows: A > B is equivalent to A = mB A B \ A B - n or pA = qB or = It is evident from these transformations and from the prepositional equiva- 26 A Survey of Symbolic Logic lents of the "inequalities" that the following is the full expression of these equations : (1) A = mB: All A is B and some B is not A. (2) nA = B: Some A is not B and all B is A. (3) mA = nB: Some, but not all, A is B, and some, but not all, B is A. (4) - = - : No A is 5. TO w The first noticeable defect here is that A/m = B/n is transformable into nA = mB and (4) can mean nothing different from (3). Lambert has, in fact, only four different propositions, if he sticks to the laws of his calculus: (1) A = B: All A is all B. (2) A = mB: All A is some B. (3) nA = B: Some A is all B. (4) mA = nB: Some A is some B. These are the four forms which become, in Hamilton's and De Morgan's treatises, the four forms of the affirmative. A little scrutiny will show that Lambert's treatment of negatives is a failure. For it to be consistent at all, it is necessary that " fractions" should not be transformed. But Lambert constantly makes such transformations, though he carefully re- frains from doing so in the case of expressions like A/m = B/n which are supposed to represent universal negatives. His method further requires that TO and n should behave like positive coefficients which are always greater than and such that m ={= n. This is unfortunate. It makes it impossible to represent a simple proposition without "entangling alliances". If he had taken a leaf from Leibniz's book and treated negative propositions as affirmatives with negative predicates, he might have anticipated the calculus of De Morgan. In symbolizing syllogisms, Lambert always uses A for the major term, B for the middle term, and C for the minor. The perfectly general form of proposition is : mA nB p q Hence the perfectly general syllogism will be : 46 mA nB Major -103. " Ibid., p. 107. p q Ibid., pp. 102-103. The Development of Symbolic Logic 27 nC vB Minor = Hn mv Conclusion C = A irq pp The indeterminates in the minor are always represented thus by Greek letters. The conclusion is de ved from the premises as follows : The major premise gives B = A. np The minor gives B = C. TTV mq up Hence - A = C. np irv and therefore C = A. n-q pp The above being the. general form of the syllogism, Lambert's scheme of moods in the first figure is the following: it coincides with the traditional classification only so far as indicated by the use of the traditional names: B = mA nB = mA I. VII. C = vB U,C = B Barbara Lilii C = mvB n/j.C = mA B = A II. C B VIII. B = A 7T U.C = nB Canerent C A Magogos uC = nA 7T p III. B A IX. B = A Decane q c P = vB Negligo q c P = vB sive sive C vA n vA Celarent Ferio _ = q P q P 28 A Survey of Symbolic Logic nB = A IV. C _B Fideleo 7T p nC _A 7T p X. Pilosos nB = A C = B n/j.C = A V. B = mA Gabini nC = vB sive U,C = mvA Darii nB = mA VI. C = B Hilario nC = mA nB = mA XL C = B Romano nC = mA nB = mA XII. uC = B Somnio * n/jiC = mA The difficulty about "division" does not particularly affect this scheme, since it is only required that if one of the premises involve "fractions", the conclusion must also. It will be noted that the mood Hilario is identi- cal in form with Romano, and Lilii with Somnio. The reason for this lies in the fact that nB = mA has two partial meanings, one affirmative and one negative (see above). Hilario and Lilii take the affirmative interpretation, as their names indicate; Romano and Somnio, the negative. Into the discussion of the other three figures, the reader will probably not care to go, since the manner of treatment is substantially the same as in the above. There are various other attempts to devise a convenient symbolism and method for formal logic; 48 but these are of the same general type, and they meet with about the same degree and kind of success. Two brief passages in which there is an anticipation of the logic of relatives possess some interest. 49 Relations, Lambert says, are "external attributes", by which he means that they do not belong to the object an sich. "Metaphysical" (i. e., non-logical) relations are represented by Greek letters. For example if / = fire, h = heat, and a = cause, f=a::h The symbol : : represents a relation which behaves like multiplication: 48 See in Seeks Versuche, v and vi. Also fragments "Uber die Vernunftlehre", in Logische und Philosophische Abhandlungen, i, xix and xx; and Anlage zur Architektonik, p. 190 ff. Versuche, pp. 19, 27 ff. The Development of Symbolic Logic 29 a : : h is in fact what Peirce and Schroder later called a "relative product". Lambert transforms the above equation into : f a -- = - Fire is to heat as cause to effect. h f Ji = - Fire is to cause as heat to effect. a - = Heat is to fire as effect to cause. / a The dot here represents Wirkuny (it might be, Wirklichkeit, in consonance with the metaphysical interpretation, suggestive of Aristotle, which he gives to Ursache). It has the properties of 1, as is illustrated elsewhere 50 by the fact that 7 may be replaced by this symbol. Lambert also uses powers of a relation. If a = (p : : b, and b =

2 : : c, a la v> = \- *c And more to the same effect. No use is made of this symbolism; indeed it is difficult to see how Lambert could have used it. Yet it is interesting that he should have felt that the powers of a relation ought to be logically important, and that he here hit upon exactly the concept by which the riddles of "mathematical induction" were later to be solved. Holland's attempt at a logical calculus is contained in a letter to Lam- bert. 51 He himself calls it an "unripe thought", and in a letter some three years later 52 he expresses a doubt if logic is really a purely formal discipline capable of mathematical treatment. But this study is of particular interest because it treats the logical classes in extension the only attempt at a symbolic logic from the point of view of extension from the time of Leibniz to the treatise of Solly in 1839. Holland objects to Lambert's method of representing the relation of concepts by the relation of lines, one under the other, and argues that the 6J Ibid., p. 21. 51 Johan. Lamberts deutscher Gelehrten Brief wechsel, Brief in, pp. 16 ff, 52 See Ibid., Brief xxvn, pp. 259 ff.

1, then the possible forms of judg- ment are as follows: (1) ^ = ; All S is all P. (2) | = y All S is some P. Now expresses negatively what I/* expresses positively. To say that an infinitely small part of a curved line is straight, means exactly : No part of a curved line is straight. (3) - = All S is not P. 1 oo (4) ^ = ^ Some S is all P. J o p (5) - = -T Some S is some P. j j S P (6) - = Some S is not P. J (7) = ?- All not-S is all P. oo See Ibid., Brief iv. The Development of Symbolic Logic 31 S P (8) = -j All not-S is some P. 00 / S P (9) = - All not-S is all not-P. oo oo (1), (2), and (9) Holland says are universal affirmative propositions; (3), (7), and (8), universal negatives; (4) and (5), particular affirmatives; (6), a particular negative. As Venn has said, this notation anticipates, in a way, the method of Boole. If instead of the fraction we take the value of the numerator indicated by it, the three values are where < v < 1, and S/oo = Q-S. But the differences between this and Boole's procedure are greater than the resemblances. The fractional form is a little unfortunate in that it suggests that the equations may be cleared of fractions, and this would give results which are logically uninterpretable. But Holland's notation can be made the basis of a completely successful calculus. That he did not make it such, is apparently due to the fact that he did not give the matter sufficient attention to elaborate the extensional point of view. He gives the following examples : Example 1 . All men H are mortal M All Europeans E are men H 7T M Ergo, E = [All Europeans are mortal] pir Example 2. All plants are organisms P = - A All plants are no animals P = 00 A Ergo, = [Some organisms are not animals] p oo 32 A Survey of Symbolic Logic R Example 3. All men are rational H = - P p All plants are not rational P = - 00 pH Ergo, All plants are no men P = oo In this last example, Holland has evidently transformed H = R/p into pH = R, which is not legitimate, as we have noted. pH = R would be "Some men are all the rational beings". And the conclusion P = pHj is also misinterpreted. It should be, "All plants are not some men". A correct reading would have revealed the invalid operation. Lambert replied vigorously to this letter, maintaining the superiority of the intensional method, pointing out, correctly, that Holland's calculus would not distinguish the merely non-existent from the impossible or contradictory (no calculus in extension can), and objecting to the use of oo in this connection. It is characteristic of their correspondence that each pointed out the logical defects in the logical procedure of the other, and neither profited by the criticism. Castillon's essay toward a calculus of logic is contained in a paper presented to the Berlin Academy in 1803. 54 The letters S, A, etc., represent concepts taken in intension, M is an indeterminate, S + M represents the "synthesis" of S and M, S M, the withdrawal or abstraction of M from S. S M thus represents a genus concept in w^hich S is subsumed, M being the logical "difference" of S in S M . Consonantly S + M, symbolizing the addition of some "further specification" to S, represents a species concept which contains (in intension) the concept S. The predicate of a universal affirmative proposition is contained in the subject (in intension). Thus "All S is A" is represented by S = A + M The universal negative "No S is A" is symbolized by S = - A + M = (- A) + M The concept S is something, M, from which A is withdrawn is no A. Particular propositions are divided into two classes, "real" and "il- lusory". A real particular is the converse of a universal affirmative; the 54 "Memoire sur un nouvel algorithme logique", in Memoires de I'Academie des Sciences de Berlin, 1803, Classe de philosophic speculative, pp. 1-14. See also his paper, "Reflexions sur la Logique", loc. cit., 1802. The Development of Symbolic Logic 33 illusory particular, one whose converse also is particular. The real particu- lar affirmative is A = S - M since this is the converse of S = A + M . The illusory particular affirmative is represented by S = A =F M Castillon's explanation of this is that the illusory particular judgment gives us to understand that some S alone is A, or that S is got from A by ab- straction (S = A M), when in reality it is A which is drawn from S by abstraction (S = M + A). Thus this judgment puts M where it should put + M; qne can, then, indicate it by S = A ^ M . The fact is, of course, that "Some S is A " indicates nothing about the relations of the concepts S and A except that they are not incompatible. This means, in intension, that if one or both be further specified in proper fashion, the results will coincide. It might w r ell be symbolized by S + N = A + M. We suspect that Castillon's choice of S = A ^ M is really governed by the consideration that S = A + M may be supposed to give S = A ^ M, the universal to give its subaltern, and that A = S M will also give S = A ^ M , that is to say, the real particular which is "All A is S" will also give S = A =F M. Thus "Some S is A" may be derived both from "All S is A" and from "All A is S", which is a de- sideratum. The illusory negative particular is, correspondingly, S = - A =F M Immediate inference works out fairly well in this symbolism. The universal affirmative and the real particular are converses. S = A + M gives A = S M , and vice versa. The universal negative is directly convertible. S = A + M gives A = S + M , and vice versa. The illusory par- ticular is also convertible. S = A =F M gives - A = S ^ M. Hence A = S ^ M , which comes back to S = A ^ M . A universal gives its subaltern S = A + M gives S = A =F M , and S = -A + M gives S = - A =F M. And a real particular gives also the converse illusory particular, for A = S M gives S = A + M, 4 34 A Survey of Symbolic Logic which gives its subaltern, S = A ^ M, which gives A = S ^ M . All the traditional moods and figures of the syllogism may be symbolized in this calculus, those which involve particular propositions being valid both for the real particular and for the illusory particular. For example: All M is A M = A + N AD /Sis M S = M + P All Sis ,4 :. 8 = A + (N + P) No M is A M = - A + N All Sis M S = M + P No SisA :. S=-A + (N + P) All Mia A M = A + N Some S is M S = M =F P or S = M - P .'. Some S is A .'. S = (A + N) =F P or S = (A + N) - P This is the most successful attempt at a calculus of logic in intension. The difficulty about "subtraction" in the XIX Fragment of Leibniz, and in Lambert's calculus, arises because M P does not mean " M but not P" or " M which is not P ". If it mean this, then [ + ] and [ ] are not true inverses. If, on the other hand, M P indicates the abstraction from the concept M of all that is involved in the concept P, then M P is difficult or impossible to interpret, and, in addition, the idea of negation cannot be represented by [ ]. How does it happen, then, that Castillon's notation works out so well when he uses [ ] both for abstraction and as the sign of negation? It would seem that his calculus ought to involve him in both kinds of difficulties. The answer is that Castillon has, apparently by good luck, hit upon a method in w r hich nothing is ever added to or subtracted from a determined concept, (Fand X(>)Y Nothing both X and Y and everything one or the other. 6. X(O)Y or both X(-)Y and XQY Everything either X or Y and some things both." Each of these propositions may, with due regard for the meaning of the sign O, be read or written backward, just as the simple propositions. The rule of transformation into other equivalent forms is slightly different: Change the quantity, or distribution, of any term and replace that term by its negative. We are not required, as with the simple propositions, to change at the same time the quality of the proposition. This difference is due to the manner in which the propositions are compounded. The rules for mediate, or "syllogistic", inference for these compound propositions are as follows : 70 "If any two be joined, each of which is [of the form of] 1, 3, 4, or 6, with the middle term of different quantities, these premises yield a con- clusion of the same kind, obtained 'by erasing the symbols of the middle term and one of the symbols [Q]- Thus X}O(Y(O}Z gives X)O)Z: or if nothing be both X and Y and some things neither, and if everything be either Y or Z and some things both, it follows that all X and two lots of other things are Z's. "In any one of these syllogisms, it follows that may be written for )O) or )O( in one place, without any alteration of the conclusion, except reducing the two lots to one. But if this be done in both places, the con- clusion is reduced to j or , and both lots disappear. Let the reader examine for himself the cases in which one of the premises is cut down to a simple universal. "The following exercises will exemplify what precedes. Letters written under one another are names of the same object. Here is a universe of 12 instances of which 3 are X's and the remainder P's; 5 are F's and the remainder Q's; 7 are Z's and the remainder R's. XXX PP PP PPPPP YYY YY qq qqqqq Z Z Z Z Z Z Z RRRRR We can thus verify the eight complex syllogisms X)omo)Z P(0)Y)0)Z P(Q(Q(o)Z P(O(Q(O(R P(0)Y)0(R X)0)Y)0(R X)0(Q(0(R The Development of Symbolic Logic 43 In every case it will be seen that the two lots in the middle form the quantity of the particular proposition of the conclusion." In so much of his work as we have thus far reviewed, De Morgan is still too much tied to his starting point in Aristotelian logic. He somewhat simplifies traditional methods and makes new generalizations which include old rules, but it is still distinctly the old logic. He does not question the inference from universals to particulars nor observe the problems there involved. 71 He does not seek a method by which any number of terms may be dealt with but accepts the limitation to the traditional two. And his symbolism has several defects. The dot introduced between the parenthetic curves is not the sign of negation, so as to make it possible to read () as, "It is false that ()". The negative of () is )(, so that this simplest of all relations of propositions is represented by a complex trans- formation applicable only when no more than two terms are involved in the prepositional relation. Also, there are two distinct senses in which a term in a proposition may be distributed or "mentioned universally", and De Morgan, following the scholastic tradition, fails to distinguish them and symbolizes both the same way. This is the secret of the difficulty in reading X)(Y, which looks like "All X is all Y", and really is "Some things are neither X nor Y ". 72 Mathematical symbols are introduced but without any corresponding mathematical operations. The sign of equality is used both for the symmetrical relation of equivalent propositions and for the un- symmetrical relation of premises to their conclusion. 73 His investigation of the logic of relations, however, is more successful, and he laid the foundation for later researches in that field. This topic is suggested to him by consideration of the formal and material elements in logic. He says: 74 71 But he does make the assumption upon which all inference (in extension) of a particular from a universal is necessarily based : the assumption that a class denoted by a simple term has members. He says (F. L., pp. 110), "Existence as objects, or existence as ideas, is tacitly claimed for the terms of every syllogism". 72 A universal affirmative distributes its subject in the sense that it indicates the class to which every member of the subject belongs, i. e., the class denoted by the predicate. Similarly, the universal negative, No X is Y, indicates that every X is not-Y", every F is not-X. No particular proposition distributes a term in that sense. The particular nega- tive tells us only that the predicate is excluded from some unspecified portion of the class denoted by the subject. X)(Y distributes X and Y in this sense only. Comparison with its equivalents shows us that it can tell us, of X, only that it is excluded from some un- specified portion of not-F; and of Y, only that it is excluded from some unspecified portion of not-X. We cannot infer that X is wholly included in Y, or Y in X, or get any other relation of inclusion out of it. 73 In one passage (Carafe. Phil. Trans., x, 183) he suggests that the relation of two premises to their conclusion should be symbolized by A B < C. 74 Camb. Phil. Trans., x, 177, footnote. 44 A Survey of Symbolic Logic "Is there any consequence without /orm? Is not consequence an action of the machinery? Is not logic the science of the action of the machinery? Consequence is always an act of the mind : on every consequence logic ought to ask, What kind of act? What is the act, as distinguished from the acted on, and from any inessential concomitants of the action ? For these are of the form, as distinguished from the matter. "... The copula performs certain functions; it is competent to those functions . . . because it has certain properties, which are sufficient to validate its use. . . . The word 'is,' which identifies, does not do its work because it identifies, except insofar as identification is a transitive and convertible motion: 'A is that which is B' means 'A is B'; and 'A is B' means 'B is A'. Hence every transitive and convertible relation is as fit to validate the syllogism as the copula 'is', and by the same proof in each case. Some forms are valid when the relation is only transitive and not convertible; as in 'give'. Thus if X Y represent X and Y connected by a transitive copula, Camestres in the second figure is valid, as in Every ZY, No X Y, therefore No X Z. ... In the following chain of propositions, there is exclusion of matter, form being preserved at every step : Hypothesis (Positively true) Every man is animal Every man is Y Y has existence. Every X is Y X has existence. Every X Y - is a transitive relation. a of A" Y a is a fraction < or = 1. (Probability j8) a of X 7 )8 is a fraction < or = 1. The last is nearly the purely formal judgment, with not a single material point about it, except the transitiveness of the copula. 75 "... I hold the supreme form of the syllogism of one middle term to be as follows: There is the probability a that X is in the relation L to 7; there is the probability /3 that Y is in the relation M to Z; whence there is the probability a/3 that X is in the relation L of M to Z. 76 "... The copula of cause and effect, of motive and action, of all which post hoc is of the form and propter hoc (perhaps) of the matter, will one day be carefully considered in a more complete system of logic." 77 75 Ibid., pp. 177-78. 76 Ibid., p. 339. 77 Ibid., pp. 179-80. The Development of Symbolic Logic 45 De Morgan is thus led to a study of the categories of exact thinking in general, and to consideration of the types and properties of relations. His division of categories into logico-mathematical, logico-physical, logico- metaphysical, and logico-contraphysical, 78 is inauspicious, and nothing much comes of it. But in connection with this, and an attempt to rebuild logic in the light of it, he propounds the well-known theorem: "The con- trary [negative] of an aggregate [logical sum] is the compound [logical product] of the contraries of the aggregants: the contrary of a compound is the aggregate of the contraries of the components." 79 For the logic of relations, X, Y, and Z will represent the class names; L, M, N, relations. X . . LY will signify that X is some one of the objects of thought which stand to Y in the relation L, or is one of the L's of F. 80 X . L Y will signify that X is not any one of the L's of Y. X . . (LM) Y or X . . LM Y will express the fact that X is one of the L's of one of the M' s of Y, or that X has the relation L to some Z which has the relation M to Y. X . LM Y will mean that X is not an L of any M of Y. It should be noted that the union of the two relations L and M is what we should call today their "relative product" ; that is, X . .LY and Y . . MZ together give X . . LM Z, but X . . LY and X . . M Y do not give X . . LM Y. If L is the relation "brother of" and M is the relation "aunt of", X . . LM Y will mean " X is a brother of an aunt of F". (Do not say hastily, " X is uncle of F". "Brother of an aunt" is not equivalent to "uncle" since some uncles have no sisters.) L, or M, written by itself, will represent that which has the relation L, or M, that is, a brother, or an aunt, and LY stands for any X which has the relation L to Y, that is, a brother of I 7 . 81 In order to reduce ordinary syllogisms to the form in which the copula has that abstractness which he seeks, that is, to the form in which the copula may be any relation, or any relation of a certain type, it is necessary to introduce symbols of quantity. Accordingly LM * is to signify an L of every M, that is, something which has the relation L to every member of the class M (say, a lover of every man). L*M is to indicate an L of none but M's (a lover of none but men). The mark of quantity, * or *, always 78 See ibid., p. 190. 79 Ibid., p. 208. See also Syll., p. 41. Pp. 39-60 of Syll. present in summary the ideas of the paper, "On the Syllogism, No. 3, and on Logic in General.'' 80 Camb. Phil. Trans., x, 341. We follow the order of the paper from this point on. 81 1 tried at first to make De Morgan's symbolism more readily intelligible by intro- ducing the current equivalents of his characters. But his systematic ambiguities, such as the use of the same letter for the relation and for that which has the relation, made this impossible. For typographical reasons, I use the asterisk where he has a small accent. 46 A Survey of Symbolic Logic goes with the letter which precedes it, but L*M is read as if {*] modified the letter which follows. To obviate this difficulty, De Morgan suggests that L*M be read, "An every-!/ of M; an L of M in every way in which it is an L," but we shall stick to the simpler reading, "An L of none but M's". LM*X means an L of every M of X: L*MX, an L of none but M's of X: L*M*, an L of every M and of none but M's: LMX*, an L of an M of every X, and so on. Two more symbols are needed. The converse of L is symbolized by L~ l . If L is "lover of", L~ l is "beloved of"; if L is "aunt", L~ l is "niece or nephew ". The contrary (or as we should say, the negative) of L is symbol- ized by 1; the contrary of M by m. In terms of these relations, the following theorems can be stated : (1) Contraries of converses are themselves contraries. (2) Converses of contraries are contraries. (3) The contrary of the converse is the converse of the contrary. (4) If the relation L be contained in, or imply, the relation M , then (a) the converse of L, L~ l , is contained in the converse of M, M~ l \ and (6) the contrary of M, m, is contained in the contrary of L, I. For example, if "parent of " is contained in "ancestor of", (a) "child of" is contained in "descendent of", and (6) "not ancestor of" is contained in "not parent of". (5) The conversion of a compound relation is accomplished by converting both components and inverting their order; thus, (LM)~ l = M~ 1 L~*. If X be teacher of the child of Y, Y is parent of the pupil of X. When a sign of quantity is involved in the conversion of a compound relation, the sign of quantity changes its place on the letter; thus, (LM*)~ l = M*- 1 !.- 1 . If X be teacher of every child of Y, Y is parent of none but pupils of X. (6) When, in a compound relation, there is a sign of quantity, if each component be changed into its contrary, and the sign of quantity be shifted from one component to the other and its position on the letter changed, the resulting relation is equivalent to the original; thus LM * = l*m and L*M = 1m*. A lover of every man is a non-lover of none but non-men; and a lover of none but men is a non-lover of every non-man. The Development of Symbolic Logic 47 (7) When a compound relation involves no sign of quantity, its contrary is found by taking the contrary of either component and giving quantity to the other. The contrary of LM is IM* or L*m. "Not (lover of a man)" is "non-lover of every man" or "lover of none but non-men"; and there are two equivalents, by (6). But if there be a sign of quantity in one component, the contrary is taken by dropping that sign and taking the contrary of the other component. The contrary of LM * is IM ; of L*M is Lm. "Not (lover of every man)" is "non-lover of a man"; and "not (lover of none but men)" is "lover of a non-man". So far as they do not involve quantifications, these theorems are familiar to us today, though it seems not generally known that they are due to De Morgan. The following table contains all of them: Converse of Contrary Combination Converse Contrary Contrary of Converse LM M~ l L~ l IM* or L*M M*^l~ l or m' l L~ l * LM*orl*m M*~ l L~ l or m^l' 1 * IM M^H L*M or Im* M~ l L~ l * or m*~ l l~ l Lm m^L' 1 The sense in which one relation is said to be "contained in" or to "imply" another should be noted: L is contained in M in case every X which has the relation L to any Y has also the relation M to that F. This must not be confused with the relation of class inclusion between two rela- tive terms. Every grandfather is also a father, the class of grandfathers is contained in the class of fathers, but "grandfather of" is not contained in "father of", because the grandfather of Y is not also the father of Y. The relation "grandfather of" is contained in "ancestor of", since the grand- father of F is also the ancestor of F. But De Morgan appropriately uses the same symbol for the relation " L contained in M " that he uses for "All L is M ", where L and M are class terms, that is, L))M. In terms of this relation of relations, the following theorems can be stated : (8) If L))M, then the contrary of M is contained in the contrary of L, that is, L))M gives ra))/. Applying this theorem to compound relations, we have: (8') LM))N gives n))lM* and n))L*m. (8"} If LM))N, then L~ l n)}m and nlf- 1 ))/. Proof: If LM))N, then n)}lM*. Whence nM- l ))lM*M~ 1 . But an / of 48 A Survey of Symbolic Logic every M of an M~ l of Z must be an / of Z. Hence nM~ l ))l. Again; if LM))N, then n))Z,*m. Whence L- l n~))L- l L*m. But whatever has the relation converse-of-Z to an L of none but m's must be itself an m. Hence I- l n))m. De Morgan calls this "theorem K " from its use in Baroko and Bokardo. (9) If LM = N, then L))NM~ l and M))L~ 1 N. Proof: If LM = N, then LMM~ l = tf Jf- 1 and L~ 1 LM = L~ 1 N. Now for any X, MM~ 1 X and L~ 1 LX are classes which contain X', hence the theorem. We do not have L = NM~ l and M = L~ 1 N, because it is not generally true that MM~ 1 X = X and L~ 1 LX = X. For example, the child of the parent of X may not be X but A"'s brother : but the class " children of the parent of X" will contain X. The relation MM~ l or M~ 1 M will not always cancel out. MM' 1 and M~ 1 M are always symmetrical relations ; if XMM~ l Y then YMM~ 1 X. If X is child of a parent of Y, then Y is child of a parent of A'. But MM- 1 and M^M are not exclusively reflexive. XMM~ 1 X does not always hold. If we know that a child of the parent of X is a celebrated linguist we may not hastily assume that X is the linguist in question. With reference to transitive relations, we may quote : 82 "A relation is transitive when a relative of a relative is a relative of the same kind; as symbolized in LL)}L, whence ZZZ))ZZ))Z; and so on. "A transitive relation has a transitive converse, but not necessarily a transitive contrary: for L~ l L~ l is the converse of LL, so that ZZ))Z gives Z^Zr^Zr 1 . From these, by contraposition, and also by theorem K and its contrapositions, we obtain the following results : L is contained in LL- 1 *, Ul~ l , l~ l l*, L*~ 1 L L- 1 .......... L*L~ l , II- 1 *, l*~ l l, L~ 1 L* I ............. IL*,L*1 I- 1 ........... Z*- 1 /- 1 , HI- 1 * LL ........... L L^l, IL- 1 ...... I LI- 1 , l^L ...... H "I omit demonstration, but to prevent any doubt about correctness of printing, I subjoin instances in words: L signifies ancestor and L- 1 descendent. 82 Camb. Phil. Trans., x, 346. For this discussion of transitive relations, De Morgan treats all reciprocal relations, such as XLL~ 1 Y, as also reflexive, though not necessarily exclusively reflexive. The Development of Symbolic Logic 49 "An- ancestor is always an ancestor of all descendents, a non-ancestor of none but non-descendents, a nbn-descendent of all non-ancestors, and a descendent of none but ancestors. A descendent is always an ancestor of none but descendents, a non-ancestor of all non-descendents, a non-descend- ent of none but non-ancestors, and a descendent of all ancestors. A non- ancestor is always a non-ancestor of all ancestors, and an ancestor of none but non-ancestors. A non-descendent is a descendent of none but non- descendents, and a non-descendent of all descendents. Among non- ancestors are contained all descendents of non-ancestors, and all non- ancestors of descendents. Among non-descendents are contained all ancestors of non-descendents, and all non-descendents of ancestors." In terms of the general relation, L, or M, representing any relation, the syllogisms of traditional logic may be tabulated as follows: K 1 2 3 4 X ..LY X . LY X . .LY X .LY I Y ..LZ Y . .MZ Y . MZ Y .MZ X . . LMZ X . .IMZ X . .LmZ X . . ImZ X .LY X . .LY X . .LY X .LY II Z ..MY Z . .MY Z . MY Z .MY X . . IM~ 1 Z X . . LM~ 1 Z X . . Lm~ l Z X . . lm-U Y ..LX Y . LX Y . .LX Y .LX III Y .MZ Y . .MZ Y . .MZ Y .MZ X . . L~ l mZ X . . i-wz X . . L-WZ X . . l~ l mZ Y .LX Y . .LX Y . LX Y ..LX IV Z .MY Z . MY Z . .MY Z ..MY X 4- l mr l Z X . . L~ l m- l Z X . . 1~ 1 M~ 1 Z X . L- [ M- ] The Roman numerals here indicate the traditional figures. All the con- clusions are given in the affirmative form; but for each affirmative con- clusion, there are two negative conclusions, got by negating the relation and replacing it by one or the other of its contraries. Thus X . . LMZ gives X . IM*Z and X . L*mZ; X . . IM~ 1 Z gives X . LM~ l *Z and X . l*m~ l Z, and so on for each of the others. 83 Ibid., p. 350. 5 50 A Survey of Symbolic Logic When the copula of all three propositions is limited to the same transitive relation, L, or its converse, the table of syllogisms will be : 84 X..LY X.LY X..LY I Y..LZ Y..L~ 1 Z Y.L~ 1 Z X . . LZ X.LZ X. L-^Z X ..LY Z .LY X . L~ 1 Z X . LY X . .LY II z. .LY Z. .L-*Y X . LZ X . .LZ Y . .LX Y . LX III Y . LZ Y . .LZ X . LZ X . L~ 1 Z Y . .LX IV Z . L~ 1 Y X . LZ Y ..LX Y . . L~ 1 Z X . . L~ 1 Z Y.LX Y..LX Z..L~ 1 Y Z..LY X.L^Z X..L~ 1 Z" Here, again, in the logic of relations, De Morgan would very likely have done better if he had left the traditional syllogism to shift for itself. The introduction of quantifications and the systematic ambiguity of L, M, etc., which are used to indicate both the relation and that which has the relation, hurry him into complications before the simple analysis of rela- tions, and types of relations, is ready for them. This logic of relations was destined to find its importance in the logistic of mathematics, not in any applications to, or modifications of, Aristotelian logic. And these compli- cations of De Morgan's, due largely to his following the clues of formal logic, had to be discarded later, after Peirce discovered the connection between Boole's algebra and relation theory. The logic of relative terms has been reintroduced by the w r ork of Frege and Peano, and more especially of Whitehead and Russell, in the logistic development of mathematics. But it is there separated and has to be separated from the simpler analysis of the relations themselves. Nevertheless, it should always be remembered that it was De Morgan who laid the foundation; and if some part of his work had to be discarded, still his contribution was indispensable and of permanent value. In concluding his paper on relations, he justly remarks: 85 84 Ibid., p. 354. 85 Ibid., p. 358. The Development of Symbolic Logic 51 "And here the general idea of relation emerges, and for the first time in the history of knowledge, the notions of relation and relation of relation are symbolized. And here again is seen the scale of graduation of forms, the manner in which what is difference of form at one step of the ascent is difference of matter at the next. But the relation of algebra to the higher developments of logic is a subject of far too great extent to be treated here. It will hereafter be acknowledged that, though the geometer did not think it necessary to throw his ever-recurring principiwn et exemplum into imita- tion of Omnis homo est animal, Sortes est homo, etc., yet the algebraist was living in the higher atmosphere of syllogism, the unceasing composition of relation, before it was admitted that such an atmosphere existed." 86 V. BOOLE The beginning from which symbolic logic has had a continuous develop- ment is that made by George Boole. 87 His significant and vital contribution was the introduction, in a fashion more general and systematic than before, of mathematical operations. Indeed Boole allows operations which have no direct logical interpretation, and is obviously more at home in mathe- matics than in logic. It is probably the great advantage of Boole's w r ork that he either neglected or was ignorant of those refinements of logical theory which hampered his predecessors. The precise mathematical development of logic needed to make its own conventions and interpreta- tions; and this could not be done without sweeping aside the accumulated traditions of the non-symbolic Aristotelian logic. As we shall see, all the nice problems of intension and extension, of the existential import of uni- versals and particulars, of empty classes, and so on, return later and demand consideration. It is well that, with Boole, they are given a vacation long enough to get the subject started in terms of a simple and general procedure. Boole's first book, The Mathematical Analysis of Logic, being an Essay toward a Calculus of Deductive Reasoning, was published in 1847, on the 88 1 omit, with some misgivings, any account of De Morgan's contributions to prob- ability theory as applied to questions of authority and judgment. (See Syll, pp. 67-72; F. L., Chap, ix, x; and Camb. Phil. Trans., vm, 384-87, and 393-405.) His work on this topic is less closely connected with symbolic logic than was Boole's. The allied subject of the "numerically definite syllogism" (see Syll., pp. 27-30; F. L., Chap, vm; and Camb. Phil. Trans., x, *355-*358) is also omitted. 87 George Boole (1815-1864) appointed Professor of Mathematics in Queen's College, Cork, 1849; LL.D. (Dublin, 1852), F.R.S. (1857), D.C.L. (Oxford, 1859). For a biographi- cal sketch, by Harley, see Brit. Quart. Rev., XLIV (1866), 141-81. See also Proc. Roy. Soc., xv (1867), vi-xi. 52 , A Survey of Symbolic Logic same day as De Morgan 's Formal Logic. 68 The next year, his article, "The Calculus of Logic," appeared in the Cambridge Mathematical Journal. This article summarizes very briefly and clearly the important innovations pro- posed by Boole. But the authoritative statement of his system is found in An Investigation of the Laws of Thought, on which are founded the Mathe- matical Theories of Logic and Probability, published in 1854. 89 Boole's algebra, unlike the systems of his predecessors, is based squarely upon the relations of extension. The three fundamental ideas upon which his method depends are: (1) the conception of "elective symbols"; (2) the laws of thought expressed as rules for operations upon these symbols; (3) the observation that these rules of operation are the same which would hold for an algebra of the numbers and I. 90 For reasons which will appear shortly, the "universe of conceivable objects" is represented by 1. All other classes or aggregates are supposed to be formed from this by selection or limitation. This operation of electing, in 1, all the A"'s, is represented by l-x or x; the operation of electing all the F's is similarly represented by l-y or y, and so on. Since Boole does not distinguish between this operation of election represented by x, and the result of performing that operation an ambiguity common in mathe- maticsa- becomes, in practice, the symbol for the class of all the X's. Thus x, y, z, etc., representing ambiguously operations of election or classes, are the variables of the algebra. Boole speaks of them as " elective symbols" to distinguish them from coefficients. This operation of election suggests arithmetical multiplication: the 'suggestion becomes stronger when we note that it is not confined to 1. 1 x y or xy will represent the operation of electing, first, all the X 's in the "universe", and from this class by a second operation, all the F's. The result of these two operations will be the class whose members are both X's and Y's. Thus xy is the class of the common members of x and y; xyz, the class of those things which belong at once to x, to y, and to 2, and so on. And for any x, 1 -x = x. The operation of "aggregating parts into a whole" is represented by + . x + y symbolizes the class formed by combining the two distinct classes, x and y. It is a distinctive feature of Boole's algebra that x and y in x + y must have no common members. The relation may be read, "that which 88 See De Morgan's note to the article "On Propositions Numerically Definite", Camb. Phil. Trans., xi (1871), 396. 89 London, Walton and Maberly. 90 This principle appears for the first time in the Laws of Thought. See pp. 37-38. Work hereafter cited as L. of T. The Development of Symbolic Logic 53 is either x or y but not both". Although Boole does not remark it, x + y cannot be as completely analogous to the corresponding operation of ordinary algebra as xy is to the ordinary algebraic product. In numerical algebras a number may be added to itself: but since Boole conceives the terms of any logical sum to be " quite distinct", 91 mutually exclusive classes, x + x cannot have a meaning in his system. As we shall see, this is very awkward, because such expressions still occur in his algebra and have to be dealt with by troublesome devices. But making the relation x + y completely disjunctive has one advantage it makes possible the inverse relation of "subtraction". The "separa- tion of a part, x, from a whole, y", is represented by y x. If x + z = y, then since x and z have nothing in common, y x = z and y z = x. Hence [+ ] and [ ] are strict inverses. x + y, then, symbolizes the class of those things which are either members of x or members of y, but not of both, x-y or xy symbolizes the class of those things which are both members of x and members of y. x y repre- sents the class of the members of x which are not members of y the x's except the y's. [ = ] represents the relation of two classes which have the same members, i. e., have the same extension. These are the fundamental relations of the algebra. The entity (1 x) is of especial importance. This represents the universe except the x's, or all things which are not x's. It is, then, the supplement or negative of x. With the use of this symbolism for the negative of a class, the sum of two classes, x and y, which have members in common, can be represented by xy + x(l - y} + (1 - x)y. The first term of this sum is the class which are both x's and y's', the second, those which are x's but not y's; the third, those which are y's but not x's. Thus the three terms represent classes which are all mutually exclusive, and the sum satisfies the meaning of + . In a similar fashion, x + y may be expanded to x(l - y} + (l - x)y. Consideration of the laws of thought and of the meaning of these sym- bols will show us that the following principles hold : (1) xy = yx What is both x and y is both y and x. (2) x + y = y + x What is either x or y is either y or x. 91 See L. of T., pp. 32-33. 54 A Survey of Symbolic Logic (3) z(x + y) = zx + zy That which is both z and (either x or y) is either both z and x or both z and y. (4) z(x y) = zx zy That which is both 2 and (x but not y) is both z and a; but not both z and y. (5) If ar = y, then 22 = zi/ 2 + x = z + y x z = y z (fyx-y=-y + x This last is an arbitrary convention: the first half of the expression gives the meaning of the last half. It is a peculiarity of "logical symbols" that if the operation x, upon 1, be repeated, the result is not altered by the repetition : l-x = l-x-x = 1-X'X-x. . ,. Hence we have : (7) .-c 2 = x Boole calls this the "index law". 92 All these laws, except (7), hold for numerical algebra. It may be noted that, in logic, "If a: = y, then zx zy" is not reversible. At first glance, this may seem to be another difference between numerical algebra and the system in question. But "If zx = zy, then x = y" does not hold in numerical algebra when 2=0. Law (7) is, then, the distinguishing principle of this algebra. The only finite numbers for which it holds are and 1. All the above laws hold for an algebra of the numbers and 1. With this observation, Boole adopts the entire procedure of ordinary algebra, modified by the law x 2 = x, introduces numerical coefficients other than and 1, and makes use, on occasion, of the operation of division, of the properties of functions, and of any algebraic transformations which happen to serve his purpose. 93 This borrowing of algebraic operations which often have no logical interpretation is at first confusing to the student of logic; and commen- tators have seemed to smile indulgently upon it. An example will help: the derivation of the "law of contradiction" or, as Boole calls it, the "law of duality", from the "index law". 94 92 In Mathematical Analysis of Logic he gives it also in the form x n = x, but in L. of T. he avoids this, probably because the factors of x n x (e. g., x 3 x) are not always logically interpretable. 93 This procedure characterizes L. of T. Only and 1, and the fractions which can be formed from them appear in Math. An. of Logic, and the use of division and of fractional coefficients is not successfully explained in that book. 94 L. of T., p. 49. The Development of Symbolic Logic 55 Since x 2 = x, x x 2 = 0. Hence, factoring, x(l x) = 0. This transformation hardly represents any process of logical deduction. Whoever says "What is both x and x, x 2 , is equivalent to x; therefore what is both x and not-z is nothing" may well be asked for the steps of his reason- ing. Nor should we be satisfied if he reply by interpreting in logical terms the intermediate expression, x x 2 = 0. Nevertheless, this apparently arbitrary way of using uninterpretable algebraic processes is thoroughly sound. Boole's algebra may be viewed as an abstract mathematical system, generated by the laws we have noted, which has two interpretations. On the one hand, the "logical" or "elec- tive " symbols may be interpreted as variables whose value is either numeri- cal or numerical 1, although numerical coefficients other than and 1 are admissible, provided it be remembered that such coefficients do not obey the "index law" which holds for "elective" symbols. All the usual alge- braic transformations will have an interpretation in these terms. On the other hand, the "logical" or "elective" symbols may be interpreted as logical classes. For this interpretation, some of the algebraical processes of the system and some resultant expressions will not be expressible in terms of logic. But whenever they are interpretable, they will be valid conse- quences of the premises, and even when they are not interpretable, any further results, derived from them, which are interpretable, will also be valid consequences of the premises. It must be admitted that Boole himself does not observe the proprieties of his procedure. His consistent course would have been to develop this al- gebra without reference to logical meanings, and then to discuss in a thorough fashion the interpretation, and the limits of that interpretation, for logical classes. By such a method, he would have avoided, for example, the difficulty about x + x. We should have x + x = 2x, the interpretation of which for the numbers and 1 is obvious, and its interpretation for logical classes would depend upon certain conventions which Boole made and which will be explained shortly. The point is that the two interpretations should be kept separate, although the processes of the system need not be limited by the narrower interpretation that for logical classes. Instead of making this separation of the abstract algebra and its two interpretations, Boole takes himself to be developing a calculus of logic; he observes that its "axioms" are identical with those of an algebra of the numbers and 1; 95 95 L. of T., pp. 37-38. 56 A Survey of Symbolic Logic hence he applies the whole machinery of that algebra, yet arbitrarily rejects from it any expressions which are not finally interpretable in terms of logical relations. The retaining of non-interpretable expressions which can be transformed into interpretable expressions he compares to "the employ- ment of the uninterpretable symbol V 1 in the intermediate processes of trigonometry." 96 It would be a pretty piece of research to take Boole's algebra, find independent postulates for it (his laws are entirely insufficient as a basis for the operations he uses), complete it, and systematically investi- gate its interpretations. But neglecting these problems of method, the expression of the simple logical relations in Boole's symbolism will now be entirely clear. Classes w r il) be represented by x, y, z, etc.; their negatives, by (1 x), (1 y), etc. That which is both x and y will be xy; that which is x but not y will be a:(l y), etc. That which is x or y but not both, will be x + y, or x(l y} + (1 x)y. That which is x or y or both w r ill be x + (1 x)y i. e., that which is x or not x but yor xy + x(l - y) + (1 - x)y that w r hich is both x and y or x but not y or y but not x. 1 represents the "universe" or "everything". The logical significance of is determined by the fact that, for any y, Oy = : the only class \vhich remains unaltered by any operation of electing from it whatever is the class "nothing". Since Boole's algebra is the basis of the classic algebra of logic which is the topic of the next chapter it will be unnecessary to comment upon those parts of Boole's procedure which were taken over into the classic algebra. These will be clear to any who understand the algebra of logic in its current form or who acquaint themselves with the content of Chapter II. We shall, then, turn our attention chiefly to those parts of his method which are peculiar to him. Boole does not symbolize the relation "x is included in ?/". Conse- quently the only copula by which the relation of terms in a proposition can be represented is the relation =. And since all relations are taken in extension, x = y symbolizes the fact that x and y are classes with identical membership. Propositions must be represented by equations in which something is put = or = 1, or else the predicate must be quantified. Boole uses both methods, but mainly relies upon quantification of the predicate. This involves an awkward procedure, though one which still survives the introduction of a symbol v or w, to represent an indefinite 91 L. of T., p. 69. The Development of Symbolic Logic 57 class and symbolize "Some". Thus "All x is (some) y" is represented by x = vy: "Some x is (some) y", by wx = vy. If v, or w, were here "the indefinite class" or "any class", this method would be less objectionable. But in such cases v, or w, must be very definitely specified: it must be a class "indefinite in all respects but this, that it contains some members of the class to whose expression it is prefixed". 97 The universal affirmative can also be expressed, without this symbol for the indeterminate, as x(l y} = 0; "All x is y" means "That which is x but not y is nothing". Negative propositions are treated as affirmative propositions with a negative predi- cate. So the four typical propositions of traditional logic are expressed as follows: 98 All x is y: x = vy, or, x(l y} = 0. Xo x is y: x = v(l y), or xy = 0. Some x is y: vx = w(\ y), or, v = xy. Some x is not y: vx = iv(l y}, or, v = x(l y}. Each of these has various other equivalents which may be readily deter- mined by the laws of the algebra. To reason by the aid of this symbolism, one has only to express his premises explicitly in the proper manner and then operate upon the resultant equation according to the laws of the algebra. Or, as Boole more explicitly puts it, valid reasoning requires: 99 " 1st, That a fixed interpretation be assigned to the symbols employed in the expression of the data; and that the laws of the combination of these symbols be correctly determined from that interpretation. " 2nd, That the formal processes of solution or demonstration be con- ducted throughout in obedience to all the laws determined as above, with- out regard to the question of the interpretation of the particular results obtained. "3rd, That the final result be interpretable in form, and that it be actually interpreted in accordance with that system of interpretation which has been employed in the expression of the data." As we shall see, Boole's methods' of solution sometimes involve an uninterpretable stage, sometimes not, but there is provided a machinery by 97 L. of T., p. 63. This translation of the arbitrary v by "Some" is unwarranted, and the above statement is inconsistent with Boole's later treatment of the arbitrary coefficient. There is no reason why such an arbitrary coefficient may not be null. 98 See Math. An. of Logic, pp. 21-22; L. of T., Chap. iv. 99 L. of T., p. 68. 58 A Survey of Symbolic Logic which any equation may be reduced to a form which is interpretable. To comprehend this we must first understand the process known as the develop- ment of a function. With regard to this, we can be brief, because Boole's method of development belongs also to the classic algebra and is essentially the process explained in the next chapter. 100 Any expression in the algebra which involves x or (1 x) may be called a function of x. A function of x is said to be developed when it has the form Ax + B(l x}. It is here required that x be a "logical symbol", susceptible only of the values and 1. But the coefficients, A and B, are not so limited: A, or B, may be such a "logical symbol" which obeys the "law of duality", or it may be some number other than or 1, or involve such a number. If the function, as given, does not have the form Ax + B(l x}, it may be put into that form by observing certain interesting laws which govern coefficients. Let /(.r) = Ax + B(l - x} Then /(I) = A-l + B(l - 1) = A And /(O) = A-0 + B(l - 0) = B Hence f(x) = /(I) -x +/(0) (1 - z) Thus if /(*) = ^ , 2 x fm 1 + 1 -2- wm.. 1 + - 1 - 2^~l " 2^o - 2 Hence f(x) = 2x+ - (1 x) A developed function of two variables, x and y, will have the form: Axy + Bx(l - y) + C(l - x)y + D(l - x)(l - y) And for any function, f(x, y}, the coefficients are determined by the law: f(x,y) =/(!, l).sy+/(l,0).*(l - 30 +/(0, !)(! - x)y +/(0,0)-(1 -*)(!- y) 100 See Math. An. of Logic, pp. 60-69; L. of T., pp. 71-79; "The Calculus of Logic," Cambridge and Dublin Math. Jour., in, 188-89. That this same method of development should belong both to Boole's algebra and to the remodeled algebra of logic, in which + is not completely disjunctive, is at first surprising. But a completely developed function, in either algebra, is always a sum of terms any two of which have nothing in common. This accounts for the identity of form where there is a real and important difference in the meaning of the symbols. The Development of Symbolic Logic 59 Thus if f(x, y} = ax + 2by, /(I, 1) = o-l + 26-l = a + 26 /(I, 0) = a-1 + 26-0 = a /(O, 1) = a-0 + 26-1 = 26 /(0,0) = a-0 + 26-0 = Hence f(x, y) = (a + 2b)xy + ax(l y] + 26(1 x)y An exactly similar law governs the expansion and the determination of coefficients, for functions of any number of variables. In the words of Boole: 101 "The general rule of development will . . . consist of two parts, the first of which will relate to the formation of the constituents of the expansion, the second to the determination of their respective coefficients. It is as follows : " 1st. To expand any function of the symbols x, y, 2 Form a series of constituents in the following manner: Let the first constituent be the product of the symbols: change in this product any symbol z into 1 z, for the second constituent. Then in both these change any other symbol y into 1 y, for two more constituents. Then in the four constituents thus obtained change any other symbol x into 1 x, for four new constit- uents, and so on until the number of possible changes has been exhausted. "2ndly. To find the coefficient of any constituent If that constituent involves as a factor, change in the original function x into 1; but if it involves 1 x as a factor, change in the original function x into 0. Apply the same rule with reference to the symbols y, z, etc. : the final calculated value of the function thus transformed will be the coefficient sought." Two further properties of developed functions, which are useful in solutions and interpretations, are: (1) The product of any two constituents is 0. If one constituent be, for example, xyz, any other constituent will have as a factor one or more of the negatives, 1 x, I y, 1 z. Thus the product of the two will have a factor of the form x(l x). And where x is a "logical symbol ", susceptible only of the values and 1, x(l x) is always 0. And (2) if each constituent of any expansion have the coef- ficient 1, the sum of all the constituents is 1. All information which it may be desired to obtain from a given set of premises, represented by equations, will be got either (1) by a solution, to determine the equivalent, in other terms, of some "logical symbol" x, or 101 L. of T., pp. 75-76. 60 A Survey of Symbolic Logic (2) by an elimination, to discover what statements (equations), which are independent of some term x, are warranted by given equations which in- volve x, or (3) by a combination of these two, to determine the equivalent of x in terms of t, u, v, from equations which involve x, t, u, v, and some other "logical" symbol or symbols which must be eliminated in the desired result. " Formal " reasoning is accomplished by the elimination of "middle" terms. The student of symbolic logic in its current form knows that any set of equations may be combined into a single equation, that any equation involving a term x may be given the form Ax + B(l x) = 0, and that the result of eliminating x from such an equation is AB = 0. Also, the solution of any such equation, provided the condition AB = be satisfied, will be x = B + v(l A), where v is undetermined. Boole's methods achieve these same results, but the presence of numerical coefficients other than and 1, as well as the inverse operations of subtraction and division, necessarily complicates his procedure. And he does not present the matter of solutions in the form in which we should expect to find it but in a more complicated fashion which nevertheless gives equivalent results. We have now to trace the procedures of interpretation, reduction, etc. by which Boole obviates the difficulties of his algebra which have been mentioned. The simplest form of equation is that in which a developed function, of any number of variables, is equated to 0, as: Ax + B(l - x) = 0, or, Axy + Bx(l - y) + C(l - x)y+D(l - *)(! - y) = 0, etc. It is an important property of such equations that, since the product of any two constituents in a developed function is 0, any such equation gives any one of its constituents, whose coefficient does not vanish in the develop- ment, = 0. For example, if we multiply the second of the equations given by xy, all constituents after the first will vanish, giving Axy = 0. Whence we shall have xy = 0. Any equation in which a developed function is equated to 1 may be reduced to the form in which one member is by the law; If V = 1, 1-7-0. The more general form of equation is that in which some "logical symbol", w, is equated to some function of such symbols. For example, suppose x = yz, and it be desired to interpret z as a function of x and y. x = yz gives z = x/y; but this form is not interpretable. We shall, then, The Development of Symbolic Logic 61 develop x/y by the law f(x, y} = /(I, 1) -*y +/(!, 0) *(! - x = and if x =(= 0, x = 1 is exactly the principle which his successors added to his system when it is to be considered as a calculus of propositions. This principle would have made his system completely inapplicable to logical classes. For propositions, this principle means, " If x is not true, then x is false, and if x is not false, it is true". But careful attention to Boole's interpre- tation for "propositions" shows that in his system x = should be inter- The Development of Symbolic Logic 67 preted "x is false at all times (or in all cases)", and x = 1 should be in- terpreted "x is true at all times". This reveals that fact that what Boole calls " propositions " are what we should now call " prepositional functions ", that is, statements which may be true under some circumstances and false under others. The limitation put upon what we now call " propositions "- namely that they must be absolutely determinate, and hence simply true or false does not belong to Boole's system. And his treatment of "prepo- sitional symbols" in the application of the algebra to probability theory gives them the character of " prepositional functions" rather than of our absolutely determinate propositions. The last one hundred and seventy-five pages of the Laws of Thought are devoted to an application of the algebra to the solution of problems in probabilities. 106 This application amounts to the invention of a new method a method whereby any logical analysis involved in the problem is performed as automatically as the purely mathematical operations. We can make this provisionally clear by a single illustration : All the objects belonging to a certain collection are classified in three ways as ^4's or not, as B's or not, and as C's or not. It is then found that (1) a fraction m/n of the ^4's are also B's and (2) the C"s consist of the ^4's which are not B's together with the B's which are not A's. Required: the probability that if one of the A's be taken at random, it will also be a C. By premise (2) C = A(l - ) + (! - A)B Since A, B, and C are "logical symbols", A* = A and A(l A} = 0. Hence, AC = A\l - 5) + .4(1 - A)B = A(\ - B). The A's which are also C"s are identical with the ^4's which are not B's. Thus the probability that a given A is also a C is exactly the probability that it is not a B; or by premise (1), 1 m/n, which is the required solution. In any problem concerning probabilities, there are usually two sorts of difficulties, the purely mathematical ones, and those involved in the logical analysis of the situation upon which the probability in question depends. The methods of Boole's algebra provide a means for expressing the relations of classes, or events, given in the data, and then transforming these logical 106 Chap. 16 ff. See also the Keith Prize essay "On the Application of the Theory of Probabilities to the Question of the Combination of Testimonies or Judgments", Trans. Roy. Soc. Edinburgh, xxi, 597 ff. Also a series of articles in Phil. Mag., 1851-54 (see Bibl). An article on the related topic "Of Propositions Numerically Definite" appeared posthumously; Camb. Phil. Trans., xr, 396-411. 68 A Survey of Symbolic Logic equations so as to express the class which the quaesitum concerns as a func- tion of the other classes involved. It thus affords a method for untangling the problem often the most difficult part of the solution. The parallelism between the logical relations of classes as expressed in Boole's algebra and the corresponding probabilities, numerically expressed, is striking. Suppose x represent the class of cases (in a given total) in which the event X occurs or those which "are favorable to" the occurrence of X. 107 And let p be the probability, numerically expressed, of the event X. The total class of cases will constitute the logical "universe", or 1; the null class will be 0. Thus, if x = 1 if all the cases are favorable to X then p = 1 the probability of X is "certainty". If x = 0, then p = 0. Further, the class of cases in which X does not occur, will be expressed by 1 x; the probability that X will not occur is the numerical 1 p. Also, x+ (1 x} = 1 and p+ (1 p*) = 1. This parallelism extends likewise to the combinations of two or more events. If x represent the class of cases in which X occurs, and y the class of cases in which Y occurs, then xy will be the class of cases in which X and 7 both occur; #(1 y}, the cases in which X occurs without Y; (1 x)y, the cases in which Y occurs without X; (1 z)(l y), the cases in which neither occurs; x(l y} + y(l x), the cases in which X or 7 occurs but not both, and so on. Suppose that X and Y are " simple " and "independent" events, and let p be the probability of X, q the prob- ability of y. Then we have: Combination of events Corresponding probabilities expressed in Boole's algebra numerically expressed xy pq x(l - y} p(l - q) (1 - x}y (1 - q)p (1 - .r)(l - y) (I- p)(l - + ...) = s Wi + s^z + s W 3 + ... "A woman" is either W \ or W 2 or W 3 , etc.; "servant of a woman" is either 136 Peirce's notation for this is s w; he uses s, w for the simple logical product. The Development of Symbolic Logic 87 servant of Wi or servant of W z or servant of W 3 , etc. Similarly, " servant of every woman" is servant of W\ and servant of W 2 and servant of W 3 , etc. ; or remembering the interpretation of x , where, of course, s\ W n represents the relative product, "s of W n ," and x represents the non-relative logical product translated by "and". The above can be more briefly symbolized, following the obvious mathematical analogies, w = VW w = 2 w (s W) Unless w represent a null class, we shall have or s w cs w The class "servants of every woman" is contained in the class "servants of a woman". This law has numerous consequences, some of which are: (l\8)c(l\ 8 w} A lover of a servant of all women is a lover of a servant of a woman. J w c (J|*) A lover of every servant-of-all-women stands to every woman in the rela- tion of lover-of-a-servant of hers (unless the class s w be null). I s I c I s w A lover of every servant-of-a-woman stands to a (some) woman in the relation of lover-of-a-servant of hers. From the general principle, 136 134 The proof of this theorem is as follows: a = ab c ... +ab -c ... +a -b c ... + ..., or a = a b c . . . + P, where P is the sum of the remaining terms. Whence, if O represent any relation distributive with respect to + , mOa = mOa be ... +mOP Similarly, mQb = mOa be... +mQQ mQc = mOab c . . . +mQR Etc., etc. Now let o, b, c, etc., be respectively /(xi), /(%), /(a; 3 ), etc., and multiply together all the above equations. On the left side, we have 88 A Survey of Symbolic Logic we have also, *wc*w, or *c* A lover of a (some) servant-of-every-woman stands to every woman in the relation of lover-of-a-servant of hers. We have also the general formulae of inclusion, If Ics, then l w cs w and, If s c w, then l w c I" The first of these means: If all lovers are servants, then a lover of every woman is also a servant of every woman. The second means : If all servants are women, then a lover of every woman is also a lover of every servant. These laws are, of course, general. We have also: 137 (l\8)\W = l\(8 W) (/) = ;w The lovers of none but servants-of-none-but-women are the lovers-of- servants of none but women. l + s w = l w x s w Those who are either-lovers-or-servants of none but women are those who are lovers of none but women and servants of none but women. i *(w x v) = a w x 8 v The servants of none but those who are both women and violinists are those who are servants of none but. women and servants of none but vi7 linists. {l\a) w C (l") w Whoever is lover-of-a-servant of none but women is a lover-of-every- servant of none but women. l\ s w c <*Pw A lover of one who is servant to none but women is a lover-of-none-but- servants to none but women. l s w c l (s\w) 138 Ibid., p. 347. 139 Ibid., pp. 348 jf. 90 A Survey of Symbolic Logic Whoever stands to a woman in the relation of lover-of-nothing-but-servants of hers is a lover of nothing but servants of women. The two kinds of involution are connected by the laws : '(,) = (',) A lover of none but those who are servants of every woman is the same as one who stands to every woman in the relation of a lover of none but servants of hers. i s = -I - 14 Lover of none but servants is non-lover of every non-servant. It appears from this last that x and x y are connected through negation : -(/) = -l\s, Not a lover of every servant is non-lover of a servant. -('$) = l\-s, Not a lover of none but servants is lover of a non- servant. l -s = ~(l\s) = -I s , A lover of none but non-servants is one who is not lover-of-a-servant, a non-lover of every servant. ~ l s = -(-l\-s) = l~ s , A non-lover of none but servants is one who is not a non-lover-of-a-non-servant, a lover of every non-servant. We have the further laws governing negatives : 141 -[(Zx*)] = -(*)+ -(*<) -(+) = -(/) + -(/<) In the early paper, "On the Description of a Notation for the Logic of Relatives", negatives are treated in a curious fashion. A symbol is used for "different from" and the negative of s is represented by tt s , "differ- ent from every s". Converses are barely mentioned in this study. In the paper of 1880, converses and negatives appear in their usual notation, "relative addition" is brought in to balance "relative multiplication", and the two kinds of involution are retained. But in "The Logic of Relatives" in the Johns Hopkins Studies in Logic, published in 1882, involution has disappeared, converses and negatives and "relative addition" are retained. This last represents the final form of Peirce's calculus of relatives. We have here, (1) Relative terms, a, 6, ... x, y, z. (2) The negative of x, -x. 140 See ibid., p. 353. Not-x is here symbolized by (1 x). "i Alg. Log. 1880, p. 55. The Development of Symbolic Logic 91 (3) The converse of x, ^x. If x is "lover", ^x is "beloved"; if KE is "lover", z is "beloved". (4) Non-relative addition, a + b, "either a or 6". (5) Non-relative multiplication, a x6, or a b, "both a and b". (6) Relative multiplication, a\b, "a of a b". (7) Relative addition, a t b, "a of everything but 6's, a of every non-6 ". (8) The relations = and c , as before. (9) The universal relation, 1, "consistent with," which pairs every term with itself and with every other. (10) The null-relation, 0, the negative of 1. (11) The relation "identical with", I, which pairs every term with itself. (12) The relation "different from", N, which pairs any term with every other term which is distinct. 142 In terms of these, the fundamental laws of the calculus, in addition to those which hold for class-terms in general, are as follows : (1) w(a) = a (2) -(.) = .(-a) (3) (acfe) = (w&cwa) (4) If acb, then (a x) c(b\x) and (x a) c(x\b). (5) If a c b, then (a t x} c (b t x) and (x t a) c (x t 6). (6) x (a 1 6) = (x a)\b (7) x t (a 1 6) = (x t a) t b (8) z|(at&)c(z a) t& (9) (at 6) a; cot (6 a;) (10) (a x) + (b\x)c(a + b)\x (11) z|(at&)c(zta)(zt&) (12) (a+b)xc(a x) + (b\x) (13) (ataO(&tz)c(a|6) tz (14) -(at 6) = -a | -6 (15) -(a 1 6) = -at -6 (16) ^(a + 6) = v/a + 6 (17) (a 6) = a 6 (18) w(a 1 6) = a t 6 (19) w( |6) = a v& For the relations 1, 0, 7, and AT, the following additional formulae are given : 142 1 have altered Peirce's notation, as the reader may see by comparison. 92 A Survey of Symbolic Logic (20) Oca; (21) zcl (22) x + Q = x (23) x-1 = x (24) x + 1 = 1 (25) x-0 = (26) x t 1 = 1 (27) x = (28) 1 t x = 1 (29) x = (30) x\ N = x (31) x\I = x (32) N t x = x (33) I x = x (34) x + -x = 1 (35) x -x = (36) 7c[ztv(-aO] (37) [z | (-*)] c AT This calculus is, as Peirce says, highly multiform, and no general prin- ciples of solution and elimination can be laid down. 143 Not only the variety of relations, but the lack of symmetry between relative multiplication and relative addition, e. g., in (10)-(13) above, contributes to this multiformity. But, as we now know, the chief value of any calculus of relatives is not in any elimination or solution of the algebraic type, but in deductions to be made directly from its formulae. Peirce's devices for solution are, there- fore, of much less importance than is the theoretic foundation upon which his calculus of relatives is built. It is this which has proved useful in later research and has been made the basis of valuable additions to logistic development. This theory is practically unmodified throughout the papers dealing with relatives, as a comparison of " Description of a Notation for the Logic of Relatives" with "The Logic of Relatives" in the Johns Hopkins studies and with the paper of 1884 will indicate. "Individual" or "elementary" relatives are the pairs (or triads, etc.) of individual things. If the objects in the universe of discourse be A, B, C: etc., then the individual relatives will constitute the two-dimensional array, A : A, A : B, A : C, A : D, ... B : A, B : B, B : C, B : D, ... C:A, C :B, C : C, C :D, ... . . . Etc., etc. It will be noted that any individual thing coupled with itself is an individual relative but that in general A : B differs from B : A individual relatives are ordered couples. A general relative is conceived as an aggregate or logical sum of such 143 "Logic of Relatives" in Studies in Logic by members of Johns Hopkins University,. p. 193. The Development of Symbolic Logic 93 individual relatives. If b represent "benefactor", then b = ZiZyPM/ : /), where (&),-/ is a numerical coefficient whose value is 1 in case I is a bene- factor of J, and otherwise 0, and where the sums are to be taken for all the individuals in the universe. That is to say, b is the logical sum of all the benefactor-benefitted pairs in the universe. This is the first formulation of "definition in extension", now widely used in logistic, though seldom in exactly this form. By this definition, b is the aggregate of all the individual relatives in our two-dimensional array which do not drop out through having the coefficient 0. It is some expression of the form, b = (X: F)i+(Z: Y) 2 + (X : F),+ ... If, now, we consider the logical meaning of + , we see that this may be read, "b is either (X : F)i or (X : F) 2 or (X : 7) 3 or . . . ". To say that b repre- sents the class of benefactor-benefitted couples is, then, inexact: b repre- sents an unspecified individual relative, any one of this class. (That it should represent "some" in a sense which denotes more than one at once which the meaning of + in the general case admits is precluded by the fact that any two distinct individual relatives are ipso facto mutually exclusive.) A general relative, so defined, is what Mr. Russell calls a "real variable". Peirce discusses the idea of such a variable in a most illuminating fashion. 144 "Demonstration of the sort called mathematical is founded on suppo- sition of particular cases. The geometrician draws a figure; the algebraist assumes a letter to signify a certain quantity fulfilling the required condi- tions. But while the mathematician supposes a particular case, his hypoth- esis is yet perfectly general, because he considers no characters of the individual case but those which must belong to every such case. The ad- vantage of his procedure lies in the fact that the logical laws of individual terms are simpler than those which relate to general terms, because indi- viduals are either identical or mutually exclusive, and cannot intersect or be subordinated to one another as classes can. . . . "The old logics distinguish between individuum signatum and indi- mduum vagum. 'Julius Caesar' is an example of the former; 'a certain man', of the latter. The indimduum vagum, in the days when such con- ceptions were exactly investigated, occasioned great difficulty from its having a certain generality, being capable, apparently, of logical division. "* "Description of a Notation, pp. 342-44. 94 A Survey of Symbolic Logic If we include under indimduum vagum such a term as 'any individual man', these difficulties appear in a strong light, for what is true of any individual man is true of all men. Such a term is in one sense not an individual term; for it represents every man. But it represents each man as capable of being denoted by a term which is individual; and so, though it is not itself an individual term, it stands for any one of a class of such terms. . . . The letters which the mathematician uses (whether in algebra or in geometry) are such individuals by second intention. . . . All the formal logical laws relating to individuals will hold good of such individuals by second intention, and at the same time a universal proposition may be substituted for a proposition about such an individual, for nothing can be predicated of such an individual which cannot be predicated of the whole class." The relative b, denoting ambiguously any one of the benefactor-bene- fitted pairs in the universe, is such an individual by second intention. It is defined by means of the "prepositional function", "/ benefits J", as the logical sum of the (/ : J) couples for which "I benefits J" is true. The compound relations of the calculus can be similarly defined. If a = 2<2y(a)iy(7 : J), and b = 2i2y(6)z symbolizes "Either

z = ipZi x ifiZz x (x, y}, may be regarded as a propositional function of two variables, or as a function of the single variable, the individual rela- tive (/ : Xz + x s is true or . . . " Similarly, U x X = (fXi X Xi is true and (xi, 2/2) *t(xi, 2/2)]+ ...} /i) x t(x 2 , 2/1)] + [ (x 3 , yi) x^(z 3 , yi)] + [v(x 3 , 1/2) x^(z 3 , 2/2)] + ...} x ... Etc., etc. This expression reads directly " {Either [x t is agent of y\ and x\ is bene- factor of 2/1] or [xi is agent of 2/2 and x\ is benefactor of 2/2] or . . . } and [either [#2 is agent of 2/1 and x 2 is benefactor of 2/1] or [x 2 is agent of 2/2 and z 2 is bene- factor of 2/2] or . . .} and {either [x 3 is agent of 2/1 and x 3 is benefactor of 2/i] or [x s is agent of 2/2 and x s is benefactor of 2/2] or . . . } and . . . Etc., etc". 98 A Survey of Symbolic Logic The operator S, which is nearer the argument, or "Boolian" as Peirce calls it, indicates the operation, + , within the lines. The outside operator, II, indicates the operation, x, between the lines i. e., in the columns; and the subscript of the operator nearer the Boolian indicates the letter which varies within the lines, the subscript of the outside operator, the letter which varies from line to line. Three operators would give a three-dimen- sional array. With a little patience, the reader may learn to interpret any such expression directly from the meaning of simple logical sums and logical products. For example, with the same meanings of v(x, y) and will mean "Everyone (x) is agent of some (y) benefactor of himself". (Note the order of the variables in the Boolian.) And 2 x 2 v U z [x x x 3 + . . ., and U x B m , B n , any members of the two series of terms, and 2 A, 2 B, S (A B) logical sums of some of the A n 's, the B n 's, and the (A n 5)'s respectively.) Condition 1. No A m is A n 2. No B m is B n 3. x = S (A B) 4. a = S A 163 Loc. tit., p. 403. 102 A Survey of Symbolic Logic Condition 5. b = 2 B 6. Some A m is B n This definition is somewhat involved: the crux of the matter is that a b will, in the case described, have as many members as there are combina- tions of a member of a with a member of b. Where the members of a are distinct (condition 1) and the members of b are distinct (condition 2), these combinations will be of the same multitude as the arithmetical a X b. It is worthy of remark that, in respect both to addition and to multi- plication, Peirce has here hit upon the same fundamental ideas by means of which arithmetical relations are defined in Principia Mathematical The "second intention" of a class term is, in Principia, Nc'a; a + b, in Peirce's discussion, corresponds to what is there called the "arithmetical sum" of two logical classes, and a X & to what is called the "arithmetical product". But Peirce's discussion does not meet all the difficulties that could hardly be expected in a short paper. In particular, it does not define the arithmetical sum in case the classes summed have members in common, and it does not indicate the manner of defining the number of a class, though it does suggest exactly the mode of attack adopted in Prin- cipia, namely, that number be considered as a property of cardinally similar classes taken in extension. The method suggested for the derivation of the laws of various numerical algebras from those of the logic of relatives is more comprehensive, though here it is only the order of the systems which is derived from the order of the logic of relatives; there is no attempt to define the number or multitude of a class in terms of logical relations. 155 We are here to take a closed system of elementary relatives, every individual in which is either a T or a P and none is both. Let c = (T : T) 8 = (P:P) p = (P:T) t = (T :P) Suppose T here represent an individual teacher, and P an individual pupil: the system will then be comparable to a school in which every person is either teacher or pupil, and none is both and every teacher teaches every pupil. The relative term, c, will then be defined as the relation of one 154 See Vol. n, Section A. 155 " Description of a Notation, pp. 359 ff. The Development of Symbolic Logic 103 teacher to another, that is, "colleague". Similarly, s is (P : P), the rela- tion of one pupil to another, that is, "schoolmate". The relative term, p, is (P : T), the relation of any pupil to any teacher, that is, "pupil". And the relative term, t, is (T : P), the relation of any teacher to any pupil, that is, "teacher". Thus from the two non-relative terms, T and P, are generated the four elementary relatives, c, s, t, and p. The properties of this system will be clearer if we venture upon certain explanations of the properties of elementary relatives which Peirce does not give and to the form of which he might object. For any such relative (I : J), where the 7's and the /'s are distinct, we shall have three laws: (1) (7 : J) J = I Whatever has the (7 : J) relation to a J must be an 7: whoever has the teacher-pupil relation to a pupil must be a teacher. (2) (7 : J) 7 = Whatever has the teacher-pupil relation to a teacher (where teachers and pupils are distinct) does not exist. (3) (7:J) (#:7<0 = ((I : J)\H] : K The relation of those which have the (7 : J) relation to those which have the (H : K) relation is the relation of those-which-have-the-(7 : J)-relation- to-an-77 to a K. It is this third law which is the source of the important properties of the system. For example: t\p = (T :P)|(P : T) = [(T : P) |P] : T = (T : T) = c The teachers of any person's pupils are that person's colleagues. (Our illustration, to fit the system, requires that one may be his own colleague or his own schoolmate.) c c = (T : T) | (T : T) = [(T : T) \ T] : T = (T : T) = c The colleagues of one's colleagues are one's colleagues. t\t = (T :P)\(T : P) = [(T : P) T] : P = (0 : P) = There are no teachers of teachers in the system. p s = (P : T) (P : P) = [(P : T) |P] : P = (0 : P) = There are no pupils of anyone's schoolmates in the system. The results may be summarized in the following multiplication table, in which the multipliers are in the column at the right and the multiplicands 104 A Survey of Symbolic Logic at the top (relative multiplication not being commutative) : 156 t p s c t c t p s p s The symmetry of the table should be noted. The reader may easily in- terpret the sixteen propositions which it gives. To the algebra thus constituted may be added modifiers of the terms, symbolized by small roman letters. If f is "French", f will be a modifier of the system in case French teachers have only French pupils, and vice versa. Such modifiers are "scalars" of the system, and any expression of the form a c + b t+ c p + d s where c, t, p, and s are the relatives, as above, and a, b, c, d are scalars, Peirce calls a "logical quaternion". The product of a scalar with a term is commutative, bt = tb since this relation is that of the non-relative logical product. Inasmuch as any (dyadic, triadic, etc.) relative is resolvable into a logical sum of (pairs, triads, etc.) elementary relatives, it is plain that any general relative what- ever is resolvable into a sum of logical quaternions. If we consider a system of relatives, each of which is of the form ai + bj + ck + dl+ ... where i, j, k, I, etc. are each of the form mu + nv + o w+ ... where m, n, o, etc. are scalars, and u, v, w, etc. are elementary relatives, we shall have a more complex algebra. By such processes of complication, multiple algebras of various types can be generated. In fact, Peirce says: 157 "I can assert, upon reasonable inductive evidence, that all such [linear associative] algebras can be interpreted on the principles of the present notation in the same way as those given above. In other words, all such algebras are complications and modifications of the algebra of (156) [for which the multiplication table has been given]. It is very likely that this Ibid., p. 361. 167 Ibid., pp. 363-64. The Development of Symbolic Logic 105 is true of all algebras whatever. The algebra of (156), which is of such a fundamental character in reference to pure algebra and our logical nota- tion, has been shown by Professor [Benjamin] Peirce to be the algebra of Hamilton's quaternions." Peirce gives the form of the four fundamental factors of quaternions and of scalars, tensors, vectors, etc., with their logical interpretations as relative terms with modifiers such as were described above. One more item of importance is Peirce's modification of Boole's calculus of probabilities. This is set forth with extreme brevity in the paper, " On an Improvement in Boole's Calculus of Logic". 158 For the expression of the relations involved, we shall need to distinguish the logical relation of identity of two classes in extension from the relation of numerical equality. We may, then, express the fact that the class a has the same membership as the class b, or all a's are all 6's, by a = 6, and the fact that the number of members of a is the same as the number of members of b, by a = b. Also we must remember the distinction between the logical relations ex- pressed by a + b, ab, a \-b, and the corresponding arithmetical relations expressed by a + b, a X b, and a b. Peirce says: 169 "Let every expression for a class have a second meaning, which is its meaning in a [numerical] equation. ^Namely, let it denote the proportion of individuals of that class to be found among all the individuals examined in the long run. "Then we have If a = b a = b a-{-b = (a + b)-\-ab "Let b a denote the frequency of the 6's among the a's. Then considered as a class, if a and b are events b a denotes the fact that if a happens b happens. a X b a = a b " It will be convenient to set down some obvious and fundamental proper- ties of the function b a . a X b a = 6 X a b then x = 0". x v oo may be abbreviated to x v, a 6 v oo to a 6 v , and ?/ v to y v , c dv to c dv , etc., since it is always understood that if one term of a relation v or v is missing, the missing term is o . This convention leads to a very pretty and convenient opera- tion: v or v may be moved past its terms in either direction. Thus, (a v6) = (ab v) = ( va 6) and (%vy) = (xyv) = ( v x y) But the forms ( v a b) and ( v x y) are never used, being redundant both logically and psychologically. Mrs. Ladd-Franklin's system symbolizes the relations of the traditional logic particularly well : All a is b. a v -b, or a -b v No a is b. avb, or a 6 v Some a is b. avb, or a b v Some a is not b. a v -b, or a -6 v Thus v characterizes a universal, v a particular proposition. And any pair of contradictories will differ from one another simply by the difference between v and v . The syllogism, " If all a is 6 and all b is c, then all a is c, " will be represented by (a v -b) (b v -c) v (a v c) where v , or v , within the parentheses is interpreted for classes, and v between the parentheses takes the prepositional interpretation. This ex- pression may also be read, "'All a is b and all b is c' is inconsistent with the negative (contradictory) of 'Some a is not c'". It is equivalent to (a v -b) (b v -c) (a v -c) v 110 A Survey of Symbolic Logic "The three propositions, 'All a is &', 'All b is c, ' and 'Some a is not c', are inconsistent they cannot all three be true". This expresses at once three syllogisms: (1) (a v -b) (b v -c) v (a v -c) "If all a is b and all 6 is c, then all a is c"; (2) (a v -6) (a v -c) v (b v -c) " If all a is b and some a is not c, then some b is not c"; (3) (6 v -c) (a v -c) v (a v -6) " If all 6 is c and some a is not c, then some a is not b ". Also, this method gives a perfectly general formula for the syllogism (a v -b) (b v c) (a v c) v where the order of the parentheses, and their position relative to the sign v which stands outside the parentheses, may be altered at will. This single rule covers all the .modes and figures of the syllogism, except the illicit particular conclusion drawn from universal premises. We shall revert to this matter in Chapter III. 169 The copulas v and v have several advantages over their equivalents, = and =t= 0, or c and its negative: (1) v and v are symmetrical rela- tions whose terms can always be interchanged; (2) the operation, mentioned above, of moving v and v with respect to their terms, accomplishes trans- formations which are less simply performed with other modes of expressing the copula; (3) for various reasons, it is psychologically simpler and more natural to think of logical relations in terms of v and v than in terms of = and =|= 0. But v and v have one disadvantage as against = , 4= > and c , they do not so readily suggest their mathematical analogues in other algebras. For better or for worse, symbolic logicians have not generally adopted v and v . Of the major contributions since Peirce, the first is that of Ernst Schroder. In his Operationskreis des Logikkalkuls (1877), Schroder pointed out that the logical relations expressed in Boole's calculus by subtraction and divi- sion were all otherwise expressible, as Peirce had already noted. The meaning of + given by Boole is abandoned in favor of that which it now has, first introduced by Jevons. And the "law of duality", which con- nects theorems which involve the relation + , or + and 1, with corresponding theorems in terms of the logical product x, or x and 0, is emphasized. See below, pp. 188 ff. The Development of Symbolic Logic 111 (This parallelism of formulae had been noted by Peirce, in his first paper, but not emphasized or made use of.) The resulting system is the algebra of logic as we know it today. This system is perfected and elaborated in Vorlesungen iiber die Algebra der Logik (1890-95). Volume I of this work covers the algebra of classes; Volume II the algebra of propositions; and Volume III is devoted to the calculus of relations. The algebra of classes, or as we shall call it, the Boole-Schroder algebra, is the system developed in the next chapter. 170 We have somewhat elabo- rated the theory of functions, but in all essential respects, we give the algebra as it appears in Schroder. There are two differences of some importance between Schroder's procedure and the one we have adopted. Schroder's assumptions are in terms of the relation of subsumption, c , instead of the relations of logical product and = , which appear in our postulates. And, second, Schroder gives and discusses the various methods of his predecessors, as well as those characteristically his own. The calculus of propositions (Aussagenkalkul) is the extension of the Boole-Schroder algebra to propositions by a method which differs little from that adopted in Chapter IV, Section I, of this book. The discussion of relations is based upon the work of Peirce. But Peirce's methods are much more precisely formulated by Schroder, and the scope of the calculus is much extended. We summarize the funda- mental propositions which Schroder gives for the sake of comparison both with Peirce and with the procedure we shall adopt in Sections II and III of Chapter IV. 1) A, B, C, D, E ... symbolize "elements" or individuals. 171 These are distinct from one another and from 0. 2) I 1 = A+B + C + D+ ... I 1 symbolizes the universe of individuals or the universe of discourse of the first order. 3) i, j, k, I, m, n, p, q represent any one of the elements A, B, C, D, ... of I 1 . 4) I 1 = Zii i 170 For an excellent summary by Schroder, see Abriss der Algebra der Logik ; ed. Dr. Eugen M tiller, 1909-10. Parts i and n, covering Vols. i and n of Schroder's Vorlesungen, have so far appeared. 171 The propositions here noted will be found in Vorlesungen uber die Algebra der Logik, m, 3-42. Many others, and much discussion of theory, have been omitted. 112 A Survey of Symbolic Logic 5) i : j represents any two elements, i and j, of I 1 in a determined order. 6) (i = ;) = (i:j =j : i), (i * j) = (i j 4= j i) for every i and j. 7) i:j*0 Pairs of elements of I 1 may be arranged in a "block": A : A, A : B, A : C, A : D, ... B : A, B : B, B : C, B : D, 8) C : A, C :B, C : C, C : D, ... D : A, D : B, D : C, D : D, ... These are the "individual binary relatives". I 2 = (A:A) + (A:B) + (A:C) + ... + (B : A} + (B : B} + (B : C) + . 9) + (C : A) + (C : B) + (C : C) + . . . + I 2 represents the universe of binary relatives. 10) I 2 = 2,2, (i : j) = 2<2y (i : j) = 2,, (i : j) 9) and 10) may be summarized in a simpler notation: 1 = ?..{ : j = A : A + A : B + A : C + . . . + B :A + B :B + B :C+ . 11) + C :A + C :B + C :C+ ... + 12) i :j : h will symbolize an "individual ternary relative". 13) I 3 = 2*2/2, (i : j : h) = 2^ i : j : h Various types of ternary relatives are 14) A : A : A, B : A : A, A : B : A, A : A : B, A : B : C It is obvious that we may similarly define individual relatives of the fourth, fifth, ... or any thinkable order. The Development of Symbolic Logic 113 The general form of a binary relative, a, is a = Si/ a if (i : j) where a t -/ is a coefficient whose value is 1 for those (i : j) pairs in which i has the relation a to j, and is otherwise 0. 1 = Si/ i : j = the null class of individual binary relatives. (a 6) t -/ = a,-/ &/ (a + &),-/ = a,- + 6 t -/ -a.-/ = (-a),-/ = -(a,/) (a 1 6)i/ = 2^ a ih b hi (a 1 &)/ = U h (a ih + &*/) The general laws which govern prepositional functions, or Aussagen- schemata, such* as (ab)a, SA a ih b h j, n* (a^ + &&/), H a t -/, S a a,-/, etc., are as follows : yl u symbolizes any statement about u; U U A U will have the value 1 in case, and only in case, A u = 1 for every u; 2 U A U will have the value 1 if there is at least one u such that A u = 1. That is to say, II M ^4 M means " A u for every u", and 2 U A U means " A u for some u". a) n u A u cA v c 2 U A U , -[?, U A U ] c -A v c -fn u ^4 u ] p) H U A U ^ AvLl u jrL U i 2s u Au ^ Ay T -* n..'\ u. (The subscript u, in a and /3, represents any value of the variable u.} T) -[n u A u ] = s u -4 U , -[s^ tt ] = n M -A u 5) If ^4 W is independent of u, then II U ^4 U = A, and S u ^l u = A. e) n u (^ c5) = (A cn u 5 tt ), n(X u cfi) = (s u ^ tt c) r?) S tt (^l M c5) = (n u ^ u c5), 2 M (^c5 u ) = UcS M 5 u ) 0) S ttfl , or SS,(-4uc5 w ) = (n tt ^ tt cS,B e ) u (^ u = i) = (n M ^ u = i), n u (A u = 0) = (s u ^ M = 0) 1 U (A U = 0) = (n u ^i u = 0), ? U (A U = i) = (s tt ^ u = i) 172 We write I where Schroder has 1'; N where he has 0'; (a | 6) for (a; 6); (a f &) for (a j b); -a for a; ~a for a. 9 1 14 A Survey of Symbolic Logic \(n u A u cU u B u -)} K) H u (A u cB u )c-\ ^ c SuU, c #) (The reader should note that II u (^4 u CjB u ) is "formal implication", in Principia Mathematica, (x).(px 3^#.) X) A S U 5 U = S u A B u , A + U U B U = Il u (A + B u ) y4 n u B u = n u A B u , A + S u # u = 2 U (A + #) ) (n u ^ u )(n r 5,,) = n u , , A u B v = u u A u B u , 2 U A U + 2 V B V = Z u ,,C4u + .BO = SC4 + 5) o) s u ii c ^4 u , c lit, s u A u , v From these fundamental propositions, the whole theory of relations is developed. Though Schroder carries this much further than Peirce, the general outlines are those of Peirce's calculus. Perhaps the most inter- esting of the new items of Schroder's treatment are the use of "matrices" in the form of the two-dimensional array of individual binary relatives, and the application of the calculus of relatives to Dedekind's theory of "chains ", as contained in Was sind und was sollen die Zahlen. Notable contributions to the Boole-Schroder algebra were made by Anton Poretsky in his three papers, Sept lois fondamentales de la tkeorie des egalites logiques (1899), Quelques lois ulterieures de la theorie des egalites logiques (1901), and Theorie des non-egalites logiques (1904). (With his earlier works, published in Russian, 1881-87, we are not familiar.) Poret- sky's Law of Forms, Law of Consequences, and Law of Causes will be given in Chapter II. As Couturat notes, Schroder had been influenced overmuch by the analogies of the algebra of logic to other algebras, and these papers by Poretsky outline an entirely different procedure which, though based on the same fundamental principles, is somewhat more "natural" to logic. Poretsky 's method is the perfection of that type of procedure adopted by Jevons and characteristic of the use of the Venn diagrams. The work of Frege, though intrinsically important, has its historical interest largely through its influence upon Mr. Bertrand Russell. Although the Begriffsschrift (1879) and the Grundlagen der Arithmetik (1884) both The Development of Symbolic Logic 115 precede Schroder's Vorlesungen, Frege is hardly more than mentioned there; and his influence upon Peano and other contributors to the Formu- laire is surprisingly small when one considers how closely their ta*sk is re- lated to his. Frege is concerned explicitly with the logic of mathematics but, in thorough German fashion, he pursues his analyses more and more deeply until we have not only a development of arithmetic of unprecedented rigor but a more or less complete treatise of the logico-metaphysical problems concerning the nature of number, the objectivity of concepts, the relations of concepts, symbols, and objects, and many other subtleties. In a sense, his fundamental problem is the Kantian one of the nature of the judgments involved in mathematical demonstration. Judgments are analytic, de- pending solely upon logical principles and definitions, or they are synthetic. His thesis, that mathematics can be developed wholly by analytic judg- ments from premises which are purely logical, is likewise the thesis of Russell's Principles of Mathematics. And Frege's Grundgesefee der Arith- metik, like Principia Mathematica, undertakes to establish this thesis for arithmetic by producing the required development. Besides the precision of notation and analysis, Frege's work is important as being the first in which the nature of rigorous demonstration is suf- ficiently understood. His proofs proceed almost exclusively by substitu- tion for variables of values of those variables, and the substitution of defined equivalents. Frege's notation, it must be admitted is against him: it is almost diagrammatic, occupying unnecessary space and carrying the eye here and there in a way which militates against easy understanding. It is probably this forbidding character of his medium, combined with the unprecedented demands upon the reader's logical subtlety, which accounts for th'e neglect which his writings so long suffered. But for this, the revival of logistic proper might have taken place ten years earlier, and dated from Frege's Grundlagen rather than Peano's Formulaire. The publication, beginning in 1894, of Peano's Formulaire de Mathe- matiques marks a new epoch in the history of symbolic logic. Heretofore, the investigation had generally been carried on from an interest in exact logic and its possibilities, until, as Schroder remarks, we had an elaborated instrument and nothing for it to do. With Peano and his collaborators, the situation is reversed: symbolic logic is investigated only as the instrument of mathematical proof. As Peano puts it: m " The laws of logic contained in what follows have generally been found 173 Formulaire, i (1901), 9. 116 A Survey of Symbolic Logic by formulating, in the form of rules, the deductions which one comes upon in mathematical demonstrations." The immediate result of this altered point of view is a new logic, no less elaborate than the old destined, in fact, to become much more elabo- rate but with its elaboration determined not from abstract logical con- siderations or by any mathematical prettiness, but solely by the criterion of application. De Morgan had said that algebraists 'and geometers live in "a higher realm of syllogism": it seems to have required the mathe- matical intent to complete the rescue of logic from its traditional inanities. The outstanding differences of the logic of Peano from that of Peirce and Schroder are somewhat as follows : m (1) Careful enunciation of definitions and postulates, and of possible alternative postulates, marking an increased emphasis upon rigorous deductive procedure in the development of the system. (2) The prominence of a new relation, e, the relation of a member of a class to the class. (3) The prominence of the idea of a prepositional function and of "formal implication" and "formal equivalence", as against "material implication" and "material equivalence". (4) Recognition of the importance of "existence" and of the properties of classes, members of classes, and so on, with reference to their "existence". (5) The properties of relations in general are not studied, and "relative addition" does not appear at all, but various special relations, prominent in mathematics, are treated of. The disappearance of the idea of relation in general is a real loss, not a gain. (6) The increasing use of substitution (for a variable of some value in its range) as the operation which gives proof. We here recognize those characteristics of symbolic logic which have since been increasingly emphasized. The publication of Principia Mathematica would seem to have deter- mined the direction of further investigation to follow that general direction indicated by the work of Frege and the Formulaire. The Principia is con- cerned with the same topics and from the same point of view. But we see here a recognition of difficulties not suggested in the Formulaire, a deeper and more lengthy analysis of concepts and a corresponding complexity of procedure. There is also more attention to the details of a rigorous method of proof. 174 All these belong also to the Logica Mathematica of C. Burali Forti (Milan, 1894). The Development of Symbolic Logic 117 The method by which the mathematical logic of Principia Mathematica is developed will be discussed, so far as we can discuss it, in the concluding section of Chapter IV. We shall be especially concerned to point out the connection, sometimes lost sight of, between it and the older logic of Peirce and Schroder. And the use of this logic as an instrument of mathematical analysis will be a topic in the concluding chapter. CHAPTER II THE CLASSIC, OR BOOLE-SCHRODER, ALGEBRA OF LOGIC I. GENERAL CHARACTER OF THE ALGEBRA. THE POSTULATES AND THEIR INTERPRETATION The algebra of logic, in its generally accepted form, is hardly old enough to warrant the epithet "classic". It was founded by Boole and given its present form by Schroder, who incorporated into it certain emendations which Jevons had proposed and certain additions particularly the relation "is contained in" or "implies" which Peirce had made to Boole's system. It is due to Schroder's sound judgment that the result is still an algebra, simpler yet more powerful than Boole's calculus. Jevons, in simplifying Boole's system, destroyed its mathematical form; Peirce, retaining the mathematical form, complicated instead of simplifying the original calculus. Since the publication of Schroder's Vorlesungen iiber die Algebra der Logik certain additions and improved methods have been offered, the most notable of which are contained in the studies of Poretsky and in Whitehead's Uni- versal Algebra. 1 But if the term "classic" is inappropriate at present, still we may venture to use it by way of prophecy. As Whitehead has pointed out, this system is a distinct species of the genus "algebra", differing from all other algebras so far discovered by its non-numerical character. It is certainly the simplest mathematical system with any wide range of useful applications, and there are indications that it will serve as the parent stem from which other calculuses of an important type will grow. Already sev- eral such have appeared. The term "classic" will also serve to distinguish the Boole-Schroder Algebra from various other calculuses of logic. Some of these, like the system* of Mrs. Ladd-Franklin, differ through the use of other relations than + , x , c ,' and = , and are otherwise equivalent 1 For Poretsky 's studies, see Bibliography; also p. 114 above. See Whitehead's Uni- versal Algebra, Bk. n. Whitehead introduced a theory of "discriminants" and a treatment of existential propositions by means of umbral letters. This last, though most ingenious and interesting, seems to me rather too complicated for use; and I have not made use of "discriminants ", preferring to accomplish similar results by a somewhat extended study of the coefficients in functions. 118 The Classic, or Boole-Schroder, Algebra of Logic 119 that is to say, with a "dictionary" of equivalent expressions, any theorem of these systems may be translated into a theorem of the Boole-Schroder Algebra, and vice versa. Others are mathematically equivalent as far as they go, but partial. And some, like the calculus of classes in Principia, Mathematica, are logically but not mathematically equivalent. And, finally, there are systems such as that of Mr. MacColl's Symbolic Logic which are neither mathematically nor logically equivalent. Postulates for the classic algebra have been given by Huntington, by Schroder (in the Abriss], by Del Re, by Sheffer and by Bernstein. 2 The set here adopted represents a modification of Huntington's third set. d It has been chosen not so much for economy of assumption as for " natural- ness" and obviousness. Postulated: A class K of elements a, b, c, etc., and a relation x such that: 1 1 If a and b are elements in K, then a x b is an element in K, uniquely determined by a and 6. 1 2 For any element a, a x a = a. 1 3 For any elements a and b, a x 6 = b x a. 1 4 For any elements a, b, and c, a x (b x c) = (a x 6) x c. 1 5 There is a unique element, 0, in K such that a x = for every ele- ment a. 1 ' 6 For every element a, there is an element, -a, such that 1-61 If x x-a = 0, then x xa = x, and 1-62 If y x a = y and y x -a = y, then y = 0. The element 1 and the relations + and c do not appear in the above. These may be defined as follows: 1-7 1 = -0 Def. 1-8 a + b = -(-ax -6) Def. 1-9 a c b is equivalent to a x6 = a Def. It remains to be proved that -a is uniquely determined by a, from which it will follow that 1 is unique and that a + b is uniquely determined by a and b. 2 See Bibl. 3 See "Sets of Independent Postulates for the Algebra of Logic", Trans. Amer. Math. Soc., v (1904), 288-309. Our set is got by replacing + in Huntington's set by x, and replacing the second half of G, which involves 1, by its analogue with 0. Thus 1 can be denned, and postulates E and H omitted. Postulate J is not strictly necessary. 120 A Survey of Symbolic Logic The sign of equality in the above has its usual mathematical meaning; i. e., { = } is a relation such that if x = y and $(x} is an unambiguous function in the system, then and c a c c b. If acb, then [1-9] a 6 = a and [2-1] (a6)c = ac (1) But [1-2-3-4] (ab)c = (ba)c = b (ac) = (ac) b = [a (cc) &] = [(ac) c] 6 = (ac)(c6) = (oc)(6c) (2) Hence, by (1) and (2), if a c b, then (a c)(6 c) = a c and [1 -9] a c c 6 c. And [1-3] c a = a c and cb = b c. Hence also c a c c 6. 2-8 -(-a) = a. [2-4] -(-a) -a = 0. Hence [2-5] -(-a) ca (1) By (1), -[-(-a)] c-o. Hence [2-7] a --[-(-a)] ca-a. 124 A Survey of Symbolic Logic But [2-4] a -a = 0. Hence a --[-(-a)] cO. Hence [2-6] a -[-(-a)] = and [2-5] a c-(-a) (2) [2-2] (1) and (2) are equivalent to -(-a) = a. 3-1 a c b is equivalent to -b c -a. [2-5] a c 6 is equivalent to a -6 = 0. And [2-8] a -6 = -b a = -b -(-a). And -6 -(-a) = is equivalent to -6 c -a. The terms of any relation c may be transposed by negating both. If region a is contained in region b, then the portion of the plane not in b is contained in the portion of the plane not in a: if all a's are 6's, all non-6's are non-a's. This theorem gives immediately, by 2-8, the two corollaries: 3-12 a c-6 is equivalent to b c-a; and 3-13 -a c 6 is equivalent to -b c a. 3-2 a = 6 is equivalent to -a = -b. [2-2] a = b is equivalent to (a c6 and b ca). [3 1] a c b is equivalent to -b c -a, and b c a to -a c -6. Hence a = b is equivalent to (-a c-6 and -b c-a), which is equiva- lent to -a = -b. The negatives of equals are equals. By 2-8, we have also 3-22 a = -b is equivalent to -a = b. Postulate 1-6 does not require that the function "negative of" be unambiguous. There might be more than one element in the system having the properties postulated of -a. Hence in the preceding theorems, -a must be read "any negative of a ", -(-6) must be regarded as any one of the negatives of any given negative of b, and so on. Thus what has been proved of -a, etc., has been proved to hold for every element related to a in the manner required by the postulate. But we can now demonstrate that for every element a there is one and only one element having the properties postulated of -a. 3-3 -a is uniquely determined by a. By 1-6, there is at least one element -a for every element a. Suppose there is more than one: let -a x and - 2 represent any two such. Then [2-8] -(-ai) = a = -(-a 2 ). Hence [3-2] -aj = -o 2 . Since all functions in the algebra are expressible in terms of a, b, c, etc., the relation x , the negative, and 0, while is unique and a x b is uniquely The Classic, or Boole-Schroder, Algebra of Logic 125 * determined by a and b, it follows from 3 3 that all functions in the algebra are unambiguously determined when the elements involved are specified. (This would not be true if the inverse operations of "subtraction" and "division" were admitted.) 3-33 The element 1 is unique. [1-5] is unique, hence [3-3] -0 is unique, and [1-7] 1 = -0. 3-34 -1 = 0. [1-7] 1 = -0. Hence [3-2] Q.E.D. 3 35 If a and b are elements in K, a + b is an element in K uniquely deter- mined by a and b. The theorem follows from 3.3, 1-1, and 1-8. 3-37 If a = b, then a + c = b + c and c + a = c + b. The theorem follows from 3 35 and the meaning of = . 3-4 -(a + 6) = -a-b. [1-8] a + b = -(-a-b). Hence [3-3, 2-8] -(a + b) = -[-(-a-b)} = -a-b. 3-41 -(a b) = -a + -b. [1-8, 2-8] -a + -b = -[-(-a) -(-b)] = -(ab). 3-4 and 3-41 together state De Morgan's Theorem: The negative of a sum is the product of the negatives of the summands; and the negative of a product is the sum of the negatives of its factors. The definition 1-8 is a form of this theorem. Still other forms follow at once from 34 and 3-41, by 2- 8: 3-42 -(-a + -b) = ab. 3-43 -(a + -b) = -ab. 3.44 -(-a + b) = a-b. 3-45 -(a-b) = -a + b. 3-46 -(-ab) = a + -b. From De Morgan's Theorem, together with the principle, 3-2, "The negatives of equals are equals", the definition 1-7, 1 = -0, and theorem 3-34, -1 = 0, it follows that for every theorem in terms of x there is a corresponding theorem in terms of + . If in any theorem, each element be replaced by its negative, and x and + be interchanged, the result is a valid theorem. The negative terms can, of course, be replaced by positive, 126 A Survey of Symbolic Logic since we can suppose x = -a, y = -b, etc. Thus for every valid theorem in the system there is another got by interchanging the negatives and 1 and the symbols x and + . This principle is called the Law of Duality. This law is to be illustrated immediately by deriving from the postulates their correlates in terms of + . The correlate of 1 1 is 3 35, already proved. 4-2 a + a = a. [l'2]-a-a = -a. Hence [1-8, 3-2, 2-8] a + a = -(-a-a) = -(-a) = a. 4-3 a + 6 = 6 + a. [1-3] -a -b = -b -a. Hence [3-2] -(-a -6) = -(-b -a). Hence [1-8] Q.E.D. 4-4 a + (b + c) = (a + 6) + c. [1-4] -a(-fe-c) = (-a-fc)-c. Hence [3-2] -[-a (-b -c)] = -[(-a -6) -c]. But [3-46, 1-8] -[-a(-fc-c)] = a + -(-b-c) = a+(6 + c). And [3-45, 1-8] -[(-a-fc)-c] = -(-a-6) + c = (a + 6)+c. 4-5 a+1 = 1. [1-5] -a-0 = 0. Hence [3-2] -(-a-0) = -0. Hence [3-46] a + -0 = -0, and [1-7] a+1 = 1. 4-61 If -x + a = 1, then x a = x. If -x + a = 1, then [3-2-34-44] x -a = -(-z + a) =-1=0. And [2 5] x -a = is equivalent to x a = x. 4 612 If -x + a = 1, then z + a = a. [4-61] If -a + a; = 1, then ax = a, and [3-2] -a + -x = -a (1) By (1) and 2-8, if -x + a = 1, x + a = a. 4-62 If y + a = y and y + -a = y, then 2/ = 1. If T/ + a = y, [3-2] -?/ -a = -(y + a) = -y. And if y + -a = y, -y a = -(y + -a) = ~y. But [1-62] if -y a = -y and -# -a = -y, -y = and i/ = -0 = 1. 4-8 a + -a = 1 = -a + a. (Correlate of 2-4) [2-4] -a a = 0. Hence [3-2] a + -a = -(-a a) =-0 = 1. Thus the modulus of the operation + is 1. 4-9 -a + b = 1, a + b = b, a -b = 0, a b = a, and a c 6 are all equivalent. [2 5] a -b = 0, a b = a, and a c 6 are equivalent. [3-2] -a + b = 1 is equivalent to a -6 = -(-a + 6) =-1=0. The Classic, or Boole-Schroder, Algebra of Logic 127 [4-612] If -a + b = 1, a + b = b. And if a + b = b, [3 37] -a + b = -a + (a + 6) = (-a + a) + b = 1+6 = 1. Hence a + b = b is equivalent to -a + b = 1. We turn next to further principles which concern the relation c . 5-1 If a c b and 6 cc, then ace. [1-9] a c b is equivalent to a b = a, and b c c to 6 c = b. If a b = a and 6 c = b, a c = (a b) c = a (b c) = a b = a. But ac = a is equivalent to ace. This law of the transitivity of the relation c is called the Principle of the Syllogism. It is usually included in any set of postulates for the algebra which are expressed in terms of the relation c . 5-2 a b ca and a b cb. (a b) a = a (a 6) = (a a) b = a b. But (a 6) a a b is equivalent to ab ca. Similarly, (a b) b = a (b b) = ab, and ab cb. 5-21 a ca + b and b ca + b. [5 2] -a -b c -a and -a -b c -6. Hence [3 12] a c -(-a -6) and be -(-a -6). But -(-a -6) = (a + b). Note that 5-2 and 5-21 are correlates by the Law of Duality. In general, having now deduced the fundamental properties of both x and + , we shall give further theorems in such pairs. A corollary of 5-21 is: 5-22 a b ca + b. [5-1-2- 21] 5-3 If a c 6 and c c d, then ac cb d. [1-9] If acb and c c d, then a b = a and c d = c. Hence (a c) (b d) = (a b) (c d) = a c, and ac cb d. 5-31 If a c b and c c d, then a + c c b + d. If acb and c c d, [3-1] -be -a and -d c -c. Hence [5-3] -b-dc-a-c, and [3-1] -(-a-c) c-(-b-d). Hence [1-8] Q.E.D. By the laws, a a = a and a + a = a, 5-3 and 5-31 give the corollaries: 5-32 If a c c and b c c, then ab cc. 128 A Survey of Symbolic Logic 5-33 If a c c and b c c, then a + b c c. 5-34 If a c 6 and ace, then a c 6 c. 5-35 If a c 6 and ace, then a c 6 + c. 5-37 If acb, then a + cc6 + c. (Correlate of 2-7) [2-3] ccc. Hence [5-31] Q.E.D. 5-4 a + ab = a. [5-21] aca + ab (1) [2-3] a ca, and [5-2] a b ca. Hence [5-33] a + a 6 ca (2) [2-2] If (1) and (2), then Q.E.D. 5-41 a (a + 6) = a. [5-4] -a + -a-6 = -a. Hence [3-2] -(-a + -a-&) = -(-a) = a. But [3-4] -(-a + --&) = a--(-a-&) = a (a + 6). 5-4 and 5-41 are the two forms of the Law of Absorption. We have next to prove the Distributive Law, which requires several lemmas. 5*5 a (6 + c) = ab + ac. Lemma I: a b + ac ca (b + c). [5-2] ab ca and ac ca. Hence [5 33] a b + a c c a (1) [5-2] ab cb and ac cc. But [5-21] b cb + c and c cb + c. Hence [5-1] a b cb + c and a c cb + c. Hence [5 33] a b + a c c b + c (2) [5-34] If (1) and (2), then a b + a c ca (b + c). Lemma 2 : If p c q is false, then there is an element x, =}= 0, such that x c p and x c -q. p-q is such an element, for [5-2] p-qcp and p-qc-q; and [4-9] if p -q = 0, then p c q, hence if p c q is false, then p -q =}= 0. (This lemma is introduced in order to simplify the proof of Lemma 3.) Lemma 3 : a (b + c) cb + ac. Suppose this false. Then, by lemma 2, there is an element x, 4= 0, such that xca(b + c) (1) and x c -(b + ac) But [?-12] if x c-(b + a c), then b + acc-x (2) [5-1] If (1), then since [5-2] a (b + ac) ca, x ca (3) and also, since a(b + ac) cb + c, x cb + c (4) [5 1] If (2), then since [5 21] b c b + a c, b c -x and [3 12] x c -b (5) Also [5 1] if (2), then since [5 21] a c c b + a c, a c c -x and [3 12] x c -(a c) (6) The Classic, or Boole-Schroder, Algebra of Logic 129 From (6) and (3), it follows that x cc must be false; for if x cc and (3) x c a, then [5 34] x c a c. But if x c a c and (6) x c -(a c), then [1-62] x = 0, which contradicts the hypothesis x ={= 0. But if x c c be false, then by lemma 2, there is an element y, + 0, such that ycx (7) and 2/c-c, or [3-121 cc-?/ (8) . [5 1] If (7) and (5), then y c -b and [3 12] b c -y (9) If (8) and (9), then [5,- 33] b + c c -y and [3 -12]yc -(b + c) (10) If (7) and (4), then [5>l]ycb + c (11) [1-9] If (11), then y (b + c) = y, and if (10), y--(b + c) = y (12) But if (12), then [1-62] y = 0, which contradicts the condition, *>o. Hence the supposition that a(b + c) c b + a c be false is a false supposition, and the lemma is established. Lemma 4: a (b + c) ca b + a c. By lemma 3, a (6 + c) c b + a c. Hence [2-7] a [a (b + c)] c a (b + a c) . But a [a (6 + c)] = (a a)(6 + c) = a (b + c). And a (b + a c) = a (a c + 6). Hence a (6 + c) c a (a c + 6). But by lemma 3, a (a c + 6) c a c + a b. And ac + ab = ab + ac. Hence a (b + c) c a b + a c. Proof of the theorem : [2-2] Lemma 1 and lemma 4 are together equiva- lent to a (b + c) = a b + a c. This method of proving the Distributive Law is taken from Huntington, "Sets of Independent Postulates for the Algebra of Logic ". The proof of the long and difficult lemma 3 is due to Peirce, who worked it out for his paper of 1880 but mislaid the sheets, and it was printed for the first time in Huntington's paper. 6 5-51 (a + b)(c + d) = (a c + b c) + (a d + b d). [5 5] (a + 6) (c + d) = (a + 6) c + (a + 6) d = (a c + b c) + (a d + b d) . 5-52 a + bc = (a + 6)(a + c). (Correlate of 5 5) [5-51] (a + 6) (a + c) = (a a + b a) + (a c + b c) = [(a + a 6) + a c] + &*e. But [5-4] (a + a 6) + a c = a + a c = a. Hence Q.E.D. Further theorems which are often useful in working the algebra and which follow readily from the preceding are as follows: 6 See "Sets of Independent Postulates, etc.", loc. c,it., p. 300, footnote. 10 130 A Survey of Symbolic Logic 5-6 a-1 = a = 1-a. [1-5] a-0 = 0. Hence a--l = 0. But [1-61] if a--l = 0, then a-1 = a. 5-61 acl. [1-9] Since a-1 = a, acl. 5-62 a + = a = + a. -a--0 = -a-1 = -a. Hence [3-2] a + = -(-a--0) = -(-a) = a. 5-63 Oca. 0-a = a-0 = 0. Hence [1-9] Q.E.D. 5-64 1 ca is equivalent to a = 1. [2 2] a = 1 is equivalent to the pair, acl and lea. But [5-61] acl holds always. Hence Q.E.D. 5-65 a c is equivalent to a = 0. [2 2] a = is equivalent to the pair, a c and Oca. But [5-63] c a holds always. Hence Q.E.D. 5-7 If a + 6 = a: and a = 0, then b = x. If a = 0, a + 6 = + 6 = 6. 5-71 If a b = x and a = 1, then b = re. If a = 1, a 6 = 1-6 = 6. 5-72 a + b = is equivalent to the two equations, a = and 6 = 0. If a = and 6 = 0, then a + 6 = + = 0. And if a + 6 = 0, -a -6 = -(a + 6) = -0 = 1. But if -a -6 = 1, a = a-1 = a(-a -6) = (a -a) -6 = 0--6 = 0. And [5-7] if a + 6 = and a = 0, then 6 = 0. 5-73 a 6 = 1 is equivalent to the two equations, a = 1 and 6 = 1. If a = 1 and 6 = 1, then a 6 = 1-1 = 1. And if a 6 = 1, -a + -6 = -(a 6) =-1=0. Hence [5-72] -a = and -6 = 0. But [3-2] if -a = 0, a = 1, and if -6 = 0, 6 = 1. 5-7 and 5-72 are important theorems of the algebra. 5-7, "Any null term of a sum may be dropped", would hold in almost any system; but 5-72, "If a sum is null, each of its summands is null", is a special law characteristic of this algebra. It is due to the fact that the system con- tains no inverses with respect to + and 0. a and -a are inverses with The Classic, or Boole-Schroder, Algebra of Logic 131 respect to x and and with respect to + and 1. 5-71 and 5-73, the correlates of 5-7 and 5-72, are less useful. 5-8 a (b + -b) = a b + a -b = a. [5-5] a (b + -6) = a b + a -b. And [4-8] 6 + -b = 1. Hence a (b + -b) = a- 1 = a. 5-85 a + b = a + -ab. [5-8] b = ab + -ab. Hence a + b = a + (a b + -a b) = (a + a 6) + a -6. But [5-4] a + ab = a. Hence Q.E.D. It will be convenient to have certain principles, already proved for two terms or three, in the more general form which they can be given by the use of mathematical induction. Where the method of such extension is obvious, proof will be omitted or indicated only. Since both x and + are associative, we can dispense with parentheses by the definitions: 5-901 a + b + c=(a + b)+c Def. 5-902 abc = (ab)c Def. 5-91 a = a (b + -b)(c + -c)(d + -d~) . . . [5-8] 5-92 1 = (a + -a)(6 + -6)(c + -c)... [4-8] 5-93 a = a + ab + ac + ad+... [5-4] 5-931 a = a (a + &)( + c)(a + d). . . [5-41] 5-94 a(b + c + d+...)=ab + ac + ad+... [5-5] 5-941 a + bcd,.. = (a + 6)(a + c)(a + d). . . [5-52] 5-95 -(a + b + c + . . . ) = -a -b -c . . . If the theorem hold for n terms, so that -(oi + a 2 + . . . + a n ) = -i -a 2 . . . -a n then it will hold f or n -\- 1 terms, for by 3 4, -[(ai + a 2 + . . . + o n ) + a B+1 ] = -(ai + a 2 + . . . + a n ) --an-i-i And [3-4] the theorem holds for two terms. Hence it holds for any number of terms. 132 A Survey of Symbolic Logic 5 951 -(a bed...) = -a + -b + -c + -d + . . . Similar proof, using 3-41. 5-96 1 = a + b + c + . . . + -a -b -c . . . [4-8, 5-951] 5-97 a + 6 + c+...=Ois equivalent to the set, a = 0, b = 0, c = 0, . . . [5-72] 5-971 abed... = 1 is equivalent to the set, a = 1, b = 1, c = 1, . . . [5-73] 5-98 a -bed... = ab-ac-ad... [1-2] a a a a . . . = a. 5-981 a+(b + c + d+...~) = (a + 6) + (a + c) + (a + d) + . . . [4-2] a + a + a+ . . . = a. The extension of De Morgan's Theorem by 5-95 and 5-951 is especially important. 5-91, 5-92, and 5-93 are different forms of the principle by which any function may be expanded into a sum and any elements not originally involved in the function introduced into it. Thus any expression whatever may be regarded as a function of any given elements, even though they do not appear in the expression,- a peculiarity of the algebra. 5 92, the expression of the universe of discourse in any desired terms, or expansion of 1, is the basis of many important procedures. The theorems 5-91-5-981 are valid only if the number of elements involved be finite, since proof depends upon the principle of mathematical induction. III. GENERAL PROPERTIES OF FUNCTIONS We may use f(x), $(x, y), etc., to denote any expression which involves only members of the class K and the relations x and + . The further requirement that the expression represented by f(x) should involve x or its negative, -x, that $(x, y) should involve x or -x and y or -y, is unnecessary, for if x and -x do not appear in a given expression, there is an equivalent expression in which they do appear. By 5-91, a = a (x + -.r) = a x + a -x = (ax + a -x) (y + -y) = axy + ax-y + a-xy + a-x -y, etc. a x + a -x may be called the expansion, or development, of a with reference to x. And any or all terms of a function may be expanded with reference to x, the result expanded with reference to y, and so on for any elements and any number of elements. Hence any expression involving only ele- The Classic, or Boole-Schroder, Algebra of Logic 133 ments in K and the relations x and + may be treated as a function of any elements whatever. If we speak of any a such that x = a as the " value of x ", then a value of x being given, the value of any function of x is determined, in this algebra as in any other. But functions of x in this system are of two types: (1) those whose value remains constant, however the value of x may vary, and (2) those such that any value of the function being assigned, the value of x is thereby determined, within limits or completely. Any function which is symmetrical with respect to x and -x will belong to the first of these classes; in general, a function which is not completely symmetrical with respect to x and -x will belong to the second. But it must be remembered, in this connection, that a symmetrical function may not look symmetrical unless it be completely expanded with reference to each of the elements involved. For example, a + -a b + -b is symmetrical with respect to a and -a and with respect to b and -b. Ex- panding the first and last terms, we have a (b + -6) + -a b + (a + -a) -b = a b + a -b + -a b + -a -b = 1 whatever the value of a or of b. Any function in which an element, x, does not appear, but into which it is introduced by expanding, will be symmetrical with respect to x and -x. The decision w r hat elements a given expression shall be considered a function of is, in this algebra, quite arbitrary except so far as it is deter- mined by the form of result desired. The distinction between coefficients and "variables" or "unknowns" is not fundamental. In fact, we shall frequently find it convenient to treat a given expression first as a function- say of x and y, then as a function of z, or of x alone. In general, coef- ficients will be designated by capital letters. The Normal Form of a Function. Any function of one variable, f(x), can be given the form A x + B -x where A and B are independent of x. This is the normal form of functions of one variable. 6-1 Any function of one variable, f(x}, is such that, for some A and some B which are independent of x, f(x} = A x + B -x 134 A Survey of Symbolic Logic Any expression which involves only elements in the class K and the relations x and + will consist either of a single term a single element, or elements related by x or of a sum of such terms. Only four kinds of such terms are possible: (1) those which involve x, (2) those which involve -x, (3) those which involve both, and (4) those which involve neither. 7 Since the Distributive Law, 5-5, allows us to collect the coefficients of x, of -x and of (x -a-), the most general form of such an expres- sion is p x + q -x + r (x -x) + s where p, q, r, and s are independent of x and -x. But [2-4] r(x-x] = r-0 = 0. And [5-9] s = s x + s -x. Hence p x + q-x + r (x -x) + s = (p + s) x + (q + s} -x. Therefore, A = p + s, B q + s, gives the required reduction. The normal form of a function of n -\- 1 variables, l, X Z , ... X n , Xn+l may be defined as the expansion by the Distributive Law of f(Xi, Xz, ... )&+! +/ '(Xi, a- 2 , ... Xn) OTn+l where / and / ' are each some function of the n variables, x\, x 2 , ... x n , and in the normal form. This is a "step by step" definition; the normal form of a function of two variables is defined in terms of the normal form of functions of one variable; the normal form of a function of three variables in terms of the normal form for two, and so on. 8 Thus the normal form of a function of two variables, $>(x, y), will be found by expanding (A x + B -x) y+(Cx + D -x) -y It will be, Axy + B-xy + Cx-y + D-x-y The normal form of a function of three variables, ^?(x, y, z), will be A x y z + B -x y z + C x-y z + D -x -y z + E xy -z + F -x y -z + G x -y -z + H -x -y -z And so on. Any function in the normal form will be fully developed with 7 By a term which "involves" x is meant a term which either is x or has x "as a factor". But "factor" seems inappropriate in an algebra in which h x is always contained in x, h x ex. 8 This definition alters somewhat the usual order of terms in the normal form of func- tions. But it enables us to apply mathematical induction and thus prove theorems of a generality not otherwise to be attained. The Classic, or Boole-Schroder, Algebra of Logic 135 reference to each of the variables involved that is, each variable, or its negative, will appear in every term. 6-11 Any function may be given the normal form. (a) By 6-1, any function of one variable may be given the normal form. (6) If functions of n variables can be given the normal form, then functions of n + 1 variables can be given the normal form, for, Let $(xi, x 2 , ... x n , x n +i) be any function of n + 1 variables. By definition, its normal form will be equivalent to f(Xi, X Z , ... X n ) 'Xn+i + / '(Xi, Xz, ... Z) Xn+l wLere / and / ' are functions of Xi, x 2 , ... x n and in the normal form. By the definition of a function, 3>(xi, x z , ... x n , x n +i) may be re- garded as a function of x n +\. Hence, by 6-1, for some A and some B which are independent of x n ^ i $(&i, Xz, ... X n , Xn+l) = A X n+l + B -X n +l Also, by the definition of a function, for some / and some / ' A = f(xi, Xz, ... x n ) and B = f '(x l} x 2 , ... x n ) Hence, for some/ and/ ' which are independent of x n+i i, Xz, ... X n , X n+i ) = Therefore, if the functions of n variables, / and / ', can be given the normal form, then $(x 1} x z , . . . x n , x n+i ) can be given the normal form. (c) Since functions of one variable can be given the normal form, and since if functions of n variables can be given the normal form, functions of n + 1 variables can be given the normal form, therefore functions of any number of variables can be given the normal form. The second step, (6), in the above proof may seem arbitrary. That it is valid, is due to the nature of functions in this algebra. 6- 12 For a function of n variables, $(xi, x 2 , ... .r n ), the normal form will be a sum of 2" terms, representing all the combinations of Xi, positive or negative, with x 2 , positive or negative, with . . . with x n , positive or negative, each term having its coefficient. 136 A Survey of Symbolic Logic (a) A normal form function of one variable has two terms, and by definition of the normal form of functions of n + 1 variables, if functions of k variables have 2 k terms, a function of k + 1 variables will have 2* + 2 k , or 2 h+} , terms. (6) A normal form function of one variable has the further character described in the theorem; and if normal form functions of k variables have this character, then functions of A' -f- 1 variables will have it, since, by definition, the normal form of a function of k + 1 variables will consist of the combinations of the (k + l)st variable, positive or negative, with each of the combinations repre- sented in functions of k variables. Since any coefficient may be 0, the normal form of a function may con- tain terms which are null. Where no coefficient for a term appears, the coefficient is, of course, 1. The order of terms in the normal form of a function will vary as the order of the variables in the argument of the function is varied. For example, the normal form of (!, 1, 1) -x y z + $(0, 1, 1) --x y z + (1, 0, 1) -x -y z + $(0, 0, 1) x -yz + $(1, 1, 0) -x y -z + $(0, 1, 0)xy-z + $(1, 0, 0) -x -y -z + $(0, 0, 0) x -y -z We can prove that this method of determining the coefficients extends to functions of any number of variables. 138 A Survey of Symbolic Logic .Ti .To .T 3 . . . X n \ 6-24 If G r be any term of (i, # 2 , z 3 , ... x n ), then r will be the coefficient, C. U, U, U, . . . U J (a) By 6 23, the theorem holds for functions of one variable. (6) If the theorem hold for functions of k variables, it will hold for functions of k + 1 variables, for, By 6-11, any function of k + 1 variables, &(xi, x 2 , ... x k , x k +i), is such that, for some / and some / ', "t:::!} . <" i, i, ... 1,1 , fi, i, ... il ,Ji, i, ... il And *o,o,...o, } =/ io ) o,...o}- +/ io ) o,...or 1 o,o,...o Therefore, if every term of / be of the form 1, 1, ... 1 1 f .TI x z ... x k f ' 1 0, 0, ... J I -T! -xz . . . -x k then every term of $ in which a^+i is positive will be of the form l, 1, ... ll f .T! Xz... x k and the coefficient of any such term will be / -j r , which, l_ u, u, ... u j *>*->*-) 1 oo o 1 Vy \J j . , . W y And similarly, if every term of / ' be of the form fi, i, ... 11 r x, xz... x k \ J I 0, 0, ... j l-x l -xz . . . -x k J then every term of $ in which .T/t+i is negative will be of the form , i, ... IT r x, xz... x k f A 0, 0, ... J l-Xi -.T 2 . . . -x k The Classic, or Boole-Schroder, Algebra of Logic 139 and the coefficient of any such term will be / ' by(2) ' is *{oo"'o' \_ \J, \J, . . . \J, Hence every term of will be of the form 1, 1, ... 1, 1 1 f Zi X 2 . . . X k Xk+i 0, 0, ... 0, j i -Xi -x z . . . -x k -x k+ i (c) Since the theorem holds for functions of one variable, and since if it hold for functions of k variables, it will hold for functions of k + 1 variables, therefore it holds for functions of any number of variables. For functions of one variable, further laws of the same type as 6-23 but less useful have been given by Peirce and Schroder. If /(a?) = Ax + B-x: 6-25 /(I) =f(A+B) = f(-A+-B). 6-26 /(O) = f(A-B) =f(-A-B). 6-27 f(A) =A + B= /(-) = f(A-B) = f(A + -B) 6-28 /() = A-B =f(-A] =f(-A-B) =f(-A + B) = /(l)-/(0) =f(x]-f(-x). The proofs of these involve no difficulties and may be omitted. In theorems to be given later, it will be convenient to denote the coef- ficients in functions of the form $(xi, x%, . . . x n } by A\, AI, AS, . . . Az n , or by Ci, C 2 , C s , . . ., etc. This notation is perfectly definite, since the order of terms in the normal form of a function is fixed. If the argument of any function be (xi, x 2 , ... x n ), then any one of the variables, Xk, will be positive in the term of which C m is the coefficient in case p-2 k ~ l < m ^ (p + l)^*- 1 where p = any even integer (including 0). Otherwise Xk will be negative in the term. Thus it may be determined, for each of the variables in the function, whether it is positive or negative in the term of which C m is the coefficient, and the term is thus completely specified. We make no use of this law, except that it validates the proposed notation. Occasionally it will be convenient to distinguish the coefficients of those terms in a function in which some one of the variables, say Xk, is positive from the coefficients of terms in which Xk is negative. We shall do this 140 A Survey of Symbolic Logic by using different letters, as PI, P 2 , PS, . . . , for coefficients of terms in which Xk is positive, and Qi, Qz, Qa, ... for coefficients of terms in which Xk is negative. This notation is perfectly definite, since the number of terms, for a function of n variables, is always 2", the number of those in which Xk is positive is always equal to the number of those in which it is negative, and the distribution of the terms in which Xk is positive, or is negative, is determined by the law given above. The sum of the coefficients, AI + A z + A 3 + . . ., will frequently be indi- cated by ^A or ^A h ', the product, Ai-Az'A 3 - . . . by H^4 or 11^. h h Since the number of coefficients involved will always be fixed by the func- tion which is in question, it will be unnecessary to indicate numerically the range of the operators ^ and IJ . The Limits of a Function. The lower limit of any function is the prod- uct of the coefficients in the function, and the upper limit is the sum of the coefficients. 6-3 A BcAx + B-xcA + B. Hence [1 -9] A B c A x + B -x. And (A x + B -x)(A + B) = Ax + AB-x + ABx + B-x = (AB + A)x + (AB + B) -x. But [5-4] A B + A = A, and A B + B = B. Hence (A x + B -x)(A + B) = Ax + B -x, and [1-9] A x + B -x cA + B. 6-31 f(B) C/(.T) cf(A). [6-3 and 6-26, 6-27] 6-32 If the coefficients in any function, F(XI, x 2 , ... x n ), be C if C 2 , C 3 , . . . , then (a) By 6 3, the theorem holds for functions of one variable. (6) Let &(xi, x 2 , ... Xk, Xk+i) be any function of k + 1 variables. By 6 1 1 , for some / and some / ', <>(zi, a? 2 , ... Xk, afc+i) = f(x lt x*, . . . x^-Xk+i +/'0ri, x z , ... Xk)'-x k+ i (1) Since this last expression may be regarded as a function of jt + i in which the coefficients are the functions / and / ', [6 3] z , . . . x k ) x/ '(xi, x z , ... x k ) c (.ri, x z , . . . x k , The Classic, or Boole-Schroder, Algebra of Logic 141 Let AI{$], A z {<&}, AS{$}, etc., be here the coefficients in <; Ai{f}, AZ{/}, A s {f}, etc., the coefficients in/; and A\{f'}, Az{f'}, A s {f'}, etc., the coefficients in / '. If IU{/} c/and lUm c/', then [6-3] and, by (1), IU{/} x TLA {/'} c *. But since (1) holds, any coefficient in $ will be either a coefficient in / or a coefficient in / ', and hence Hence if the theorem hold for functions of k variables, so that zi, 3 2 , ... %k) and then ^{$} $(&!, ar 2 , x k , x k+ i). Similarly, since (1) holds, [6-23] $<=/+/'. Hence if/c^>{/} and /' c ^A{f f }, then [5-31] But since any coefficient in $ is either a coefficient in / or a coef- ficientin/', Hence $ c Thus if the theorem hold for functions of k variables, it will hold for functions of k + 1 variables. (c) Since the theorem holds for functions of one variable, and since if it hold for functions of k variables, it will hold for functions of k + 1 variables, therefore it holds generally. As we shall see, these theorems concerning the limits of functions are the basis of the method by which eliminations are made. Functions of Functions. Since all functions of the same variables may be given the same normal form, the operations of the algebra may frequently be performed simply by operating upon the coefficients. 6-4 If /Or) = A x + B -x, then -[f(x}\ = -Ax + -B -x. [3 -4] -(A x + B -x) = -(A x) -(B -x) = (-A + -X)(-B + x) = -A-B + -AX + -B -x - (-A -B + -4) x + (-A -B + -B} -x But [5-4] -A -B + -A = -A and -A -B + -B = -B. Hence -(A x + B -x) = -A x + -B -x. 142 A Survey of Symbolic Logic 6-41 The negative of any function, in the normal form, is found by re- placing each of the coefficients in the function by its negative. (a) By 6-4, the theorem is true for functions of one variable. (6) If the theorem hold for functions of k variables, then it will hold for functions of k + 1 variables. Let F(XI, Xz, ... x k , x k+ i) be any function of k + 1 variables. Then by 6-11 and 3 2, for some / and some / ', ... x k , + /'(&!, Xz, ... Xk)'-X k+l ] But f(xi, Xz, ... Xk)-Xk+i+f'(xi, Xz, ... Xk)--Xk+i may be regarded as a function of Hence, by 6-4, ~[f(Xi, X*, X^'Xk+i +/ '(Xi, Xz, ... X k } '- -[f'(Xi, Xz, ... Hence if the theorem be true for functions of k variables, so that the negative of / is found by replacing each of the coefficients in / by its negative and the negative of / ' is found by replacing each of the coefficients in / ' by its negative, then the negative of F will be found by replacing each of the coefficients in F by its negative, for, as has just been shown, any term of z, ... x k , in which x k +i is positive is such that its coefficient is a coefficient in and any term of ~[F(xi, x 2 , ... x in which x k +i is negative is such that its coefficient is a coefficient in (c) Since (a) and (6) hold, therefore the theorem holds generally. Since a difference in the order of terms is not material, 6-41 holds not only for functions in the normal form but for any function which is com- pletely expanded so that every element involved appears, either positive or negative, in each of the terms. It should be remembered that if any term of an expanded function is missing, its coefficient is 0, and in the negative of the function that term will appear with the coefficient 1. The Classic, or Boole-Schroder, Algebra of Logic 143 6-42 The sum of any two functions of the same variables, (xi, x z , ... x n ) and ty(xi, Xz, . . . x n }, is another function of these same variables, F(Xi, Xz, . . . X n }, such that the coefficient of any term in F is the product of the coeffi- cients of the corresponding terms in $ and ty. {/> <"V -V /v rvt /y Xi Xz . . . X n I J Xi Xz . . . X n I , r and B k ~\ r be any two Xi Xz . . . X' n J L Xi Xz . . . X n J corresponding terms in and SK Xz ... X n \ D f Xi Xz . . . X n X D I Xi Xz ~Xn J L Xi Xz . Xn r Xi Xz x n -Xi -Xz . . . ~X n By 6 15, $ and ty do not differ except in the coefficients, and by 6-17, whatever the coefficients in the normal form of a function, the product of any two terms is null. Hence all the cross-products of terms in $ and S^ will be null, and- the product of the functions will 144 A Survey of Symbolic Logic be equivalent to the sum of the products of their corresponding terms, pair by pair. Since in this algebra two functions in which the variables are not the same may be so expanded as to become functions of the same variables, these theorems concerning functions of functions are very useful. IV. FUNDAMENTAL LAWS OF THE THEORY OF EQUATIONS We have now to consider the methods by which any given element may be eliminated from an equation, and the methods by which the value of an "unknown" may be derived from a given equation or equations. The most convenient form of equation for eliminations and solutions is the equation with one member 0. Equivalent Equations of Different Forms. If an equation be not in the form in which one member is 0, it may be given that form by multiplying each side into the negative of the other and adding these two products. 7-1 a = b is equivalent to a-b + -a 6 = 0. [2-2] a = b is equivalent to the pair, a c b and b c a. [4-9] a cb is equivalent to a -b = 0, and b c a to -a b 0. And [5 72] a -b = and -a b = are together equivalent to a -b + -a b = 0. The transformation of an equation with one member 1 is obvious: 7-12 a = 1 is equivalent to -a = 0. [3-2] By 6-41, any equation of the form f(xi, x 2 , ... #) = 1 is reduced to the form in which one member is simply by replacing each of the coefficients in / by its negative. Of especial interest is the transformation of equations in which both members are functions of the same variables. 7-13 If &(xi, xz, ... x n ) and ^f(xi, x z , ... x n ) be any two functions of the same variables, then x- 2 , . . . x n is equivalent to F(XI, xz, ... x n ) = 0, where F is a function such that if A i, AZ, A 3 , etc., be the coefficients in <, and BI, BZ, B 3 , etc., be the coef- ficients of the corresponding terms in ^, then the coefficients of the corre- sponding terms in F will be (Ai -Bi + -Ai J5i), (A 2 -B 2 + -A 2 BZ], (A S -B 3 + -A 3 B 3 ), etc. The Classic, or Boole-Schroder, Algebra of Logic 145 By 7- 1, $ = ^ is equivalent to ( x-) + (- x ) =0. By 6-41, - and -^ are functions of the same variables as and ^. Hence, by 6-43, x-ty and -< x ^ will each be functions of these same variables, and by 6-42, ($ x-^) + (-$ x>J>) will also be a function of these same variables. Hence $, SF, -<, -^, $ x-SI>, - x SI>, and ($ x-^) + (-< x ^) are all functions of the same variables and, by 6-15, will not differ except in the coefficients of the terms. If Ak be any coefficient in <, and Bk the corresponding coefficient in ^, then by 6-41, the corresponding coefficient in -$ will be -Ak and the corresponding coefficient in -ty will be -Bk- Hence, by 6-43, the corresponding coefficient in $x->3> will be Ak -Bk, and the corresponding coefficient in -$ x ^ will be -A k B k . Hence, by 6-42, the corresponding coefficient in ($ x-SI>) + (-$ x >J>) will be A k -B k + -A k B k . Thus (< x -^) + (-< x ^) is the function F, as described above, and the theorem holds. By 7-1, for every equation in the algebra there is an equivalent equation in the form in which one member is 0, and by 7 13 the reduction can usually be made by inspection. One of the most important additions to the general methods of the algebra which has become current since the publication of Schroder's work is Poretsky's Law of Forms. 9 By this law, given any equation, an equiva- lent equation of which one member may be chosen at will can be derived. 7-15 a = is equivalent to t = a -t + -a t. If a = 0, a-t + -at = Qt+l-t = t. And if t = a-t + -at, then [7 1] (a -t + -a t) -t + (a t + -a -t) I = = a-t + at = a Since t may here be any function in the algebra, this proves that every equation has an unlimited number of equivalents. The more general form of the law is : 7-16 a = b is equivalent to t = (a b + -a -6) t + (a -b + -a b) -L [7 1] a = b is equivalent to a -b + -a b = 0. And [6 4] -(a -b + -a 6) = ab + -a -b. Hence [7 -15] Q.E.D. The number of equations equivalent to a given equation and expressible 9 See Sept lois fondamentales de la theorie des egalites logiques, Chap. I. 11 146 A Survey of Symbolic Logic in terms of n elements will be half the number of distinct functions which can be formed from n elements and their negatives, that is, 2~ n /2. The sixteen distinct functions expressible in terms of two elements, a and b, are: a, -a, b, -b, (i. e., a -a, b-b, etc.), 1 (i. e., a + -a, b + -b, etc.), ab, a -b, -a b, -a -b, a + b, a + -b, -a + b, -a + -b, a b + -a -b, and a -b + -a b. In terms of these, the eight equivalent forms of the equation a = b are: a = b; -a = -b; = a -b + -a b; 1 = ab + -a -b; ab = a + b; a -b = -ab; -a-b = -a + -b; and a + -b = -a + b. Each of the sixteen functions here appears on one or the other side of an equation, and none appears twice. For any equation, there is such a set of equivalents in terms of the elements which appear in the given equation. And every such set has what may be called its "zero member" (in the above, = a-b + -ab) and its "whole member" (in the above, 1 = ab + -a-b). If we observe the form of 7-16, we shall note that the functions in the "zero member" and "whole member" are the functions in terms of which the arbitrarily chosen t is determined. Any t = the t which contains the function { = 0} and is contained in the function { = 1 } . The validity of the law depends simply upon the fact that, for any t, ct c 1, i. e., t = l-t + Q--t. It is rather surprising that a principle so simple can yield a law so powerful. Solution of Equations in One Unknown. Every equation which is pos- sible according to the laws of the system has a solution for each of the un- knowns involved. This is a peculiarity of the algebra. We turn first to equations in one unknown. Every equation in x, if it be possible in the algebra, has a solution in terms of the relation c . 7-2 A x + B -x = is equivalent to B c x c -A. [5-72] A x + B -x = is equivalent to the pair, A x = and B -x = 0. [4 9] B -x = is equivalent to B c x. And A x = is equivalent to x -(-A) = 0, hence to x c-A. 7-21 A solution in the form // c x c K is indeterminate whenever the equa- tion which gives the solution is symmetrical with respect to x and -x. First, if the equation be of the form A x + A -x = 0. The solution then is, A ex c-A. But if A x + A -x 0, then A = A (x + -x) = A x + A -x = 0, and -A = 1. The Classic, or Boole-Schroder, Algebra of Logic 147 Hence the solution is equivalent to Oczcl, which [5-61-63] is satisfied by every value of x. In general, any equation symmetrical with respect to x and -x which gives the solution, H c x c K, will give also H c -x c K. But if H c x and H c -x, then [4 9] H x = H and H -x = H. Hence [1- 62] # = 0. And if xcK and -xcK, then [5-33] x + -xcK, and [4-8, 5-63] K = 1. Hence H c x c K will be equivalent to c x c 1. It follows directly from 7-21 that if neither x nor -x appear in an equa- tion, then although they may be introduced by expansion of the functions involved, the equation remains indeterminate with respect to x. 7 22 An equation of the form A x + B -x = determines x uniquely when- ever A = -B, B = -A. [3-22] A = -B and -A = B are equivalent; hence either of these conditions is equivalent to both. [7 21 A x + B -x = is equivalent to B c x c -A. Hence if B = -A, it is equivalent to B ex cB and to -A ex c-A, and hence [2-2] to x = B = -A. In general, an equation of the form A x + B -x = determines x be- tween the limits B and -A. Obviously, the solution is unique if, and only if, these limits coincide; a,nd the solution is wholly indeterminate only when they are respectively and 1, the limiting values of variables generally. 7-221 The condition that an equation of the form A x + B -x = be pos- sible in the algebra, and hence that its solution be possible, is A B = 0. By 6-3, ABcAx + B-x. Hence [5-65] if A x + B -x = 0, then AB = 0. Hence if A B + 0, then A x + B -x = must be false for all values of x. And A x + B -x = and the solution B ex c -A are equivalent. A B = is called the "equation of condition" of A x + B -x = 0: it is a necessary, not a sufficient condition. To call it the condition that A x + B -x = have a solution seems inappropriate : the solution B ex c -A is equivalent to A x + B -x = 0, whether A x + B -x = be true, false, or impossible. The sense in which A B = conditions other forms of the solution of A x + B -x = will be made clear in what follows. The equation of condition is frequently useful in simplifying the solution. 148 A Survey of Symbolic Logic (In this connection, it should be borne in mind that A B = follows from A x + B -x = 0.) For example, if a b x + (a + 6) -x = then (a + 6) car c-(a &). But the equation of condition is a b (a + 6) = a b = 0, or, -(a 6) = 1 Hence the second half of the solution is indeterminate, and the complete solution may be written a + b ex However, this simplified form of the solution is equivalent to the original equation only on the assumption that the equation of condition is satisfied and a b = 0. Again suppose ax + b -x + c = Expanding c with reference to x, and collecting coefficients, we have (a + c) x + (b + c) -x = and the equation of condition is (a + c) (6 + c) = ab + a c + b c + c = ab + c = The solution is b + c c x c -a -c But, by 5-72, the equation of condition gives c = 0, and hence -c 1. Hence the complete solution may be written b ex c -a But here again, the solution b ex c-a is equivalent to the original equation only on the assumption, contained in the equation of condition, that c = 0. This example may also serve to illustrate the fact that in any equation one member of which is 0, any terms which do not involve x or -x may be dropped without affecting the solution for x. If a x + b -x + c = 0, then by 5-72, a x + b -x = 0, and any addition to the solution by retaining c will be indeterminat^. All terms which involve neither the unknown nor its negative belong to the "symmetrical constituent" of the equation to be explained shortly. Poretsky's Law of Forms gives immediately a determination of x which is equivalent to the given equation, whether that equation involve x or not. 7-23 A x + B -x = is equivalent to x = -A x + B -x. [7 15] A x + B -x = is equivalent to x = (A x + B -x~) -x + (-A x + -B -x) x = B -x + -A x The Classic, or Boole-Schroder, Algebra of Logic 149 This form of solution is also the one given by the method of Jevons. 10 Although it is mathematically objectionable that the expression which gives the value of x should involve x and -x, this is in reality a useful and logically simple form of the solution. It follows from 7-2 and 7-23 that x -A x + B -x is equivalent to B c x c -A . Many writers on the subject have preferred the form of solution in which the value of the unknown is given in terms of the coefficients and an undetermined (arbitrary) parameter. This is the most "mathematical" form. 7-24 If A B = 0, as the equation A x + B -x = requires, then A x + B -x = is satisfied by x = B -u + -A u, or x = B + u -A, where u is arbitrary. And this solution is complete because, for any x such that A x + B -x = there is some value of u such that x = B -u + -A u = B + u-A. (a) By 6 4, if x = B -u + -A u, then -x = -B -u + A u. Hence if x = B -u + -A u, then A x + B -x = A (B -u + -B u} + B (-B -u + Au) = AB-u + ABu = AB Hence if A B = and x = B -u + -A u, then whatever the value of u, A x + B -x = 0. (6) Suppose x known and such that A x + B -x = 0. Then if x = B -u + -A u, we have, by 7 1, (B -u + -A u) -x + (-B -u + A u) x = (A x + -A -x} u+ (B -x + -B x) -u = The condition that this equation hold for some value of u is, by 7 221, (AX + -A -x}(B -X+-BX) = A-BX+-AB-X = o This condition is satisfied if A x + B -x = 0, for then A (B + -B) x + (A + -A} B -x = A B + A - x + -A B -x = and by 5 -72, A -B x + -A B -x = 0. (c) If A B = 0, then B -u + -A u = B + u -A, for: If A B = 0, then A B u = 0. Hence B -u + -A u = B -u + -A (B + -B) u + A B u = B -u + (A + -A) B u + -A-B u = B (-u + u) + -A -B u = B + -A-BU. But [5-85] B + -A -B u = B + u -A. 10 See above, p. 77. 150 A Survey of Symbolic Logic Only the simpler form of this solution, x = B + u -A, will be used hereafter. The above solution can also be verified by substituting the value given for x in the original equation. We then have A (B -u + -A u} + B (-B -u + A u) = A B -u + A B u = A B And if A B = 0, the solution is verified for every value of u. That the solution, x = B -u + -A u = B + u -A, means the same as Bc.xc.-A, will be clear if we reflect that the significance of the arbitrary parameter, u, is to determine the limits of the expression. If u = 0, B -u + -A u = B + u -A = B. If u = 1, B-u + -Au = -A and B + u-A = B + -A. But when AB = 0, B + -A = -A B + -A = -A. Hence x B -u + -A u = B + u -A simply expresses the fact, otherwise stated by B ex c-A, that the limits of x are B and -A. The equation of condition and the solution for equations of the form C x + D -x = 1, and of the form A x + B -x = C x + D -x, follow readily from the above. 7 25 The equation of condition that C x + D -x = 1 is C + D = 1, and the solution of C x + D -x = 1 is -D c x c C. (a) By 6-3, Cx + D -x c C + D. Hence if there be any value of x for which Cx + D-x = 1, then necessarily C + D = 1 . (6) If Cx + D-x = 1, then [6-4] -Cx + -D-x = 0, and [7-2] -DcxcC. 7-26 If C + D = 1, then the equation Cx + D-x = 1 is satisfied by x = -D + uC, where u is arbitrary. Since [6-4] C x + D -x = 1 is equivalent to -C x + -D -x = 0, and C + D = 1 is equivalent to -C -D = 0, the theorem follows from 7-24. 7-27 If A x + B -x = C x + D -x, the equation of condition is (A-C + -A C)(B-D + -BD) = and the solution is B -D + -B D c x c A C + -A -C, or x = B -D + -B D + u (A C + -A -C), where u is arbitrary. By 7' 13, A x + B -x = C x + D -x is equivalent to (A-C + -ACx + B-D + -Bd-x = 0. The Classic, or Boole-Schroder, Algebra of Logic 151 Hence, by 7-221, the equation of condition is as given above. And by 7 2 and 7 24, the solution is B-D + -BDcxc-(A-C + -A C), or x = B -D + -B D + u--(A -C + -A C), where u is arbitrary. And [6-4] -(A-C + -AC) = AC + -A-C. The subject of simultaneous equations is very simple, although the clearest notation we have been able to devise is somewhat cumbersome. 7-3 The condition that n equations in one unknown, A 1 x + B l -x = 0, A*x + B 2 -x = 0, ... A n x + B n -x = 0, may be regarded as simultaneous, is the condition that (A* B k ) = h, k And the solution which they give, on that condition, is ^B k cxc H-A k k k or x = 23 B k + u- II -.4*, where u is arbitrary. k k By 6-42 and 5-72, A l x + B 1 -x = 0, A 2 x + B 2 -x = 0, ... A n x + B n -x = 0, are together equivalent to (A 1 + A 2 + . . . + A n )x + (B 1 + B 2 + . . . + B n ) -x = or A k * + Z B k -x = k k By 7 23, the equation of condition here is 2 A k x^B k .= k k But Z A k x Z B k = (A 1 + A 2 + . . . + A n )(B l + B z + . . . + B n ) k k = A 1 B 1 + A 1 B z + . . . + A 1 B n + A* B 1 + A 2 B 2 + . . . + A 2 B n + A 3 B l + A 3 B 2 +...+A 3 B n +...+A n B l +...+A n B n = ^(AB k }. h, k And by 7 2 and 7 24, the solution here is or k k And by 5-95, -{ 2 A k ] = JI ~A k . k k It may be noted that from the solution in this equation, n z partial solu 152 A Survey of Symbolic Logic tions of the form B h ex c-A' can be derived, for B h c B k and H -A" c -A*. k k Similarly, 2 2n 1 partial solutions can be derived by taking selections of members of X^ B k an ^ II -A k . k k Symmetrical and Unsymmetrical Constituents of Equations. Some of the most important properties of equations of the form A x + B -x = are made clear by dividing the equation into two constituents the most comprehensive constituent which is symmetrical with respect to x and -x, and a completely unsymmetrical constituent. For brevity, these may be called simply the "symmetrical constituent" and the "unsymmetrical constituent ". In order to get the symmetrical constituent complete, it is necessary to expand each term with reference to every element in the function, coefficients included. Thus in A x + B -x = it is necessary to expand the first term with respect to B, and the second with respect to A. A(B + -B)x + (A + -A) B -x = A B x + A B -x + A -B x + -A B -x = By 5 72, this is equivalent to the two equations, A B (x + -x) = AB = and A -B x + -A B -x = The first of these is the symmetrical constituent; the second is the unsym- metrical constituent. The symmetrical constituent will always be the equa- tion of condition, while the unsymmetrical constituent will give the solution. But the form of the solution will most frequently be simplified by con- sidering the symmetrical constituent also. The unsymmetrical constituent will always be such that its equation of condition is satisfied a priori. Thus the equation of condition of A -B x .+ -A B-x = is (A -B)(-A B) = 0, which is an identity. By this method of considering symmetrical and unsymmetrical con- stituents, equations which are indeterminate reveal that fact by having no unsymmetrical constituent for the solution. Also, the method enables us to treat even complicated equations by inspection. Remembering that any term in which neither x nor -x appears belongs to the symmetrical constituent, as does also the product of the coefficients of x and -x, the separation can be made directly. For example, (c + x) d + -c -d + (-a + -a;) 6 = The Classic, or Boole-Schroder, Algebra of Logic 153 will have as its equation of condition c d + -c -d + -ab + b d = and the solution will be bc.xc.-d Also, as we shall see shortly, the symmetrical constituent is always the complete resultant of the elimination of x. The method does not readily apply to equations which do not have one member 0. But these can always be reduced to that form. How it extends to equations in more than one unknown will be clear from the treatment of such equations. Eliminations. The problem of elimination is the problem, what equa- tions not involving x or -x can be derived from a given equation, or equa- tions, w r hich do involve x and -x. In most algebras, one term can, under favorable circumstances, be eliminated from two equations, two terms from three, n terms from n + 1 equations. But in this algebra any number of terms (and their negatives) can be eliminated from a single equation; and the terms to be eliminated may be chosen at will. The principles whereby such eliminations are performed have already been provided in theorems concerning the equation of condition. 7-4 A B = contains all the equations not involving x or -x which can be derived from A x + B -x = 0. By 7 24, the complete solution of A x + B -x = is x = B -u + -A u Substituting this value of x in the equation, w r e have A (B -u + -A u) + B (-B -u + A u) = A B -u + A B u = A B = Hence A B = is the complete resultant of the elimination of x. It is at once clear that the resultant of the elimination of x coincides with the equation of condition for solution and with the symmetrical con- stituent of the equation. 7-41 If n elements, Xi, x 2 , x s , ... x n , be eliminated from any equation, F(XI, Xz, x s , ... Xn) = 0, the complete resultant is the equation to of the product of the coefficients in F(XI, x 2 , x 3 , ... x n ). (a) By 6-1 and 7-4, the theorem is true for the elimination of one element, x, from any equation, f(x) = 0. (6) If the theorem hold for the elimination of k elements, x\, x z , 154 A Survey of Symbolic Logic ... Xk, from any equation, $(x\, Xz, ... Xk) = 0, then it will hold for the elimination of k + 1 elements, x\, x 2 , ... Xk, Xk+i, from any equation, ty(xi, x z , ... Xk, Xk+i) = 0, for: By 6- 11, V(xi, xz, . . . x k , ar/fc+i) = f(xi, Xz, ... x k } -x k +i And the coefficients in SF will be the coefficients in / and / '. By 7-4, the complete resultant of eliminating x k +i from f(xi, x z , ... Xk) -x k+ i +/ r (xi, x 2 , ... Xk)--x k +i = is /Oi, x 2 , ... Xk) x/ '(xi, xz, . . . x k ) = And by 6-43, f(xi, x 2 , ... x k ] *f'(xi, x 2 , ... x k ) = is equivalent to Q(XI, x 2 , ... Xk) = 0, where $ is a function such that if the coefficients in/ be PI, P 2 , P 3 , etc., and the corresponding coefficients in/' be Qi, Qz, Qs, etc., then the corresponding coefficients in $ will be PiQi, PzQz, PsQs, etc. Hence if the theorem hold for the elimina- tion of k elements, x\, x 2 , ... Xk, from 3>(xi, x 2 , ... x k ) = 0, this elimination will give (P 1 Q 1 )(P 2 Q 2 )(P 3 Q 3 ). . . == (PiP 2 P 3 . ..QiQ&. . .) = 0, where PiP 2 P 3 . . .QiQzQs- is the product of the coefficients in <, or in/ and/ ' i. e., the product of the coefficients in ^. Hence if the theorem hold for the elimination of k elements, x\, xz, ... Xk, from Q(XI, Xz, ... x k ) = 0, it will hold for the elimination of k + 1 elements, xi, xz, ... x k , x k+ \, from ^(a;i, Xz, ... Xk, x k+ i) = 0, provided Xk+\ be the first eliminated. But since the order of terms in a function is immaterial, and for any order of elements in the argument of a function, there is a normal form of the function, ^ + i in the above may be any of the A: + 1 elements in ^, and the order of elimination is immaterial. (c) Since (a) and (6) hold, therefore the theorem holds for the elimination of any number of elements from the equation to of any function of these elements. By this theorem, it is possible to eliminate simultaneously any number of elements from any equation, by the following procedure: (1) Reduce the equation to the form in which one member is 0, unless it already have that form; (2) Develop the other member of the equation as a normal-form function of the elements to be eliminated; (3) Equate to the product of the coefficients in this function. This will be the complete elimination resultant. The Classic, or Boole-Schroder, Algebra of Logic 155 Occasionally it is convenient to have the elimination resultant in the form of an equation with one member 1, especially if the equation which gives the resultant have that form. 7-42 The complete resultant of eliminating n elements, x\, x%, ... x n , from any equation, F(x\, x 2 , ... x n ) = 1, is the equation to 1 of the sum of the coefficients in F(XI, x 2 , ... #). Let AI, A 2 , AS, etc., be the coefficients in F(XI, x 2 , ... x n ). F(XI, x 2 , ... Xn) = 1 is equivalent to -[F(xi, x 2 , ... x n }\ = 0. And by 6-41, -[F(xi, x 2 , ... #)] is a function, $(xi, x 2 , ... x n ), such that if any coefficient in F be A k , the corresponding coefficient in $ will be -Ak. Hence, by 7-41, the complete resultant of eliminating x i} x z , ... x n , from F(XI, Xz, ... Xn) = 1 is IL-A=0, or -{11-4} =1 But [5-95] -{ 11-^1 = T.A. Hence Q.E.D. For purposes of application of the algebra to ordinary reasoning, elimina- tion is a process more important than solution, since most processes of reasoning take place through the elimination of "middle" terms. For example : If all 6 is x, 6 ex, b -x = and no a is x, a x = 0, then a x + b -x = 0. Whence, by elimination, a b = 0, or no a is b. Solution of Equations in more than one Unknown. The complete solu- tion of any equation in more than one unknown may be accomplished by eliminating all the unknowns except one and solving for that one, repeating the process for each of the unknowns. Such solution will be complete because the elimination, in each case, will give the complete resultant which is independent of the unknowns eliminated, and each solution will be a solution for one unknown, and complete, by previous theorems. How- ever, general formulae of the solution of any equation in n unknowns, for each of the unknowns, can be proved. 7 5 The equation of condition of any equation in n unknowns is identical with the resultant of the elimination of all the unknowns; and this resultant is the condition of the solution with respect to each of the unknowns sepa- rately. (a) If the equation in n unknowns be of the form F(XI, x z , . . . Xn) = 0: 156 A Survey of Symbolic Logic Let the coefficients in F(x\, x z , ... #) be A\, At, As, etc. Then, by 6 -32, II A c F(xi, xz, . . . x n } and [5 65] H A = is a condition of the possibility of F(XI, xz, ... x n ) = And [7-41] II A = is the resultant of the elimination of x\, Xz, ... x n , from F(XI, x 2) ... x n ) = 0. (6) If the equation in n unknowns have some other form than F(XI, x 2 , ... Xn) = 0, then by 7-1, it has an equivalent which is of that form, and its equation of condition and its elimination resultant are the equivalents of the equation of condition and elimination resultant of its equivalent which has the form F(Xi, Xz, ... X n ) = (c) The result of the elimination of all the unknowns is the equation of condition with respect to any one of them, say Xk, because : (1) The equation to be solved for Xk will be the result of eliminat- ing all the unknowns but Xk from the original equation; and (2) The condition that this equation, in which Xk is the only unknown, have a solution for Xk is, by (a) and (6), the same as the result of eliminating Xk from it. Hence the equation of condition with respect to Xk is the same as the result of eliminating, from the original equation, first all the other unknowns and then Xk- And by 7-41 and (6), the result of eliminating the unknowns is independent of the order in which they are eliminated. Since this theorem holds, it will be unnecessary to investigate separately the equation of condition for the various forms of equations; they are already given in the theorems concerning elimination. 7-51 Any equation in n unknowns, of the form F(XI, x 2 , ... x n ) = 0, provided its equation of condition be satisfied, gives a solution for each of the unknowns as follows: Let Xk be any one of the unknowns; let PI, PZ, P 3 , etc., be the coefficients of those terms in F(XI, x z> ... x n ) in which Xk is positive, and Qi, Qz, Qs, etc., the coefficients of those terms in which Xk is negative. The solution then is JJ Q c x k c ^ -P, or Xk = II Q + u ^ -P, where u is arbitrary. The Classic, or Boole-Schroder, Algebra of Logic 157 (a) By 6-11, for some / and some /', F(x\., x 2 , ... x n ) = is equivalent to f(xi, x 2 , ... x n -i) -x n +f '(xi, x 2 , ... x n -i}--x n = 0. Let the coefficients in / be PI, P 2 , PS, etc., in / ' be Qi, Q 2 , Qs, etc. Then PI, P 2 , PS, etc., will be the coefficients of those terms in F in which Xk is positive, Qi, Q 2 , Q s , the coefficients of terms in F in which Xk is negative. If f(xi, xz, ... x n -i)-x n be regarded as a function of the variables, x i} Xz, ... x n -i, its coefficients will be P\x n , P 2 x n , P 3 x n , etc. And if f'(xi, x 2 , ... x n -i)--x n be regarded as a function of Xi, x 2 , . . . x n -i, its coefficients will be Qi -x n , Q 2 -x n , Q s -x n , etc. Hence, by 6-42, f(Xi, X Z , . . . X n -j_) -X n +f '(Xi, X 2) ... Xr^i)"X n = is equivalent to ^(xi, x 2 , . . . x n -\) = 0, where ^ is a function in which the coefficients are (Pi x n + Qi -x n }, (Pz x n + Q 2 -x n ), (P 3 x n + Qs -Xn), etc. And SlK^i, x z , ... n-i) = is equivalent to F(XI, x 2 , . . . x n ) = 0. By 7-41, the complete resultant of the elimination of Xi, x 2 , ... x n -i from ^(#1, Xz, ... Xn-i) = will be the equation to of the product of its coefficients, Qr-xJ =0 But any expression of the form PrX n + Q T -x n is a normal form func- tion of x n . Hence, by 6 43, II (P#n + Qr -Xn) = H P#n + II Qr ~X n By 7-2 and 7-24, the solution of H P^n+ TlQr -x n = i s , or x n = + u. And [5-951] -{UP} = Z-J- (6) Since the order of terms in a function is immaterial, and for any order of the variables in the argument of a function there is a normal form of the function, x n in the above may be any one of the variables in F(XI, x 2 , ... x n ), and f(xi, x 2 , ... x n -J and f'(xi, x 2 , . . . x n -i) each some function of the remaining n 1 variables. Therefore, the theorem holds for any one of the variables, Xk- That a single equation gives a solution for any number of unknowns is another peculiarity of the algebra, due to the fact that from a single equation any number of unknowns may be eliminated. 158 A Survey of Symbolic Logic As an example of the last theorem, we give the solution of the exemplar equation in two unknowns, first directly from the theorem, then by elimina- tion and solution for each unknown separately. (1) A x y + B -x y + C x -y + D -x -y = has the equation of condition, ABCD = Provided this be satisfied, the solutions for x and y are B DC. re -A +-C, or x = B D + u (-A + -C) CDcyc-A+-B, or y = C D + u (-A + -B) (2) A x y + B -x y + C x -y + D -x -y = is equivalent to (a) (A x + B -) y -f (C x + D -x) -y = and to (6) (A y + C -y} x + (B y + D -y} -x = Eliminating y from (a), we have The equation of condition with respect to x is, then, (AC)(BD) = ABCD = And the solution for x is B D ex c-(A C), or x = B D + ir(A C). And -(.4 C) = -A + -C Eliminating x from (6), we have (Ay+C-y)(By + D-y) = ABy+CD-y = The equation of condition with respect to y is, then, ABCD = 0. And the solution for y is CDcyc-(AB), or y = CD+v-(AB). And -(A B} = -A + -B Another method of solution for equations in two unknowns, x and y, would be to solve for y and for -y in terms of the coefficients, with x and u as undetermined parameters, then eliminate y by substituting this value of it in the original equation, and solve for x. By a similar substitution, x may then be eliminated and the resulting equation solved for y. This method may inspire more confidence on the part of those unfamiliar with this algebra, since it is a general algebraic method, except that in other algebras more than one equation is required. The solution of A x y + B -x y+ C x -y + D -x -y = for y is y = (C x + D -x) + u--(A x + B -x} = (C + u -A) x + (D + u -B) -x The Classic, or Boole-Schroder, Algebra of Logic 159 The solution for -y is -y = (Ax + B -x} + v-(C x + D -x) = (A+v -C) x + (B + v -D} -x Substituting these values for y and -y in the original equation, (Ax + B -x)[(C + u-A)x + (D + u -B) -x] + (C x + D -x)[(A + v-C)x + (B + v -D) -x\ = A (C + u -A] x + B (D + u -B} -x + C (A + v -C) x + D (B + v -D) -x = ACx + BD-x = 0. Hence BDcxc-A+-C. Theoretically, this method can be extended to equations in any number of unknowns: practically, it is too cumbersome and tedious to be used at all. 7 52 Any equation in n unknowns, of the form F(XI, x 2 , ... x n ) = f(xi, xz, ... x n ) gives a solution for each of the unknowns as follows: Let x k be any one of the unknowns; let PI, P 2 , PS, Qi, Qz, Qs, be the coefficients in F, and MI, MI, M s , . . . NI, NZ, N 3 , . . . the coefficients of the corresponding terms in /, so that P r and M r are coefficients of terms in which x k is positive, and Q r and N r are coefficients of terms in which Xk is negative. The solu- tion for Xk then is II (Qr -N r + -Q r N r ) c x k c (P r M r + -P r -MJ r r Or X k = II (Qr -N r + -Qr N r )+U-^ (P r M r + -P r ~M r ) r r By 7-13, F(XI, x 2 , ... Xn) = f(xi, x 2 , ... x n ) is equivalent to is a function such that if A r and B r be coefficients of any two corresponding terms in F and /, then the coefficient of the corresponding term in $ will be A r -B r + -A r B r . Hence, by 7-51, the solution will be II (Q, -N r + -Qr N r ~) C X k C -(P r ~M r + ~P r M r ) r r Or X k = II (Qr -N r + -Q r N r ) + U ^ "(Pr "M r + -P r Jf r ) r r And [6-4] -(P r -i r + -P r i r ) = (P r M r + -P r - M r ) . 7-53 The condition that m equations in n unknowns, each of the form F(XI, Xz, ... x n ) = 0, may be regarded as simultaneous, is as follows: Let the coefficients of the terms in F 1 , in the equation F l (x i} x 2 , ... x n } = 0, be P^, P 2 S Pg 1 , . . . Q! 1 , Q 2 J , Q s l , ...; let the coefficients of the corre- 160 A Survey of Symbolic Logic spending terms in F 2 , in the equation F 2 (xi, a* 2 , ... x n ) = 0, be Pi 2 , P 2 2 , PS, Qi 2 , Qz 2 , Qs 2 , .', the coefficients of the corresponding terms in F m , in the equation F m (xi, x 2 , ... x n ) = 0, be Pi" 1 , P 2 m , P 3 m , . . . Qi m , Q 2 m , Q 3 m , .... The condition then is r A r h Or if C/ be any coefficient, whether P or Q, in P A , the condition is r A And the solution which n such equations give, on this condition, for any one of the unknowns, x k , is as follows: Let PA P 2 A , Pz h , ... be the coef- ficients of those terms, in any one of the equations F h = 0, in which Xk is positive, and let Qi h , Q 2 h , Q,3 h , ... be the coefficients of those terms, in F h = 0, in which a*& is negative. The solution then is or r h r h By 6-42, m equations in n unknowns, each of the form F(x\, z 2 , . . . x n ) = 0, are together equivalent to the single equation $(xi, x 2 , ...)= 0, where each of the coefficients in $ is the sum of the corresponding coefficients in F 1 , F 2 , F 3 , ... F m . That is, if P r l , P r 2 , . . . P r m be the coefficients of corresponding terms in F 1 , F 2 , ... F m , then the coefficient of the corresponding term in < will be Pr 1 + P r 2 + . . . + P r >, Or X) Pr k A and if Q r l , Q r 2 , . . . Q r m be the coefficients of corresponding terms in F 1 , F 2 , ... F m , then the coefficient of the corresponding term in $ will be The equation of condition for $ = 0, and hence the condition that F 1 = 0, F 2 = 0, ... F m = may be regarded as simultaneous, is the equation to of the product of the coefficients in <; that is, 2 P!* X P 2 " X P 3 " X . . . X 2 <3l" >< Z QS >< Z #3* X . . . =0 A A A A A A or r h And by 7-51, the solution of $(0:1, 2 , . . . .r n ) = for Xk is The Classic, or Boole-Schroder, Algebra of Logic 161 or afc= r h r h And by 5-95, -[ P P ] = -PA /i A 7 54 The condition that m equations in n unknowns, each of the form F(XI, x 2 , ... x n ) = f(xi, x 2 , ... Xn) may be regarded as simultaneous, is as follows: Let the coefficients in F 1 , in the equation F 1 =f l , be Pi 1 , Pa 1 , Ps 1 , . . . Qi 1 , & 1 , Qa 1 , . . ., and let the coefficients of the corresponding terms in / l , in the equation F 1 = f l , be Mi 1 , Mz l , M 3 l , . . . Ni 1 , N 2 l , Ns 1 , . . .; let the coefficients of the corresponding terms in F 2 , in the equation F 2 = f 2 , be P^, P 2 2 , P 3 2 , . . . Qi 2 , Q 2 2 , Q 3 2 , and let the coefficients of the corresponding terms in / 2 be M i 2 , M 2 2 , M 3 2 , . . . Ni 2 , Nz z , N s 2 , . . . ; let the coefficients of the corresponding terms in F m , in the equation F m = / m , be Pi m , P 2 m , P 3 TO , . . . Qi m , Qz m , Qs m , -, and let the coefficients of the corresponding terms in / m be Mi m , M z m , M s m , . . . Ni m , N 2 m , N 3 m , .... The condition then is II I E (P/ ~Mr h + -P r h M r h }} X H [ E (&* -# r* + -Qr* ^r*)] = r h r h or if A r h represent any coefficient in F h , whether P or Q, and B r h represent the corresponding coefficient in / h , whether M or N, the condition is II [E(^r*-Br* + -^r*5r*)] =0 r h And the solution which m such equations give, on this condition, for any one of the unknowns, Xk, is as follows: Let P r h and M r h be the coefficients of those terms, in any one of the equations F h = f h , in which x k is positive, and let Q r h and N r h be the coefficients of the terms, in F h = f h , in which Xk is negative. The solution then is II [ E (Qr k -Nr k + ~Qr h N*)] C X k C E til (P^ M r h + ~P r h ~M T h }} r h r h Or X k = I r By 7-13, .F A (i, X 2 , ... ) = / A (a;:, x z , ... ^ n ) is equivalent to ty(xi, x 2 , ... a; n ) = 0, where ^ is a function such that if Q r h and N r h be coefficients of corresponding terms in F h and / h , the coefficient of the corresponding term in ^ will be Q r h -N r h + -Q r h N r h , and if P r * and M r h be coefficients of corresponding terms in F h and / h , the coefficient of the corresponding term in ^ will be P r * -M r h + -P r h M r h . And -(P r h -M r h + -P r h M r h ) = P r h M r h + -P r * -3f r \ Hence the theorem follows from 7 53. 12 162 A Survey of Symbolic Logic F(XI, x 2 , . . . Xn) = f(x\, #2, ... x n } is a perfectly general equation, since F and / may be any expressions in the algebra, developed as functions of the variables in question. 7-54 gives, then, the condition and the solution of any number of simultaneous equations, in any number of unknowns, for each of the unknowns. This algebra particularly lends itself to generaliza- tion, and this is its most general theorem. It is the most general theorem concerning solutions in the whole of mathematics. Boole's General Problem. Boole proposed the following as the general problem of the algebra of logic. 11 Given any equation connecting the symbols x, y, ... w, z, .... Re- quired to determine the logical expression of any class expressed in any way by the symbols x, y, ... in terms of the remaining symbols w, z, .... We may express this: Given t = f(x, y, . . .) and <(, y, . . . ) = ^(w, z, . . . ) ; to determine t in terms of w, z, .... This is perfectly general, since if x, y, ... and w, z, ... are connected by any number of equations, there is, by 7-1 and 5-72, a single equation equivalent to them all. The rule for solution may be stated: Reduce both t = f(x, y, . . .) and $(x, y, . . .) = ^ (w, z, . . . ) to the form of equations with one member 0, combine them by addition into a single equation, eliminate x, y, . . . , and solve for t. By 7-1, the form of equation with one member is equivalent to the other form. And by 5-72, the sum of two equations with one member is equivalent to the equations added. Hence the single equation resulting from the process prescribed by our rule will contain all the data. The result of eliminating will be the complete resultant which is independent of these, and the solution for t will thus be the most complete determination of t in terms of w, z, ... afforded by the data. Consequences of Equations in General. A word of caution with refer- ence to the manipulation of equations in this algebra may not be out of place. As compared with other algebras, the algebra of logic gives more room for choice in this matter. Further, in the most useful applications of the algebra, there are frequently problems of procedure which are not resolved simply by eliminating this and solving for that. The choice of method must, then, be determined with reference to the end in view. But the following general rules are of service: (1) Get the completest possible expression = 0, or the least inclusive possible expression = 1. a + b + c+ . . . =0 gives a = 0, 6 = 0, c = 0, . . . , a + 6 = 0, a + c = 0, "Laws of Thought, p. 140. The Classic, or Boole-Schroder, Algebra of Logic 163 etc. But a = will not generally give a + 6 = 0, etc. Also, a = 1 gives a + b = 1, a+ . . . = 1, but a + b = 1 will not generally give a = 1. (2) Reduce any number of equations, with which it is necessary to deal, to a single equivalent equation, by first reducing each to the form in which one member is and then adding. The various constituent equations can always be recovered if that be desirable, and the single equation gives other derivatives also, besides being easier to manipulate. Do not forget that it is possible so to combine equations that the result is less general than the data. If w T e have a = and 6 = 0, we have also a b, or a b = 0, or a + 6 = 0, according to the mode of combination. But a + b = is equivalent to the data, while the other two are less comprehensive. A general method by which consequences of a given equation, in any desired terms, may be derived, was formulated by Poretsky, 12 and is, in fact, a corollary of his Law of Forms, given above. We have seen that this law may be formulated as the principle that if a = b, and therefore a-b + -ab = and ab + -a-b = 1, then any t is such that a -b + -a b c t and tcab + -a-b, or any t = the t which contains the "zero member" of the set of equations equivalent to a b, and is contained in the " whole member" of this set. Now if x c t, u x c t, for any u whatever, and thus the "zero member" of the Law of Forms may be multiplied by any arbitrarily chosen u which we choose to introduce. Similarly, if t c y, then t c y + v, and the "whole member" in the Law of Forms may be increased by the addition of any arbitrarily chosen v. This gives the Law of Consequences. 7-6 If a = b, then t = (a b + -a -b + u) t + v (a -b + -a b) -t, where u and v are arbitrary. [7 1 12J If a = b, then a -b + -a b = and ab + -a-b = 1. Hence (a b + -a -b + u) t + v (a -b + -a 6) -t = (1 + u) t + v -i --= t. This law includes all the possible consequences of the given equation. First, let us see that it is more general than the previous formulae of elimina- tion and solution. Given the equation A x + B -x = 0, and choosing A B for t, we should get the elimination resultant. If A x + B -x = 0, then A B = (-A x + -B -x + u} A B + v(Ax + B-x)(-A+-B) = u A B + v (A -B x + -A B -x). Since u and v are both arbitrary and may assume the value 0, there- fore AB = 0. 12 Sept lois, etc., Chap. xn. 164 A Survey of Symbolic Logic But this is only one of the unlimited expressions for A B which the law gives. Letting u = 0, and v = 1, we have A B = A -B x + -A B -x Letting u = A and v = B, we have AB = AB + -A B -x And so on. But it will be found that every one of the equivalents of A B which the law gives will be null. Choosing x for our t, we should get the solution. If A x + B -x = 0, then x = (-A x + -B -x + ii) x + v (A x + B -x) -x = (-A + w) x + v B -x. Since u and v may both assume the value 0, x = -A x, or x c -A (1) And since u and v may both assume the value 1, x = x + B -x, or B -x c x But if -a; c x, then B -x = (B -x) x = 0, or B c x (2) Hence, (1) and (2), Bex c-A. When u = and v = 1, the Law of Consequences becomes simply the Law of Forms. For these values in the above, x = -A x + B -x which is the form which Poretsky gives the solution for x. The introduction of the arbitraries, u and v, in the Law of Consequences extends the principle stated by the Law of Forms so that it covers not only all equivalents of the given equation but also all the non-equivalent inferences. As the explanation which precedes the proof suggests, this is accomplished by allowing the limits of the function equated to t to be expressed in all possible ways. If a = b, and therefore, by the Law of Forms, t = (a b + -a -6) t + (a -b + -a 6) -t the lower limit of t, 0, is expressed as a -b + -a b, and the upper limit of t, 1, is expressed as a b + -a -b. In the Law of Consequences, the lower limit, 0, is expressed as v (a-b + -ab), that is, in all possible w r ays which can be derived from its expression as a-b + -ab; and the upper limit, 1, is expressed as a b + -a -b + u, that is, in all possible ways which can be derived from its expression as a b + -a -6. Since an expression of the form t = (a b + -a -b) t + (a -b + -a b) -t The Classic, or Boole-Schroder, Algebra of Logic 165 or of the form t = (a b + -a -b + w) t + v (a -b + -a 6) -t determines t only in the sense of thus expressing its limits, and the Law of Consequences covers all possible ways of expressing these limits, it covers all possible inferences from the given equation. The number of such inferences is, of course, unlimited. The number expressible in terms of n elements will be the number of derivatives from an equation with one member and the other member expanded with reference to n elements. The number of constituent terms of this expanded member will be 2 n , and the number of combinations formed from them will be 2 2?l . Therefore, since p\ + p 2 + PS + . . . =0 gives pi = 0, p 2 = 0, p 3 = 0, etc., this is the number of consequences of a given equation which are expressible in terms of n elements. As one illustration of this law, Poretsky gives the sixteen determinations of a in terms of the three elements, a, b, and c, which can be derived from the premises of the syllogism in Barbara: 13 If all a is b, a -b 0, and all b is c, b -c = 0, then a-b + b -c = 0, and hence, a = a (b + -c) = a (b + c) = a (-b + c) = a + b -c = a b = a (b c + -b -c) = b -c + a (b c + -b -c) =ac = b-c + ac = a (-6 + c) + -a b -c = ab c = b -c + ab c = a (b c + -b -c) + -a b -c = a c + -a b -c = a b c + -a b -c The Inverse Problem of Consequences. Just as the Law of Conse- quences expresses any inference from a = b by taking advantage of the fact that if a-b + -ab = 0, then (a-b + -ab)v = 0, and if ab + -a-b = 1, then a b + -a -b + u = 1 ; so the formula for any equation which will give the inference a = b can be expressed by taking advantage of the fact that if v (a b + -a -b) = 1, then ab + -a-b = 1, and if a -b + -a b + u = 0, then a-b + -ab = 0. We thus get Poretsky's Law of Causes, or as it would be better translated, the Law of Sufficient Conditions. 14 7 7 If for some value of u and some value of v t = v (a b + -a -6) t + (a -b + -a b + u) -t, then a = b. If t = v (a b + -a -6) t + (a -b + -a b + u) -t, then [7-1, 5-72] [v (a b + -a -b) t + (a -b + -a b + it) -t] -t = = (a -b + -a b + u) -t = (a -b + -a 6) -t + u -t = 13 Ibid., pp. QBff. 14 Ibid., Chap. xxm. 166 A Survey of Symbolic Logic Hence (a -b + -a b) -t = (1) Hence also [5-7] t = v (ab + -a-b) t, and [4-9] t -[v (a b + -a -b)] = = t (-v + a -b + -a 6) = t -v + (a -6 + -a 6) t Hence [5 72] (a -6 + -a 6) t = (2) By (1) and (2), (a -6 + -a 6) (t + -t) = = a -6 + -a b. Hence [7-1] a = b. Both the Law of Consequences and the Law of Sufficient Conditions are more general than the Law of Forms, which may be derived from either. Important as are these contributions of Poretsky, the student must not be misled into supposing that by their use any desired consequence or sufficient condition of a given equation can be found automatically. The only sense in which these laws give results automatically is the sense in which they make it possible to exhaust the list of consequences or conditions expressible in terms of a given set of elements. And since this process is ordinarily too lengthy for practical purposes, these laws are of assistance principally for testing results suggested by some subsidiary method or by "intuition ". One has to discover for himself what values of the arbitraries u and v will give the desired result. V. FUNDAMENTAL LAWS OF THE THEORY OF INEQUATIONS In this algebra, the assertory or copulative relations are = and c . The denial of a = b may conveniently be symbolized in the customary way: 8-01 a 3= b is equivalent to "a = b is false ". Def. We might use a symbol also for "a c b is false ". But since a c b is equiva- lent to a b = a and to a -b = 0, its negative may be represented by a b 4= a or by a -6 4= 0. It is less necessary to have a separate symbolism for "acb is false ", since "a is not contained in b" is seldom met with in logic except where a and b are mutually exclusive, in which case a b 0. For every proposition of the form "If P is true, then Q is true ", there is another, " If Q is false, then P is false ". This is the principle of the reductio ad absurdum, or the simplest form of it. In terms of the relations = and 4=> the more important forms of this principle are: (1) "If a = b, then c = d", gives also, "If c 4= d, then a 4= 6 ". (2) "If a = b, then c = dandh = k", gives also, " If c 4= Athena =|= 6", and "If h =1= Mhena * b". (3) "If a = b and c = d, then h = k", gives also, "If a = b and h =|= k, then c 4= d", and " If c = d and h =(= A-, then a 4= b ". The Classic, or Boole-Schroder, Algebra of Logic 167 (4) "a = b is equivalent to c = d", gives also, "a 4= b is equivalent toe 4= d". (5) "a = b is equivalent to the set, c = d, h = k, . . .," gives also, "a =}= b is equivalent to 'Either c =j= ^ or A =|= k, or ...'". 16 The general forms of these principles are themselves theorems of the "calculus of propositions" the application of this algebra to propositions. But the calculus of propositions, as an applied logic, cannot be derived from this algebra without a circle in the proof, for the reasoning in demon- stration of the theorems presupposes the logical laws of propositions at every step. We must, then, regard these laws of the reductio ad absurdum, like the principles of proof previously used, as given us by ordinary logic, which mathematics generally presupposes. In later chapters, 16 we shall discuss another mode of developing mathematical logic the logistic method w r hich avoids the paradox of assuming the principles of logic in order to prove them. For the present, our procedure may be viewed simply as an application of the reductio ad absurdum in ways in which any mathe- matician feels free to make use of that principle. Since the propositions concerning inequations follow immediately, for the most part, from those concerning equations, proof will ordinarily be unnecessary. Elementary Theorems. The more important of the elementary propo- sitions are as follows: 8-1 If a c =%= b c, then a 4= b. [2-1] 8-12 If a + c 4= b + c, then a 4= b. [3-37] 8-13 a =1= b is equivalent to -a =)= -b. [3-2] 8-14 a + b 4= b, a b =4= a, -a + b =f= 1, and a -b 4= are all equivalent. [4-9] 8-15 If a + b = x and b =f= %, then a 4= [5-7] 8-151 If a = and b 4= x, then a + b 4= % [5-7] 8-16 If a b = x and b 3= x, then a 4= 1. [5-71] 15 "Either ... or ..." is here to be interpreted as not excluding the possibility that both should be true. 18 Chap, iv, Sect, vi, and Chap. v. 168 A Survey of Symbolic Logic 8- 161 If a = 1 and 6 4= x, then ab = x. [5-71] 8-17 If a + b 4= and a = 0, then b ={= 0. [5-72] 8-18 If a 6 4= 1 and a = 1, then 6 4= 1. [5-73] 8-17 allows us to drop null terms from any sum 4= 0. In this, it gives a rule by which an equation and an inequation may be combined. Suppose, for example, a + b 4= and x = 0. a + b = (a + 6) (x + -x) = ax+bx+a-x+b -x. Hence a x + b x + a -x + b -x 4= 0. But if x = 0, then a x = and b x = 0. Hence [8 17] a -x + b -x 4= 0. 8-2 If a 4= 0, then a + b 4= 0. [5-72] 8-21 If a 4= 1, then ab 4= 1. [5-73] 8-22 If a b 4= 0, then a 4= and 6 4= 0. [1-5] 8-23 If a+b 4= 1, then a 4= 1 and b 4= 1. [4-5] 8-24 If a 6 4= # and a = x, then b =]= a:. [1-2] 8-25 If a 4= and a c 6, then 6 =1= 0. [1-9] If a c 6, then a b = a. Hence if a 4= and a c 6, then a 6 4= 0. Hence [8-22] 6 + 0. 8-26 a + 6 4= is equivalent to "Either a 4= or b 4= ". [5-72] 8-261 di + a z + a 3 + . . . 4= is equivalent to "Either a x 4= or a 2 4= or o 8 4= 0, or . . . ". 8-27 a 6 4= 1 is equivalent to "Either a 4= 1 or 6 4= 1 ". [5-73] 8-271 ai a 2 3 . . . 4= 1 is equivalent to "Either a x 4= 1 or a 2 4= 1 or a, 4= 1 or . . . ". The difference between 8-26 and 8-27 and their analogues for equa- tions 5 -72 a + b is equivalent to the pair, a = and 6 = 0, and The Classic, or Boole-Schroder, Algebra of Logic 169 5-73 a b = 1 is equivalent to the pair, a = 1 and b = 1 points to a neces- sary difference between the treatment of equations and the treatment of inequations. Two or more equations may always be combined into an equivalent equation; two or more inequations cannot be combined into an equivalent inequation. But, by 8-2, a+ b =(= is a consequence of the pair, a =f= and b 4= 0. Equivalent Inequations of Different Forms. The laws of the equiva- lence of inequations follow immediately from their analogues for equations. 8-3 a =}= b is equivalent to a -b + -a b =(= 0. [7-1] 8-31 a 4= 1 is equivalent to -a 4= 0. [7-12] 8-32 If 3>(xi, x, then the coefficients of the corresponding terms in F will be A\ -Bi + -Ai BI, A z -B 2 + -A 2 B 2 , A s -B 3 + -A s B 3> etc. [7-13] Poretsky's Law of Forms for inequations will be : 8-33 a =J= is equivalent to t 4= a ~t+ -a t. [7-151 Or in more general form : 8 34 a 4 1 b is equivalent to t + (ab + -a -6) t + (a -b + -a b) -t. [7-16] Elimination. The laws governing the elimination of elements from an inequation are not related to the corresponding laws governing equations by the reductio ad absurdum. But these laws follow from the same theorems concerning the limits of functions. 8-4 If Ax + B-x 4= 0, then ^4 + 5 + 0. [6 3] A x + B -x c A + B. Hence [8 251 Q.E.D. 8-41 If the coefficients in any function of n variables, F(XI, x 2 , ... #), 170 A Survey of Symbolic Logic be Ci, C'2, Cz, etc., and if F(x lt .r 2 , . . . .r n ) =|= 0, then EC*O [6-32] F(xi, .TO, ... x n )cC. Hence [8-25] Q.E.D. Thus, to eliminate any number of elements from an inequation with one member 0, reduce the other member to the form of a normal function of the elements to be eliminated. The elimination is then secured by putting =(= the sum of the coefficients. The form of elimination resultants for inequations of other types follows immediately from the above. It is obvious that they will be analogous to the elimination resultants of equa- tions as follows: To get the elimination resultant of any inequation, take the elimination resultant of the corresponding equation and replace = by 4= > and x by + . A universal proposition in logic is represented by an equation: "All a is b " by a -b = 0, " No a is b " by a b = 0. Since a particular proposition is always the contradictory of some universal, any particular proposition may be represented by an inequation: "Some a is b" by a b =f= 0, "Some a is not b" by a -b =(= 0. The elimination of the "middle" term from an equation which represents the combination of two universal premises gives the equation which represents the universal conclusion. But elimina- tion of terms from inequations does not represent an analogous logical process. Two particulars give no conclusion: a particular conclusion requires one universal premise. The drawing of a particular conclusion is represented by a process which combines an equation with an inequation, by 8-17, and then simplifies the result, by 8-22. For example, All a is b, a -b = 0. .'. a -b c = 0. Some a is c, a c + 0. .'. a b c + a -b c ={= 0. .*. abc * 0. [8-17] Some b is c. .'. be ^ 0. [8-22] " Solution " of an Inequation. An inequation may be said to have a solution in the sense that for any inequation involving x an equivalent inequation one member of which is x can always be found. 8-5 A x + B -x 4= is equivalent to x 4= -A x + B -x. [7-23] 8-51 A x + B -x =f= is equivalent to "Either B-x^OorAx^O ", i. e., to "Either B ex is false or x c-A is false ". [7-2! The Classic, or Boole-Schroder, Algebra of Logic 171 Neither of these "solutions" determines x even within limits. "Bex is false " does not mean " B is excluded from x"; it means only " B is not wholly within x ". "Either Bex is false or xc-A is false" does not determine either an upper or a lower limit of x; and limits x only by ex- cluding B + u -A from the range of its possible values. Thus " solutions " of inequations are of small significance. Consequences and Sufficient Conditions of an Inequation. By Poret- sky's method, the formula for any consequence of a given inequation follows from the Law of Sufficient Conditions for equations. 17 If for some value of u and some value of v, t = v (a b + -a -6) t + (a -b + -a b + u) -t then a = b. Consequently, we have by the reductio ad absurdum: 8-52 If a =(= 6, then t ^ v (ab + -a -b) t + (a -b + -a b + u) -t, where u and v are arbitrary. [7-7] The formula for the sufficient conditions of an inequation similarly fol- lows from the Law of Consequences for equations. If a = b, then t = (a b + -a -b + u) t + v (a -b + -a b) -t where u and v are arbitrary. Consequently, by the reductio ad absurdum: 8-53 If for some value of u and some value of v, t =^ (a b + -a -b + u) t + v (a -b + -a 6) -t then a =(= b. [7-6] System of an Equation and an Inequation. If we have an equation in one unknown, x, and an inequation which involves x, these may be combined in either of two ways: (1) each may be reduced to the form in which one member is and expanded with reference to all the elements involved in either. Then all the terms which are common to the two may, by 8-17, be dropped from the inequation; (2) the equation may be solved for x, and this value substituted for x in the inequation. 8-6 If A x + B -x = and C x + D -x 4= 0, then -A C x + -B D -x 4= 0. [5-8] If Cx + D-x 4= 0, then A C x + -A C x + B D -x + -B D -x 4= 17 See Poretsky, Theorie des non-egalites logiques, Chaps. 71, 76. 172 A Survey of Symbolic Logic [5-72] If Ax + B-x = 0, then A x = and B -x = 0, and hence A C x = and B D -x = 0. Hence [8-17] -A C x + -B D -x 4= 0. The result here is not equivalent to the data, since for one reason the equation ACx + BD-x = is not equivalent to A x + B -x = 0. Nevertheless this mode of combination is the one most frequently useful. 8-61 The condition that the equation Ax + B-x = and the inequation C x + D -x 4= may be regarded as simultaneous is, A B = and -A C + -B D 4= 0, and the determination of x which they give is x 4= (-A -C + A-D)x + (BC + -B D) -x [7-23] A x + B -x = is equivalent to x = -A x + B -x. Substi- tuting this value of x in the inequation, C (-A x + B -x) + D (A x + -B -x} 4= or (-AC + AD)x+(BC + -BD)-x + 0. [8-4] A condition of this inequation is (-AC + AD) + (BC + -BD) 4= 0, or (-A+B) C + (A + -B) D =j= 0. But the equation A x + B -x = requires that A B = 0, and hence that -A + B = -A and -B + A = -B. Hence if the equation be possible and A B = 0, the condition of the inequation reduces to -A C + -B D 4= 0. [8-4] If the original inequation be possible, then C + D 4= 0. But this condition is already present in -A C + -B D 4= 0, since -A C cC and hence [8-25] if -A C 4= 0, then (7 + 0, and -BDcD and hence if -B D 4= 0, then D 4= 0, while [8-26] C + D 4= is equivalent to " Either C 4= or D 4= ", and -A C + -B D 4= is equivalent to " Either -A C 4= or -B D 4= ". Hence the entire condition of the system is expressed by AB = and -AC + -BD*Q And [8-5] the solution of the inequation, (-AC + AD)x + (BC + -BD)-x 4= 0, is x 4= (-A-C + A-D)x + (BC + -BD)-x This method gives the most complete determination of x, in the form of an inequation, afforded by the data. The Classic, or Boole-Schroder, Algebra of Logic 173 VI. NOTE ON THE INVERSE OPERATIONS, "SUBTRACTION" AND "DIVISION" It is possible to define "subtraction" { } and "division" { : } in the algebra. Let a b be x such that b + x = a. And let a : b be y such that b y = a. However, these inverse operations are more trouble than they are worth, and should not be admitted to the system. In the first place, it is not possible to give these relations a general meaning. We cannot have in the algebra: (1) If a and b are elements in K, then a : b is an element in K; nor (2) If a and b are elements in K, then a b is an element in K. If a : b is an element, y, then for some y it must be true that b y = a. But if b y = a, then, by 2 2, acby and, by 5-2, a c b. Thus if a and b be so chosen that a c b is false, then a : 6 cannot be any element in K. To give a : b a general meaning, it would be required that every element be contained in every element that is, that all elements in K be identical. Similarly, if a b be an element, x, in K, then for some x, it must be true that b + x = a. But if b + x = a, then, by 2-2, b + xca and, by 5-21, be a. Thus if a and b be so chosen that & c a is false, then a b cannot be any element in K. Again, a b and a : b are ambiguous. It might be expected that, since a + -a = 1, the value of 1 a would be unambiguously -a. But 1 a = x is satisfied by any x such that -a c x. For 1 a = x is equiva- lent to x + a = 1, which is equivalent to -(x + a) = -1 = = -a -x And -a -x = is equivalent to -a c x. Similarly, it might be expected that, since a -a = 0, the value of : a would be unambiguously -a. But : a = y, or a y = 0, is satisfied by any y such that y c -a. a y = and y c -a are equivalent. Finally, these relations can always be otherwise expressed. The value of a : b is the value of y in the equation, b y = a. b y = a is equivalent to -a b y + a -b + a -y = The equation of condition here is a -b 0. And the solution, on this condition, is y = a + u (a + -&) = ab + u-a-b, where u is undetermined. The value of a b is the value of x in the equation, 6 + x = a. b + x = a is equivalent to -a b + -a x + a -b -x = 174 A Survey of Symbolic Logic The equation of condition here is, -a b = 0. And the solution, on this condition, is x = a-b + v a = a-b + v ab, where v is undetermined. In each case, the equation of condition gives the limitation of the meaning of the expression, and the solution expresses the range of its possible values. CHAPTER III APPLICATIONS OF THE BOOLE-SCHRODER ALGEBRA There are four applications of the classic algebra of logic which are commonly considered: (1) to spatial entities, (2) to the logical relations of classes, (3) to the logical relations of propositions, (4) to the logic of relations. The application to spatial entities may be made to continuous and discontinuous segments of a line, or to continuous and discontinuous regions in a plane, or to continuous and discontinuous regions in space of any dimensions. Segments of a line and regions in a plane have both been used as diagrams for the relations of classes and of propositions, but the application to regions in a plane gives the more workable diagrams, for obvious reasons. And since it is only for diagrammatic purposes that the application of the algebra to spatial entities has any importance, we shall confine our attention to regions in a plane. I. DIAGRAMS FOR THE LOGICAL RELATIONS OF CLASSES For diagrammatic purposes, the elements of the algebra, a, b, c, etc., will denote continuous or discontinuous regions in a given plane, or in a circumscribed portion of a plane. 1 represents the plane (or circumscribed portion) itself. is the null-region which is supposed to be contained in every region. For any given region, a, -a denotes the plane exclusive of a, i. e., not-a. The "product", a x6 or a b, is that region which is com- mon to a and b. If a and b do not "overlap ", then a fy is the null-region, 0. The "sum", a + b, denotes the region which is either a or b (or both). In determining a + b, the common region, a b, is not, of course, counted twice over. a + b = a-b + ab + -ab. This is a difference between + in the Boole-Schroder Algebra and the + of arithmetic. The equation, a = b, signifies that a and 6 denote the same region, a c b signifies that a lies wholly within b, that a is included or contained in b. It should be noted that whenever a = b, a c b and b c a. Also, a c a holds always. Thus the relation c is analogous not to < in arithmetic but to . 175 176 A Survey of Symbolic Logic While the laws of this algebra hold for regions, thus denoted, however those regions may be distributed in the plane, not every supposition about their distribution is equally convenient as a diagram for the relations of classes. All will be familiar with Euler's diagrams, invented a century earlier than Boole's algebra. "All a is b" is represented by a circle a wholly within a circle b; " No a is b " by two circles, a and b, which nowhere intersect; "Some a is b" and "Some a is not b" by intersecting circles, sometimes with an asterisk to indicate that division of the diagram \vhich represents the proposition. The defects of this style of diagram are obvious: All a is 6 No a is b Some a is 6 Some a is not b FIG. 1 the representation goes beyond the relation of classes indicated by the propo- sition. In the case of "All a is b", the circle a falls within b in such wise as to suggest that we may infer "Some b is not a", but this inference is not valid. The representation of "No a is b" similarly suggests "Some things are neither a nor b", 'which also is unwarranted. With these dia- grams, there is no way of indicating whether a given region is null. But the general assumption that no region of the diagram is null leads to the misinterpretations mentioned, and to others which are similar. Yet Euler's diagrams were in general use until the invention of Venn, and are still doing service in some quarters. The Venn diagrams were invented specifically to represent the relations of logical classes as treated in the Boole-Schroder Algebra. 1 The principle of these diagrams is that classes be represented by regions in such relation to one another that all the possible logical relations of these classes can be indicated in the same diagram. That is, the diagram initially leaves room for any possible relation of the classes, and the actual or given relation can then be specified by indicating that some particular region is null or is not- null. Initially the diagram represents simply the "universe of discourse", or 1. For one element, a, 1 a + -a. 2 For two elements, a and b, 1 = (a + -a) (6 + -6) = a b + a -b + -a b + -a -b 1 See Venn, Symbolic Logic, Chap. v. The first edition of this book appeared before Schroder's Algebra der Logik, but Venn adopts the most important alteration of Boole's original algebra the non-exclusive interpretation of a + b. 2 See above, Chap, u, propositions 4-8 and 5-92. Applications of the Boole-Schroder Algebra 177 For three elements, a, b, and c, 1 = (a + -a) (b + -6) (c + -c) = a a-bc+-abc + a-b-c + -a b -c + -a -b c + -a -b -c Thus the "universe of discourse" for any number of elements, n, must correspond to a diagram of 2" divisions, each representing a term in the expansion of 1. If the area within the square in the diagram represent -a -a-b FIG. 2 the universe, and the area within the circle represent the element a, then the remainder of the square will represent its negative, -a. If another element, b, is to be introduced into the same universe, then b may be repre- sented by another circle whose periphery cuts the first. The divisions, (1) into a and -a, (2) into b and -b, will thus be cross-divisions in the uni- verse. If a and b be classes, this arrangement represents all the possible subclasses in the universe; a b, those things which are both a and b; a -b, those things which are a but not b ; -ab, those things which are b but not a; -a -b, those things which are neither a nor 6. The area which represents the product, a b, will readily be located. We have enclosed by a broken line, in figure 2, the area which represents a + b. The negative of any entity is always the plane exclusive of that entity. For example, -(a b + -a -6), in the above, will be the sum of the other two divisions of the diagram, a-b + -a b. If it be desired t introduce a third element, c, into the universe, it is necessary to cut each one of the previous subdivisions into two one part which shall be in c and one part which shall be outside c. This can be be accomplished by introducing a third circle, thus It is not really necessary to draw the square, 1, since the area given to the figure, or the whole page, may as well be taken to represent the universe. But when the square is omitted, it must be remembered that the unenclosed 13 178 A Survey of Symbolic Logic area outside all the lines of the figure is a subdivision of the universe the entity -a, or -a -6, or -a -b -c, etc., according to the number of elements involved. -a-b-c FIG. 3 If a fourth element, d, be introduced, it is no longer possible to repre- sent each element by a circle, since a fourth circle could not be introduced in figure 3 so as to cut each previous subdivision into two parts one part in d and one part outside d. But this can be done with ellipses. 3 Each FIG. 4 3 We have deformed the ellipses slightly and have indicated the two points of junction. This helps somewhat in drawing the diagram, which is most easily done as follows: First, draw the upright ellipse, a. Mark a point at the base of it and one on the left. Next, Applications of the Boole-Schroder Algebra 179 one of the subdivisions in figure 4 can be "named" by noting whether it is in or outside of each of the ellipses in turn. Thus the area indicated by 6 is a b c -d, and the area indicated by 12 is -a -b c d. With a diagram of four elements, it requires care, at first, to specify such regions as a + c, a c + b d, b + -d. These can always be determined with certainty by developing each term of the expression with reference to the missing ele- ments. 4 Thus ac + b d = ac (b + -b) (d + -d) + b d (a + -a) (c + -c) = abcd+abc-d + a-bcd+a-bc-d + ab -ab -c d The terms of this sum, in the order given, are represented in figure 4 by the divisions numbered 10, 6, 9, 5, 14, 11, 15. Hence ac + bd is the region which combines these. With a little practice, one may identify such regions without this tedious procedure. Such an area as b + -d is more easily identified by inspection: it comprises 2, 3, 6, 7, 10, 11, 14, 15, and 1, 4, 5, 8. Into this diagram for four elements, it is possible to introduce a fifth, e, if we let e be the region between the broken lines in figure 5. The principle of the " square diagram" (figure 6) is the same as Venn's: it represents all FIG. 5 draw the horizontal ellipse, d, from one of these points to the other, so that the line con- necting the two points is common to a and d. Then, draw ellipse b from and returning to the base point, and ellipse c from and returning to the point on the left. If not done in this way, the first attempts are likely to give twelve or fourteen subdivisions instead of the required sixteen. 4 See Chap, u, 5-91. 180 A Survey of Symbolic Logic the subclasses in a universe of the specified number of elements. No diagram is really convenient for more than four elements, but such are 5 -b a~b -ab aI) a b FIG. 6 frequently needed. The most convenient are those made by modifying slightly the square diagram of four terms, at the right in figure 6. 5 Figure 7 -a -e -h FIG. 7 -e gives, by this method, the diagrams for five and for six elements. We give also the diagram for seven (figure 8) since this is frequently useful and not easy to make in any other way. The manner in which any function in the algebra may be specified in a diagram of the proper number of divisions, has already been explained. We must now consider how any asserted relation of elements any inclu- 6 See Lewis Carroll, Symbolic Logic, for the particular form of the square diagram which we adopt. Mr. Dodgson is able, by this method, to give diagrams for as many as 10 terms, 1024 subdivisions (p. 176). Applications of the Boole-Schroder Algebra 181 sion, a c b, or any equation, a = b, or inequation, a + b may be repre- sented. Any such relation, or any set of such relations, can be completely specified in these diagrams by taking advantage of the fact that they -e FIG. 8 can always be reduced either to the form of an expression = or to the form of an expression =|= 0. Any inclusion, a c b, is equivalent to an equa- tion, a -b = O. 6 And every equation of the form a = b is equivalent to one of the form a-b + -ab = O. 7 Thus any inclusion or equation can be represented by some expression = 0. Similarly, any inequation of the form a =f= b is equivalent to one of the form a-b + -ab =j= O. 8 Thus any asserted relation whatever can be specified by indicating that some region (continuous or discontinuous) either is null, { = 0}, or is not-null, {4= 0}. We can illustrate this, and at the same time indicate the manner in which such diagrams are useful, by applying the method to a few syllogisms* Given : All a is b, and All b is c, 8 See Chap. 11, 4-9. 7 See Chap, n, 6-4. 8 See Chap, n, 7-1. a cb, b cc, a -b = 0. b -c = 0. 182 A Survey of Symbolic Logic We have here indicated (figure 9) that a -b the a which is not b is null by striking it out (with horizontal lines). Similarly, we have indicated that all b is c by striking out b -c (with vertical lines). Together, the two operations have eliminated the whole of a -c, thus indicating that a -c = 0, or "All aisc". FIG. 9 FIG. 10 For purposes of comparison, we may derive this same conclusion by algebraic processes. 9 Since a -b = = a -b (c + -c) = a -b c + a -b -c, and b -c = = b -c (a + -a) = a b -c + -a b -c, therefore, a -b c + a -b -c + a b -c + -a b -c = 0, and [5-72] a b -c + a -b -c = = a -c (b + -b) = a -c. The equation in the third line, which combines the two premises, states exactly the same facts which are represented in the diagram. The last equation gives the conclusion, which results from eliminating the middle term, b. Since a diagram will not perform an elimination, we must there "look for" the conclusion. One more illustration of this kind : Given: All a is b, a-b = 0. and No 6 is c, b c = 0. The first premise is indicated (figure 10) by striking out the area a -b (with horizontal lines), the second by striking out be (with vertical lines). To- gether, these operations have struck out the whole of a c, giving the con- clusion a c = 0, or "No a is c". 9 Throughout this chapter, references in square brackets give the number of the the- orem in Chap, n by which any unobvious step in proof is taken. Applications of the Boole-Schroder Algebra 183 In a given diagram where all the possible classes or regions in the uni- verse are initially represented, as they are by this method of diagramming, we cannot presume that a given subdivision is null or is not-null. The actual state of affairs may require that some regions be null, or that some be not-null, or that some be null and others not. Consequently, even when we have struck out the regions which are null, we cannot presume that all the regions not struck out are not-null. This would be going beyond the premises. All we can say, when we have struck out the null-regions, is that, so far as the premises represented are concerned, any region not struck out may be not-null. If, then, we wish to represent the fact that a given region is definitely not-null that a given class has members, that there is some expression =}= we must indicate this by some distinctive mark in the diagram. For this purpose, it is convenient to use asterisks. That a b =}= 0, may be indicated by an asterisk in the region a b. But here a further difficulty arises. If the diagram involve more than two elements, say, a, b, and c, the region a b will be divided into two parts, a b c and a b -c. Now the inequation, a b =f= 0, does not tell us that a b c + 0, and it does not tell us that a b -c =t= 0. It tells us only that ab c + ab -c 4= 0. If, then, we wish to indicate a b =}= by an asterisk in the region a b, we shall not be warranted in putting it either inside the circle c or outside c. It belongs in one or the other or both that is all we know. Hence it is convenient to indicate a b 4= by placing an asterisk in each of the divisions of a 6 and connecting them by a broken line, to signify that at least one of these regions is not-null (figure 11). FIG. 11 We shall show later that a particular proposition is best interpreted by an inequation; "Some a is b", the class ab has members, by a b 4= 0. 184 A Survey of Symbolic Logic Suppose, then, we have: Given: All a is b, a-b = 0. and Some a is c, a c =|= 0. The conclusion, "Some b is c", is indicated (figure 12) by the fact that one of the two connected asterisks must remain the whole region a b c + a-bc cannot be null. But one of them, in a -b c, is struck out in indi- cating the other premise, a -b = 0. Thus a 6 c 4= 0, and hence a c 4= 0. FIG. 12 The entire state of affairs in a universe of discourse may be represented by striking out certain regions, indicating by asterisks that certain regions are not-null, and remembering that any region which is neither struck out nor occupied by an asterisk is in doubt. Also, the separate subdivisions of a region occupied by connected asterisks are in doubt unless all but one of these connected asterisks occupy regions which are struck out. And any regions which are left in doubt by a given set of premises might, of course, be made specifically null or not-null by an additional premise. In complicated problems, the use of the diagram is often simpler and more illuminating than the use of transformations, eliminations, and solu- tions in the algebra. All the information to be derived from such opera- tions, the diagram gives (for one who can "see" it) at a glance. Further illustrations will be unnecessary here, since we shall give diagrams in con- nection with the problems of the next section. II. THE APPLICATION TO CLASSES The interpretation of the algebra for logical classes has already been explained. 10 a, b, c, etc., are to denote classes taken in extension; that is 10 Chap, n, pp. 121-22. Applications of the Boole-Schroder Algebra 185 to say, c signifies, not a class-concept, but the aggregate of all the objects denoted by some class-concept. Thus if a = b, the concept of the class a may not be a synonym for the concept of the class b, but the classes a and b must consist of the same members, have the same extension, a c b sig- nifies that every member of the class a is also a member of the class 6. The "product", a 6, denotes the class of those things which are both mem- bers of a and members of 6. The "sum ", a + b, denotes the class of those things which are either members of a or members of b (or members of both). denotes the null-class, or class without members. Various concepts may denote an empty class "immortal men", "feathered invertebrates", "Julius Caesar's twin," etc. but all such terms have the same extension; they denote nothing existent. Thus, since classes are taken in extension, there is but one null-class, 0. Since it is a law of the algebra that, for every x, ex, we must accept, in this connection, the convention that the null-class is contained in every class. All the immortal men are mem- bers of any class, since there are no such. 1 represents the class "every- thing ", the "universe of discourse ", or simply the "universe ". This term is pretty well understood. But it may be defined as follows: if a n be any member of the class a, and X represent the class-concept of the class x, then the "universe of discourse" is the class of all the classes, x, such that "a n is an X" is either true or false. If "The fixed stars are blind" is neither true nor false, then "fixed stars" and the class "blind" do not belong to the same universe of discourse. The negative of a, -a, is a class such that a and -a have no members in common, and a and -a between them comprise everything in the universe of discourse: a -a = 0, "Nothing is both a and not-a", and a + -a = 1, "Everything is either a or -a". Since inclusions, a c b, equations, a = b, and inequations, a =(= b, repre- sent relations which are asserted to hold between classes, they are capable of being interpreted as logical propositions. And the operations of the algebra transformations, eliminations, and solutions are capable of interpretation as processes of reasoning. It would hardly be correct to say that the operations of the algebra represent the processes of reasoning from given premises to conclusions: they do indeed represent processes of reasoning, but they seldom attain the result by just those operations which are supposed to characterize the customary processes of thinking. In fact, it is the greater generality of the symbolic operations which makes their application to reasoning valuable. 186 A Survey of Symbolic Logic The representation of propositions by inclusions, equations, and in- equations, and the interpretation of inclusions, equations, and inequations in the algebra as propositions, offers certain difficulties, due to the fact that the algebra represents relations of extension only, while ordinary logical propositions quite frequently concern relations of intension. In discussing the representation of the four typical propositions, we shall be obliged to consider some of these problems of interpretation. The universal affirmative, "All a is b", has been variously represented as, (1) a = a b, (2) acb, (3) a = v b, where v is undetermined, (4) a -b = 0. All of these are equivalent. 11 The only possible doubt concerns (3) a = v b, where v is undetermined. But its equivalence to the others may be demon- strated as follows: [7 1] a = v b is equivalent to a -(v b) + -a v b = 0. But a--(v 6) + -a v b = a (-v + -b) + -avb = a -v + a -b + -a % b. Hence [5-72] if a = v b, then a -b = 0. And if a a b, then for some value of v (i. e., v = a), a = v b. These equivalents of " All a is b " would most naturally be read : (1) The a's are identical with those things which are a's and 6's both. (2) a is contained in b: every member of a is also a member of b. (3) The class a is identical with some (undetermined) portion of the class 6. (4) The class of those things which are members of a but not members of b is null. If we examine any one of these symbolic expressions of "All a is b", we shall discover that not only may it hold when a = 0, but it always holds when a = 0. = 0-6, cb, and 0--6 = 0, will be true for every element b. And "0 = 06 for some value of v" is always true for v = 0. Since a = means that a has no members, it is thus clear that the algebra requires that "All a is b" be true whenever no members of a exist. The actual use of language is ambiguous on this point. We should hardly say that "All sea serpents have red wings, because there aren't any sea ser- pents"; yet we understand the hero of the novel who asserts "Whoever 11 See Chap, n, 4-9. Applications of the Boole-Schroder Algebra 187 enters here must pass over my dead body". This hero does not mean to assert that any one will enter the defended portal over his body: his desire is that the class of those who enter shall be null. The difference of the two cases is this: the concept "sea serpent" does not necessarily involve the concept "having red wings", while the concept of "those who enter the portal" as conceived by the hero does involve the concept of passing over his body. We readily accept and understand the inclusion of an empty class in some other when the concept of the one involves the concept of the other when the relation is one of intension. But in this sense, an empty class is not contained in any and every class, but in some only. In order to understand this law of the algebra, "For every x, ex", we must bear in mind two things: (1) that the algebra treats of relations in extension only, and (2) that ordinary language frequently concerns relations of intension, and is usually confined to relations of intension where a null class is involved. The law does not accord with the ordinary use of language. This is, however, no observation upon its truth, for it is a necessary law of the relation of classes in extension. It is an immediate consequence of the principle, "For every y, y c 1", that is, "All members of any class, y, are also members of the class of all things". One cannot accept this last without accepting, by implication, the principle that, in extension, the null- class is contained in every class. The interpretation of propositions in which no null-class is involved is not subject to any corresponding difficulty, both because in such cases the relations predicated are frequently thought of in extension and because the relation of classes in extension is entirely analogous to their relation in intension except where the class or the class 1 is involved. But the interpretation of the algebra must, in all cases, be confined to extension. In brief: "All a is b" must always be interpreted in the algebra as stating a relation of classes in extension, not of class-concepts, and this requires that, whenever a is an empty class, "All a is 6" should be true. The proposition, "No a is b", is represented by a b = "Nothing is both a and b", or "Those things which are members of a and of b both, do not exist". Since "No a is 6" is equivalent to "All a is not-6", it may also be represented by a -b = -b, a c -b, b c -a, or a = v -b, where v is undetermined. In the case of this proposition, there is no discrepancy between the algebra and the ordinary use of language. The representation of particular propositions has been a problem to symbolic logicians, partly because they have not clearly conceived the 188 A Survey of Symbolic Logic relations of classes and have tried to stretch the algebra to cover traditional relations which hold in intension only. If "Some a is b" be so interpreted that it is false when the class a has no members, then "Some a is b" will not follow from "All a is 6", for "All a is 6" is true whenever a = 0. But on the other hand, if "Some a is 6" be true when a = 0, we have two diffi- culties: (1) this does not accord with ordinary usage, and (2) "Some a is b" will not, in that case, contradict "No a is b". For whenever there are no members of a (when a = 0), "No a is b" (a b = 0) will be true. Hence if "Some a is 6" can be true when a = 0, then "Some a is b" and "No a is b" can both be true at once. The solution of the difficulty lies in observing that "Some a is 6" as a relation of extension requires that there be some a that at least one member of the class a exist. Hence, when propositions are interpreted in extension, "Some a is b" does not follow from "All a is b", precisely because whenever a = 0, "All a is b" will be true. But "Some a is b" does follow from "All a is b, and members of a exist". To interpret properly "Some a is b", we need only remember that it is the contradictory of "No a is b". Since "No a is 6" is interpreted by a b = 0, "Some a is b" will be a b =(= 0, that is, "The class of things which are members of a and of b both is not-null". It is surprising what blunders have been committed in the representation of particular propositions. "Some x is y" has been symbolized by x y = v, where v is undetermined, and by u x = v y, where u and v are undetermined. Both of these are incorrect, and for the same reason : An " undetermined " element may have the value or the value 1 or any other value. Conse- quently, both these equations assert precisely nothing at all. They are both of them true a priori, true of every x and y and in all cases. For them to be significant, u and v must not admit the value 0. But in that case they are equivalent to x y =|= 0, which is much simpler and obeys well- defined laws which are consonant with its meaning. Since we are to symbolize "All a is b" by a -b = 0, it is clear that its contradictory, "Some a is not b", will be a -b =(= 0. To sum up, then: the four typical propositions will be symbolized as follows : A. All a is b, a-b = 0. E. No a is b, a b = 0. I. Some a is b, a b 4= 0. O. Some a is not b, a-b ^ 0. Each of these four has various equivalents : 12 12 See Chap, u, 4-9 and 8-14. Applications of the Boole-Schroder Algebra 189 A. a -b = 0, a a b, -a+b = 1, -a + -b = -a, a c b, and -b c -a are all equivalent. E. a b = 0, a = a -b, -a + -b = 1, -a + b = -a, ac-b, and be -a are all equivalent. I. a b + 0, a 4= fl ~&> -a + -b =1= 1, and -a + b =t= -ct are all equivalent. O. a -6 4= 0, a =1= a b, -a + b =f= 1, and -a + -b =f= -a are all equivalent. The reader will easily translate these equivalent forms for himself. With these symbolic representations of A, E, I and O, let us investigate the relation of propositions traditionally referred to under the topics, "The Square of Opposition", and "Immediate Inference". That the traditional relation of the two pairs of contradictories holds, is at once obvious. If a -b = is true, then a -b =1= is false; if a -b = is false, then a -b =j= is true. Similarly for the pair, a b = and a b 4= 0. The relation of contraries is defined: Two propositions such that both may be false but both cannot be true are "contraries". This relation is traditionally asserted to hold between A and E. It does not hold in ex- tension: it fails to hold in the algebra precisely whenever the subject of the two propositions is a null-class. If a = 0, then a -b = and a b = O. 13 That is to say, if no members of a exist, then from the point of view of extension, "All a is b" and "No a is b" are both true. But if it be assumed or stated that the class a has members (a =}= 0), then the relation holds. a = a (b + -6) = a b + a -b. Hence if a =f= 0, then a b + a -b =1= 0. [8 17] If a b + a -6 4= and a -b = 0, then a b + 0. (1) And if a b + a -b 4= and a b - 0, then a -b 4= 0. (2) We may read the last two lines : (1) If there are members of the class a and all a is b, then "No a is b" is false. (2) If there are members of the class a and no a is b, then "All a is b" is false. By tradition, the particular affirmative should follow from the universal affirmative, the particular negative from the universal negative. As has been pointed put, this relation fails to hold when a = 0. But it holds when- ever a 4= 0. We can read a b 4= 0, in (1) above, as "Some a is b" instead of "'No a is 6' is false", and a -b 4= 0, in (2), as "Some a is not 6" instead of " ' All a is b ' is false ". We then have : 13 See Chap, n, 1-5. 190 A Survey of Symbolic Logic (1) If there are members of a, and all a is b, then some a is 6. (2) If there are members of a, and no a is 6, then some a is not b. " Subcontraries " are propositions such that both cannot be false but both may be true. Traditionally "Some a is 6" and "Some a is not b" are subcontraries. But whenever a = 0, a b =f= and a -b =j= are both false, and the relation fails to hold. When a =f= 0, it holds. Since a b = is "'Some a is b' is false ", and a -b = is "'Some a is not b' is false", we can read (1) and (2) above: (1) If there are members of a, and "Some a is b" is false, then some a is not b. (2) If there are members of a and "Some a is not 6" is false, then some a is 6. To sum up, then: the traditional relations of the " square of opposition " hold in the algebra whenever the subject of the four propositions denotes a class which has members. When the subject denotes a null-class, only the relation of the contradictories holds. The two universal propositions are, in that case, both true, and the two particular propositions both false. The subject of immediate inference is not so well crystallized by tradi- tion, and for the good reason that it runs against this very difficulty of the class without members. For instance, the following principles would all be accepted by some logicians: "No a is b" gives "No 6 is a". "No b is a" gives "All b is not-a". "All b is not-a" gives "Some b is not-a". "Some b is not-a" gives "Some not-a is b". Hence "No a is 6" gives "Some not-a is b". "No cows (a) are inflexed gasteropods (6) " implies "Some non-cows are inflexed gasteropods": "No mathematician (a) has squared the circle (6)" implies "Some non-mathematicians have squared the circle". These infer- ences are invalid precisely because the class b inflexed gasteropods, suc- cessful circle-squarers is an empty class; and because it was presumed that "All b is not-a" gives "Some b is not-a". Those who consider the algebraic treatment of null-classes to be arbitrary will do well to consider the logical situation just outlined with some care. The inference of any particular proposition from the corresponding universal requires the assumption that either the class denoted -by the subject of the particular proposition or the class denoted by its predicate ("not-6" regarded as the predicate of "Some a is not b") is a class which has members. Applications of the Boole-Schroder Algebra 191 The "conversion " of the universal negative and of the particular affirma- tive is validated by the law a b = b a. " No a is b ", a b = 0, gives b a = 0, "No b is a". And "Some a is b", a b + 0, gives b a 4= 0, "Some b is a". Also, "Some a is not b", a -b =t= 0, gives -b a =j= 0, "Some not-6 is a". The "converse" of the universal affirmative is simply the "converse" of the corresponding particular, the inference of which from the universal has already been discussed. What are called "obverses" i. e., two equivalent propositions with the same subject and such that the predicate of one is the negative of the predicate of the other are merely alternative readings of the same equation, or depend upon the law, -(-a) = a 14 . Since xy = is " No x is y ", a -b = 0, which is "All a is b", is also "No a is not-6". And since a b = is equiva- lent to a -(-b) = 0, "No a is b" is equivalent to "All a is not-6". A convenient diagram for immediate inferences can be made by putting S (subject) and P (predicate) in the center of the circles assigned to them, -S between the two divisions of -S, and -P between its two constituent divisions. The eight arrows indicate the various ways in which the dia- Gtiyen Prop, Converse FIG. 13 gram may be read, and thus suggest all the immediate inferences which are valid. For example, the arrow marked "converse" indicates the two terms which will appear in the converse of the given proposition and the order in which they occur. In this diagram, we must specify the null and "See Chap, n, 2-8. 192 A Survey of Symbolic Logic not-null regions indicated by the given proposition. And we may if we wish add the qualification that the classes, S and P, have members. If "No S is P", and S and P have members: SP = 0, 5. -S-P FIG. 14 Reading the diagram of figure 14 in the various possible ways, we have: 1. No S is P, and 1. Some S is not P. (According as we read what is indicated by the fact that S P is null, or what is indicated by the fact that S -P is not-null.) 2. All S is not-P, and 2. Some S is not-P. 3. All P is not-S, and 3. Some P is not-S. 4. No P is S, and 4. Some P is not-S. 5. Wanting. 6. Some not-S is P. 7. Some not-P is S. 8. Wanting. Applications of the Boole-Schroder Algebra 193 Similarly, if "All S is P", and S and P have members: S-P=.Q, S*Q, P + S. 3. A .6* FIG. 15 Reading from the diagram (figure 15), we have: 1. All S is P, and 1. Some S is P. 2. No S is not-P. 3. Wanting. 4. Some P is S. 5. Some not-S is not-P. 6. Wanting. 7. No not-P is S. 8. All not-P is not-S, and 8. Some not-P is not-S. The whole subject of immediate inference is so simple as to be almost trivial. Yet in the clearing of certain difficulties concerning null-classes the algebra has done a real service here. The algebraic processes which give the results of syllogistic reasoning have already been illustrated. But in those examples we carried out the 14 194 A Survey of Symbolic Logic operations at unnecessary lengths in order to illustrate their connection with the diagrams. The premises of any syllogism give information which concerns, altogether, three classes. The object is to draw a conclusion which gives as much of this information as can be stated independently of the "middle" term. This is exactly the kind of result which elimination gives in the algebra. And elimination is very simple. The result of eliminating x from A x + B -x = is A B = O. 15 Whenever the conclusion of a syllogism is universal, it may be obtained by combining the premises in a single equation one member of which is 0, and eliminating the "middle" term. For example : No x is y, x y = 0. All z is x, z-x = 0. Combining these, x y + z -x = 0. Eliminating x, y z = 0. Hence the valid conclusion is "No y is z", or "No z is y". Any syllogism with a universal conclusion may also be symbolized so that the conclusion follows from the law, "If a cb and b cc, then ace". By this method, the laws, -(-a) = a and "If a cb, then -b c-a", are some- times required also. 16 For example: No x is y, x c -y. All z is x, z ex. Hence z c-y, or "No z is y", and y c-z, or "No y is z". There is no need to treat further examples of syllogisms with universal conclusions: they are all alike, as far as the algebra is concerned. Of course, there are other ways of representing the premises and of getting the con- clusion, but the above are the simplest. When a syllogism has a particular premise, and therefore a particular conclusion, the process is somewhat different. Here we have given one equation { = } and one inequation { =f= } . W T e proceed as follows : (1) expand the inequation by introducing the third element; (2) multiply the equation by the element not appearing in it; (3) make use of the prin- ciple, " If a + b 4= and a = 0, then 6 4= 0", to obtain an inequation with only one term in the literal member; (4) eliminate the element representing the "middle term" from this inequation. Take, for example, A 1 1 in 15 See Chap, n, 7-4. "See Chap, n, 2-8 and 3-1. Applications of the Boole-Schroder Algebra 195 the third figure: All x is z, x -z = 0. Some x is y, x y =f= 0. x y = x y (z + -z) = x y z + xy -z. Hence, x y z + xy -z 4= 0. [1 5] Since x -z = 0, x y -z = 0. [8 17] Since xy z + xy -z =f= and x y -x = 0, therefore x y z ={= 0. Hence [8-221 yz + 0, or "Some y is z". An exactly similar process gives the conclusion for every syllogism with a particular premise. We have omitted, so far, any consideration of syllogisms with both premises universal and a particular conclusion those with "weakened" conclusions, and A A I and E A in the third and fourth figures. These are all invalid as general forms of reasoning. They involve the difficulty which is now familiar: a universal does not give a particular without an added assumption that some class has members. If we add to the premises of such syllogisms the assumption that the class denoted by the middle term is a class with members, this makes the conclusion valid. Take, for example, A A I in the third figure : All x is y, x-y = 0, and x has members, x =t= 0. All x is z, x-z = 0. Since x 4= 0, x y + x -y 4= 0, and since x -y = 0, x y 4= 0. Hence x y z + x y -z 4= 0. (1) Since x -z = 0, x y -z = 0. (2) By (1) and (2), x y z + 0, and hence y z =(= 0, or "Some y is z". Syllogisms of this form are generally considered valid because of a tacit assumption that we are dealing with things which exist. In symbolic reasoning, or any other which is rigorous, any such assumption must be made explicit. An alternative treatment of the syllogism is due to Mrs. Ladd-Franklin. 17 If we take the two premises of any syllogism and the contradictory of its conclusion, we have what may be called an "inconsistent triad" three propositions such that if any two of them be true, the third must be false. For if the two premises be true, the conclusion must be true and its con- 17 See "On the Algebra of Logic", in Studies in Logic by members of Johns Hopkins University, ed. by Peirce; also articles listed in Bibl. We do not follow Mrs. Franklin's symbolism but give her theory in a modified form, due to Josiah Royce. 196 A Survey of Symbolic Logic tradictory false. And if the contradictory of the conclusion be true, i. e., if the conclusion be false, and either of the premises true, then the other premise must be false. As a consequence, every inconsistent triad corre- sponds to three valid syllogisms. Any two members of the triad give the contradictory of the third as a conclusion. For example: Inconsistent Triad 1. All x is y 2. Ally is z 3. Some x is not z. Valid Syllogisms 1. All x is y 1. All x is y 2. All y is 2 2. All y is z 3. Some x is not z 3. Some x is not 2 .'. All x is z. .'. Some y is not z. .'. Some x is not y. Omitting the cases in which two universal premises are supposed to give a particular conclusion, since these really have three premises and are not syllogisms, the inconsistent triad formed from any valid syllogism will consist of two universals and one particular. For two universals will give a universal conclusion, whose contradictory will be a particular; while if one premise be particular, the conclusion will be particular, and its contradictory w r ill be the second universal. Representing universals and particulars as we have done, this means that if we symbolize any incon- sistent triad, we shall have two equations { = 0} and one inequation { 4= 0}. And the two universals { = } must give the contradictory of the particular as a conclusion. This means that the contradictory of the particular must be expressible as the elimination resultant of an equation of the form ax + b-x = 0, because we have found all conclusions from two universals to be thus obtainable. Hence the two universals of any inconsistent triad will be of the form a x = and b -x respectively. The elimination resultant of a x + b -x = is a b = 0, whose contradictory will be a b 4= 0. Hence every inconsistent triad will have the form : ax = 0, b-x = 0, a 6 4= where a and b are any terms whatever positive or negative, and x is any positive term. The validity of any syllogism may be tested by expressing its proposi- tions in the form suggested, contradicting its conclusion by changing it from { = 0} to { = 0} or the reverse, and comparing the resulting triad Applications of the Boole-Schroder Algebra 197 with the above form. And the conclusion of any syllogism may be got by considering how the triad must be completed to have the required form. Thus, if the two premises are No x is y, x y = and All not-2 is y, -z -y = the conclusion must be universal. The particular required to complete the triad is x -z 4= 0. Hence the conclusion is x -z = 0, pr "All x is 2". (Incidentally it may be remarked that this valid syllogism is in no one of the Aristotelian moods.) Again, if the premises should be x y = and y z = 0, no conclusion is possible, because these two cannot belong to the same inconsistent triad. We can, then, frame a single canon for all strictly valid syllogistic reason- ing: The premises and the contradictory of the conclusion, expressed in symbolic form, { = } or { 4= } , must form a triad such that (1) There are two universals { = 0} and one particular { 4= 0}. (2) The two universals have a term in common, which is once positive and once negative. (3) The particular puts =(= the product of the coefficients of the com- mon term in the two universals. A few experiments with traditional syllogisms will make this matter clear to the reader. The validity of this canon depends solely upon the nature of the syllogism three terms, three propositions and upon the law of elimination resultants, "If a x + b -x = 0, then a b = 0". Reasoning which involves conditional propositions hypothetical argu- ments, dilemmas, etc. may be treated by the same process, if we first reduce them to syllogistic form. For example, we may translate "If A is B, then C is D" by "All x is y", where x is the class of cases in which A is B, and y the class of cases in which C is D i. e., "All cases in which A is B are cases in which C is D". And we may translate "But A is B" by " All z is x ", where 2 is the case or class of cases under discussion. Thus the hypothetical argument: "If A is B, C is D. But A is B. Therefore, C is D", is represented by the syllogism: "All cases in which A is B are cases in which C is Z). " But all the cases in question are cases in which A is B. "Hence all the cases in question are cases in which C is D." And all other arguments of this type are reducible to syllogisms in some similar fashion. Thus the symbolic treatment of the syllogism extends to 198 A Survey of Symbolic Logic them also. But conditional reasoning is more easily and simply treated by another interpretation of the algebra the interpretation for propositions. The chief value of the algebra, as an instrument of reasoning, lies in its liberating us from the limitation to syllogisms, hypothetical arguments, dilemmas, and the other modes of traditional logic. Many who object to the narrowness of formal logic still do not realize how arbitrary (from the logical point of view) its limitations are. The reasons for the syllogism, etc., are not logical but psychological. It may be worth while to exemplify this fact. We shall offer two illustrations designed to show, each in a different way, a wide range of logical possibilities undreamt of in formal logic. The first of these turns upon the properties of a triadic relation whose significance was first pointed out by Mr. A. B. Kempe. 18 It is characteristically human to think in terms of dyadic relations: we habitually break up a triadic relation into a pair of dyads. In fact, so ingrained is this disposition that some will be sure to object that a triadic relation is a pair of dyads. It would be exactly as logical to maintain that all dyadic relations are triads with a null member. Either statement is correct enough : the difference is simply one of point of view psychological preference. If there should be inhabitants of Mars whose logical sense coincided with our own, so that any conclusion which seemed valid to us would seem valid to them, and vice versa, but whose psychology otherwise differed from ours, these Martians might have an equally fundamental prejudice in favor of triadic relations. We can point out one such which they might regard as the elementary relation of logic as we regard equality or inclusion. In terms of this triadic relation, all their reasoning might be carried out with complete success. Let us symbolize by (ac/b), a -b c + -a b -c = 0. This relation may be diagrammed as in figure 16, since a -b c + -a b -c = is equivalent to a c c b c (a + c) . (Note that (ac/b) and (ca/b) are equivalent, since a -be f -a b -c is symmetrical with respect to a and c.) This relation (ac/b) represents precisely the information which we habitually discard in drawing a syllogistic conclusion from two universal premises. If all a is b and all b is c, we have a -b = and b -c = Hence a -b (c + -c) + (a + -a) b -c = 0, 18 See his paper "On the Relation of the Logical Theory of Classes and the Geometrical Theory of Points,", Proc. London Math. Soc., xxi, 147-82. But the use we here make of this relation is due to Josiah Royce. For a further discussion of Kempe's triadic relation, see below, Chap, vi, Sect. iv. Applications of the Boole-Schroder Algebra 199 Or, a -b c + a -b -c + a b -c + -a b -c = 0. [5 72] This equation is equivalent to the pair, (1) a-b-c + ab -c = a -c (b + -b) = a -c = 0, and (2) a -b c + -a b -c = 0. (1) is the syllogistic conclusion, "All a is c"; (2) is (ac/6). Perhaps most of us would feel that a syllogistic conclusion states all the information given by the premises: the Martians might equally well feel that precisely FIG. 16 what we overlook is the only thing worth mentioning. And yet with this curious "illogical" prejudice, they would still be capable of understanding and of getting for themselves any conclusion which a syllogism or a hypo- thetical argument can give, and many others which are only very awkwardly stateable in terms of our formal logic. Our relation, a cb, or "All a is b", would be, in their terms, (06/a). (Ofe/a) is equivalent to l-a-b + 0--ab = = a -b Hence the syllogism in Barbara would be " (Ob/a) and (Oc/6), hence (Oc/a) ". This would, in fact, be only a special case of a more general principle which is one of those we may suppose the Martians would ordinarily rely upon for inference: "If (xb/a) and (xc/b), then (xc/a)". That this general principle holds, is proved as follows: (xbja} is "X a -b + x -a b = (xc/b) is -x b -c + x -b c = These two together give: -x a-b (c + -c) + x -a b (c + -c) + -x b -c (a + -a) + x -b c (a + -a) = 0, or, -x a -b c + -x a -b -c + x -a b c + x -a b -c + -x a b -c + -x -a b -c + xa-b c + x-a-bc = 0. 200 A Survey of Symbolic Logic [5-72] This equation is equivalent to the pair, (1) ~x a b -c + -x a -b -c + x -a b c + x -a -b c = -x a-c (b + -b) + x -a c (b + -6) = -x a -c + x -a c = 0. (2) x -a b -c + -x -a b -c + x a -b c + -x a -b c = -a b -c (x + -x) + a -b c (x + -x) = -a b -c + a -b c =0. (1) is (xc/a), of which our syllogistic conclusion is a special case; (2) is a similar valid conclusion, though one which we never draw and have no language to express. Thus these Martians could deal with and understand our formal logic by treating our dyads as triads with one member null. In somewhat similar fashion, hypothetical propositions, the relation of equality, syllo- gisms with a particular premise, dilemmas, etc., are all capable of state- ment in terms of the relation (acjb). As a fact, this relation is much more powerful than any dyad for purposes of reasoning. Anyone who will trouble to study its properties will be convinced that the only sound reason for not using it, instead of our dyads, is the psychological difficulty of keeping in mind at once two triads with two members in common but differently placed, and a third member which is different in the two. Our attention-span is too small. But the operations of the algebra are inde- pendent of such purely psychological limitations that is to say, a process too complicated for us in any other form becomes sufficiently simple to be clear in the algebra. The algebra has a generality and scope which " formal " logic cannot attain. This illustration has indicated the possibility of entirely valid non- traditional modes of reasoning. We shall now exemplify the fact that by modes wilich are not so remote from familiar processes of reasoning, any number of non-traditional conclusions can be drawn. For this purpose, we make use of Poretsky's Law of Forms: 19 x = is equivalent to t = t -x + -t x This law is evident enough: if x 0, then for any t, t-x = M = t, and -tx = -t-0 = 0, while t+ = t. Let us now take the syllogistic premises, "All a is 6" and "All b is c", and see what sort of results can be derived from them by this law. All a is b, a-b = 0. All b is c, b-c = 0. 19 See Chap, n, 7-15 and 7-16. Applications of the Boole-Schroder Algebra 201 Combining these, a-b + b -c = 0. And [3-4-41] -(a-b + b-c} = -(a -&)-(& -c) = (-a + 6) (-6 + c) = -a -6 + -a c + b c. Let us make substitutions, in terms of a, b, and c, for the t of this formula. a + b = (a + 6) (-a -6 + -a c + b c) + -a -6 (a-b + b -c) = abc + -abc + bc = 6c What is either a or 6 is identical with that which is both 6 and c. This is a non-syllogistic conclusion from "All a is b and all b is c". Other such conclusions may be got by similar substitutions in the formula. a + c = (a + c) (-a -6 + -a c + b c) + -a -c (a-b + b -c) = a 6 c + -a -& c + -a c + b c + -a b -c = a b c + -a (b + c) . What is either a or c is identical with that which is a, b, and c, all three, or is not a and either b or c. -6 c = -b c (-a -b + -a c + b c) + (b + -c) (a-b + b -c) = -a -b c + b -c + a -b -c = -a -b c + (a + 6) -c That which is b but not c is identical with what is c but neither a nor b or is either a or 6 but not c. The number of such conclusions to be got from the premises, "All a is b" and "All b is c", is limited only by the number of functions which can be formed with a, b, and c, and the limitation to sub- stitutions in terms of these is, of course, arbitrary. By this method, the number of conclusions which can be drawn from given premises is entirely unlimited. In concluding this discussion of the application of the algebra to the logic of classes, we may give a few examples in which problems more involved than those usually dealt with by formal logic are solved. The examples chosen are mostly taken from other sources, and some of them, like the first, are fairly historic. Example I. 20 A certain club has -the following rules: (a) The financial committee shall be chosen from among the general committee; (6) No one shall be a member both of the general and library committees unless he be also on the financial committee; (c) No member of the library committee shall be on the financial committee. Simplify the rules. 10 See Venn, Symbolic Logic, ed. 2, p. 331. 202 A Survey of Symbolic Logic Let / = member of financial committee. g= " " general 1= " " library " . The premises then become: (a) fcg, or f-g = 0. , or -l = 0. (c) / / = 0. We can discover by diagramming whether there is redundancy here. In figure 17, (a) is indicated by vertical lines, (6) by horizontal, (c) by oblique. (a) and (c) both predicate the non-existence of / -g I. To simplify the rules, unite (a), (6), and (c) in a single equation: Hence, / -g + -f g I +/ / (g + -g) = f -g + -f g I +f g I +f -g I [5-91] = f-g+(-f+f)gl=f-g + gl = Q. And [5-72] this is equivalent to the pair, f-g = and g I = 0. Thus the simplified rules will be : (') The financial committee shall be chosen from among the general committee. (6') No member of the general committee shall be on the library com- mittee. Applications of the Boole-Schroder Algebra 203 Example 2. 21 The members of a certain collection are classified in three ways as a's or not, as 6's or not, and as c's or not. It is then found that the class b is made up precisely of the a's which are not c's and the c's which are not a's. How is the class c constituted? Given : b = a -c + -a c. To solve for c. 22 b = b (c + -c) = bc + b-c. Hence, bc + b-c = a -c + -a c. Hence [7-27] a -6 + -a 6 c c c a -b + -a b. Or [2-2] c = a -6 + -a 6. The c's comprise the a's which are not 6's and the 6's which are not a's. Another solution of this problem would be given by reducing 6 = a -c + -ac to the form { = 0} and using the diagram. [7 1] 6 = a -c + -a c is equivalent to 6 -(a -c + -a c) + -6 (a -c + -a c) =0 And [6 4] -(a -c + -a c) = a c + -a -c. Hence, a 6 c + -a 6 -c + a -6 -c + -a -6 c = 0. We observe here (figure 18) not only that c = a -6 + -a 6, but that the FIG. 18 relation of a, 6, and c, stated by the premise is totally symmetrical, so that we have also a = 6 -c + -6 c. 21 Adapted from one of Venn's, first printed in an article on "Boole's System of Logic", Mind, i (1876), p. 487. 22 This proof will be intelligible if the reader understands the solution formula referred to. 204 A Survey of Symbolic Logic Example 3. M If x that is not a is the same as 6, and a that is not x is the same as c, what is x in terms of a, b, and c? Given: 6 = -ax and c = a -.r. To solve for x. [7-1] b = -ax is equivalent to -(-a x) b + -a -b x = = (a + -a:) b + -a -b x = ab + b-x + -a-bx = Q (1) And c = a -x is equivalent to -(a -x)c + a -c -x = = (-a + x) c + a -c -x Combining (1) and (2), a b + -a c + (-a -b + c) x + (b + a -c) -x = Hence [5-72] (-a -b + c) x + (b + a -c) -x = [7-221] This gives the equation of condition, (-a -b + c)(b + a-c) = b c = [7-2] The solution of (4) is (b + a-c) ex c-(-a -b + c) And by (5), -(-a -b + c) = -(-a -b + c) -c-x = Q (2) (3) (4) (5) Hence [2-2] x = b + a -c. = (a + 6) -c + 6 c = a -c + 6 (c + -c) = b + a -c FIG. 19 u See Lambert, Logische Abhandlungen, i, 14. Applications of the Boole-Schroder Algebra 205 This solution is verified by the diagram (figure 19) of equation (3), which combines all the data. Lambert gives the solution as x = (a + 6) -c This also is verified by the diagram. Example 4. 24 What is the precise point at issue between two disputants, one of whom, A, asserts that space should be defined as three-way spread having points as elements, while the other, B, insists that space should be defined as three-way spread, and admits that space has points as elements. Let s = space, t = three-way spread, p = having points as elements. A asserts : s = t p. B states : s = t and s c p. s = t p is equivalent to s--(tp)+-stp = Q = s-t + s-p + -stp = Q (1) s c p is equivalent to s -p = (2) And s = t is equivalent to s -t + -s t = (3) (2) and (3) together are equivalent to s -t + s -p + -s t = (4) (1) represents ^4's assertion, and (4) represents 5's. The difference between FIG. 20 the two is that between -s t p = and -st = 0. (See figure 20.) -st = -s tp + -st-p 24 Quoted from Jevons by Mrs. Ladd-Franklin, loc. tit., p. 52. 206 A Survey of Symbolic Logic The difference is, then, that B asserts -st-p = 0, while A does not. It would be easy to misinterpret this issue, -s t-p = is t-p cs, "Three- way spread not having points as elements, is space ". But B cannot sig- nificantly assert this, for he has denied the existence of any space not having points as elements. Both assert s = tp. The real difference is this: B definitely asserts that all three-way spread has points as elements and is space, w r hile A has left open the possibility that there should be three-way spread not having points as elements which should not be space. *> Example 5. Amongst the objects in a small boy's pocket are some bits of metal which he regards as useful. But all the bits of metal which are not heavy enough to sink a fishline are bent. And he considers no bent object useful unless it is either heavy enough to sink a fishline or is not metal. And the only objects heavy enough to sink a fishline, which he regards as useful, are bits of metal that are bent. Specifically what has he in his pocket which he regards as useful? Let x = bits of metal, y = objects he regards as useful, z = things heavy enough to sink a fishline, w = bent objects. Symbolizing the propositions in the order stated, we have xy*0 x-zc.w, or x -z -w = y w c (z + -x), or xy-zw = zy c.x w, or -x y z + y z -w = Expanding the inequation with reference to z and w, xyzw + xyz-w + xy-zw + xy-z-w 4= Combining the equations, x -z -w (y + -y~) + x y -z w + -x y z (w + w} + y z-w (x + -x) = or xy -z-w + x-y -z-w + x y -zw + -x y zw + -x y z-w + x y z-w = All the terms of the inequation appear also in this equation, with the exception of x y z w. Hence, by 8-17, x y z w 4= 0. The small boy has Applications of the Boole-Schroder Algebra 207 some bent bits of metal heavy enough to sink a fishline, which he considers useful. This appears in the diagram (figure 21) by the fact that while FIG. 21 some subdivision of x y must be not-null, all of these but x y z w is null. It appears also that anything else he may have which he considers useful may or may not be bent but is not metal. Example 6. 25 The annelida consist of all invertebrate animals having red blood in a double system of circulating vessels. And all annelida are soft-bodied, and either naked or enclosed in a tube. Suppose we wish to obtain the relation in which soft-bodied animals enclosed in tubes are placed (by virtue of the premises) with respect to the possession of red blood, of an external covering, and of a vertebral column. Let a = annelida, s = soft-bodied animals, n = naked, t = enclosed in a tube, i = invertebrate, r = having red blood, etc. Given: a = ir and acs (n + t), with the implied condition, n t = 0. To eliminate a and find an expression for s t. 25 See Boole, Laws of Thought, pp. 144-46. 208 A Survey of Symbolic Logic a = iris equivalent to -(i r) a + -a i r = a -i + a -r -a i r = (1) a cs (n + 2) is equivalent to a--(s n + s t) =0. -(* n + s f) = -(s n) --(s i) = (-s + -n)(-s + -t) = -s + -n -t. Hence, a -s + a -n -t = (2) Combining (1) and (2) and n t = 0, a ~i + a -r + -a i r + a -s + a -n ~t + n t = (3) Eliminating a, by 7*4, (-i + -r + -s + -n -t + n t) (i r + n t) = nt + ir -s + ir -n-t = The solution of this equation for s is 26 ires. And its solution for t is i r -n c t c -n. Hence [5-3] ir -nc.sic.-n, t>r st = ir-n + u--n, where u is un- determined. The soft-bodied animals enclosed in a tube consist of the invertebrates -i 26 See Chap, n, Sect, iv, "Symmetrical and Unsymmetrical Constituents of an Equa- tion ". Applications of the Boole-Schroder Algebra 209 which have red blood in a double system of circulating vessels and a body covering, together with an undetermined additional class (which may be null) of other animals which have a body covering. This solution may be verified by the diagram of equation (3) (figure 22). In this diagram, s t is the square formed by the two crossed rectangles. The lower half of this inner square exhibits the solution. Note that the qualification, -n, in ir -nest, is necessary. In the top row is a single undeleted area repre- senting a portion of i r (n) which is not contained in s t. Example 7. 27 Demonstrate that from the premises "All a is either b or c", and "All c is a", no conclusion can be drawn which involves only two of the classes, a, b, and c. Given : a c (b + c) and c c a. To prove that the elimination of any one element gives a result which is either indeterminate or contained in one or other of the premises. a c (b + c) is equivalent to a-b -c = 0. And c c a is equivalent to -a c = 0. Combining these, a -b -c + -a c = 0. Eliminating a [7-4], (-b -c) c = 0, which is the identity, = 0. Eliminating c, (a -b) -a = 0, or = 0. Eliminating b, (-a c + a -c) -a c = -ac = 0, which is the second premise. Example 8. A set of balls are all of them spotted with one or more of the colors, red, green, and blue, and are numbered. And all the balls spotted with red are also spotted with blue. All the odd-numbered blue balls, and all the even numbered balls which are not both red and green, are on the table. De- scribe the balls not on the table. Let e = even-numbered, -e = odd-numbered, r = spotted with red, b = spotted with blue, g = spotted with green, t = balls on the table. Given: (1) -r-b-g = 0. 27 See De Morgan, Formal Logic, p. 123. 15 210 A Survey of Symbolic Logic (2) r-6 = 0. (3) [-eb + e -(r g)] ct, or (-e b + e -r + e -g) -t = 0. To find an expression, x, such that -I c x, or -t x = -t. Such an expression should be as brief as possible. Consequently we must develop -t with respect to e, r, b, and g, and eliminate all null terms. (An alternative method would be to solve for -t, but the procedure suggested is briefer.) -t = -t(e + -e) (r + -r} (b + -b} (g + -g) = -t(erbg+erb-g+er-bg + e-rbg + -erbg + er-b-g + e -r b -g + -e r b -g + e -r -b g + -e -r b g + -e r -b g + e-r -b -g + -e r -b -g + -e-rb-g + -e -r -b g + -e -r -b -g) (4) From (1), (2), and (3), -t (-eb + e-r + e-g + r-b + -r -b -g) = (5) Eliminating from (4) terms involved in (5), -t -t (e rb g + -e-r -b g), or -tc(erbg + -e-r -b g) All the balls not on the table are even-numbered and spotted with all three colors or odd-numbered and spotted with green only. Applications of the Boole-Schroder Algebra 211 In the diagram (figure 23), equation (1) is indicated by vertical lines, (2) by oblique, (3) by horizontal. Example 9. 28 Suppose that an analysis of the properties of a particular class of sub- stances has led to the following general conclusions: 1st. That wherever the properties a and b are combined, either the property c, or the property d, is present also; but they are not jointly present. 2d. That wherever the properties b and c are combined, the properties a and d are either both present with them, or both absent. 3d. That wherever the properties a and b are both absent, the proper- ties -c and d are both absent also; and vice versa, where the properties c and d are both absent, a and b are both absent also. Let it then be required from the above to determine what may be con- cluded in any particular instance from the presence of the property a with respect to the presence or absence of the properties b and c, paying no regard to the property d. Given: (1) a b c (c -d + -c d). (2) be c (ad + -a-d). (3) -a -b = -c -d. To eliminate d and solve for a. (1) is equivalent to a b--(c -d + -c d) = 0. (2) is equivalent to b c--(a d + -a -d) = 0. But [6 4] -(c -d + -c d) = c d + -c -d, and -(a d + -a -d) = -a d+ a -d. Hence we have, a b (c d + -c -d} =abcd+ab-c-d = 0. (4) and b c (-a d+ a -d) = -abcd+abc-d = Q (5) (3) is equivalent to -a -b (c + d) + (a + 6) -c -d = -a -b c + -a -6 d + a -c -d + b -c -d = (6) Combining (4), (5), and (6), and giving the result the form of a function of d, (-a -b c + -a -b + a b c + -a b c) d + (-a -b c + a -c + b -c + a b -c + a b c) -d = 28 See Boole, Laws of Thought, pp. 118-20. For further problems, see Mrs. Ladd- Franklin, loc. tit., pp. 51-61, Venn, Symbolic Logic, Chap, xm, and Schroder, Algebra der Logik: Vol. i, Dreizehnte Vorlesung. 212 A Survey of Symbolic Logic Or, simplifying, by 5-4 and 5-91, (-a -b + b c) d + (-a -b c + a-c + b-c + ab c) -d = Hence [7 4] eliminating d, (- a -l + 5 c ) (-a -b c + a-c + b-c + ab c) = -a-b c + ab c = Solving this equation for a [7-2], -be c a c (-6 + -c). The property a is always present when c is present and b absent, and when- ever a is present, either b is absent or c is absent. The diagram (figure 24) combines equations (4), (5), and (6). FIG. 24 As Boole correctly claimed, the most powerful application of this algebra is to problems of probability. But for this, additional laws which do not belong to the system are, of course, required. Hence we omit it. Some- thing of what the algebra will do toward the solution of such problems will be evident if the reader imagine our Example 8 as giving numerically the proportion of balls spotted with red, with blue, and with green, and the quaesitum to be "If a ball not on the table be chosen at random, what is the probability that it will be spotted with all three colors? that it will be spotted with green?" The algebra alone, without any additional laws, answers the last question. As the reader will observe from the solution, all the balls not on the table are spotted with green. Applications of the Boole-Schroder Algebra 213 III. THE APPLICATION TO PROPOSITIONS If, in our postulates, a, b, c, etc., represent propositions, and the "prod- uct", a b, represent the proposition which asserts a and b both, then we have another interpretation of the algebra. Since a+b is the negative of -a-b, a + b will represent "It is false that a and b are both false", or "At least one of the two, a and b, is true". It has been customary to read a + b, "Either a or b", or "Either a is true or 6 is true". But this is some- what misleading, since "Either ... or ..." frequently denotes, in ordinary use, a relation which is to be understood in intension, while this algebra is incapable of representing relations of intension. For instance, we should hardly affirm "Either parallels meet at finite intervals or all men are mortal". We might well say that the "Either . . . or . . ." relation here predicated fails to hold because the two propositions are irrelevant. But at least one of the two, "Parallels meet at finite intervals" and "All men are mortal", is a true proposition. The relation denoted by + in the algebra holds between them. Hence, if we render a + b by "Either a or 6", we must bear in mind that no necessary connection of a and b, no relation of "relevance" or "logical import", is intended. The negative of a, -a, will be its contradictory, or the proposition "a is false". It might be thought that -a should symbolize the "contrary" of a as well, that if a be "All men are mortal ", then "No men are mortal" should be -a. But if the contrary as well as the contradictory be denoted by -a, then -a will be an ambiguous function of a, whereas the algebra requires that -a be unique. 29 The interpretation of and 1 is most easily made clear by considering the connection between the interpretation of the algebra for propositions and its interpretation for classes. The propositional sign, a, may equally well be taken to represent the class of cases in which the proposition a is true, a b will then represent the class of cases in which a and 6 are both true; -a, the class of cases in which a is false, and so on. The "universe", 1, will be the class of all cases, or all "actual" cases, or the universe of facts. Thus a = 1 represents "The -cases in which a is true are all cases", or "a is true in point of fact", or simply "a is true". Similarly is the class of no cases, and a = will mean "a is true in no case", or "a is false". It might well be asked: May not a, b, c, etc., represent statements which are sometimes true and sometimes false, such as "Today is Monday" or "The die shows an ace"? May not a symbolize the cases in which a is 29 See Chap. 11, 3-3. 214 A Survey of Symbolic Logic true, and these be not all but only some of the cases? And should not a = 1 be read "a is always true", as distinguished from the less com- prehensive statement, "a is true"? The answer is that the interpretation thus suggested can be made and that Boole actually made it in his chapters on "Secondary Propositions". 30 But symbolic logicians have come to distinguish between assertions which are sometimes true and sometimes false and propositions. In the sense in which "Today is Monday" is sometimes true and sometimes false, it is called a propositional function and not a proposition. There are two principal objections to interpreting the Boole-Schroder Algebra as a logic of propositional functions. In the first place, the logic of propositional functions is much more complex than this algebra, and in the second place, it is much more useful to restrict the algebra to propositions by the additional law "If a =f= 0, then a = 1, and if a =f= lj then a = 0", and avoid any confusion of propositions with asser- tions which are sometimes true and sometimes false. In the next chapter, we shall investigate the consequences of this law, which holds for proposi- tions but not for classes or for propositional functions. We need not pre- sume this law at present: the Boole-Schroder Algebra, exactly as presented in the last chapter, is applicable throughout to propositions. But w r e shall remember that a proposition is either always true or never true : if a proposi- tion is true at all, it is always true. Hence in the interpretation of the algebra for propositions, a = 1 means "a is true" or "a is always true" indifferently the two are synonymous. And a = means either "a is false" or "a is always false". The relation a c 6, since it is equivalent to a -b = 0, may be read " It is false that 'a is true and b is false'", or loosely, "If a is true, then b is true ". But a c b, like a + b, is here a relation which does not signify "relevance" or a connection of "logical import". Suppose a = "2 + 2 = 4" and b = "Christmas is a holiday". We should hardly say "If 2 + 2 = 4, then Christmas is a holiday". Yet it is false that "2 + 2 = 4 and Christmas is not a holiday": in this example a -b = is true, and hence acb will hold. This relation, a c b, is called "material implication"; it is a relation of extension, whereas we most frequently interpret "implies" as a relation of intension. But a c b has one most important property in common with our usual meaning of "a implies b" when a cb is true, the case in which a is true but b is false does not occur. If a c b holds, and a is true, then b will not be false, though it may be irrelevant. Thus "material 30 Laws of Thought, Chaps, xi-xiv. Applications of the Boole-Schroder Algebra 215 implication" is a relation which covers more than the "implies" of ordinary logic: a cb holds whenever the usual "a implies b" holds; it also holds in some cases in which "a implies b" does not hold. 31 The application of the algebra to propositions is so simple, and so resembles its application to classes, that a comparatively few illustrations will suffice. We give some from the elementary logic of conditional propo- sitions, and conclude with one taken from Boole. Example 1. If A is B, C is D. (1) And A is B. (2) Let x = A is B; y = C is Z). The two premises then are : (1) xcy, or [4-9] -x + y = 1. (2) x = 1, or -x = 0. [5-7] Since -x + y = 1 and -x = 0, y = 1. y = I is the conclusion " C is D ". Example 2. (1) If A is B, C is Z). (2) But C is not D. Let x = A is B; y = C is Z). (1) x cy, or -x + y = 1. (2) y = Q. [5-7] Since -x + y = 1 and y = 0, -x = 1. -x = 1 is the conclusion "A is 5 is false", or "A is not 5". Example 3. (1) If ^ is B, C is Z); and (2) if EhF,G is H. (3) But either ^4 is B or C is D. Let w = ^4 is B; x = C is D; y = E is F; z = G is H. (1) wcx, or [4-9] wx = w. (2) i/cz, or yz = y. (3) W + T/ = 1. 31 "Material implication" is discussed more at length in Chap, iv, Sect, i, and Chap, v, Sect. v. 216 A Survey of Symbolic Logic Since iv + y = 1, and wx = w and yz = y,wx + yz = l. Hence [4-5] w x + -w x + y z + -y z 1 + -w x + -y z = 1 . Hence x (w + -w) + z (y + -y) = x + z = 1. x+z = 1 is the conclusion "Either C is D or G is H". This dilemma may be diagrammed if we put our equations in the equivalent forms (1) w -x = 0, (2) y -z = 0, (3) -w -y = 0. In figure 25, w -x is struck FIG. 25 out with horizontal lines, y -z with vertical, -w -y with oblique , That everything which remains is either x or z is evident. Example 4. (1) Either A is B or C is not D. (2) Either C is D or E is F. (3) Either A is B or E is not F. Let z = yl is ; y = C is D; z = E is F. (1) ar + -y = 1. (2) y + z = 1, or -y -z = 0. (3) z + -z = 1, or -xz = 0. By (1), x + -y(z + -z) = x + -y z + -y -z = 1. Hence by (2), x + -y z = 1 = x + -y z (x + -x} x + x -y z + -x -y z. And by (3), -x -y z = 0. Hence x + x-y z = x 1. Thus these three premises give the categorical conclusion "A is B", indi- cating the fact that the traditional modes of conditional syllogism are by no means exhaustive. Applications of the Boole- Schroder Algebra 217 Example 5. 32 Assume the premises: 1. If matter is a necessary being, either the property of gravitation is necessarily present, or it is necessarily absent. 2. If gravitation is necessarily absent, and the world is not subject to any presiding intelligence, motion does not exist. 3. If gravitation is necessarily present, a vacuum is necessary. 4. If a vacuum is necessary, matter is not a necessary being. 5. If matter is a necessary being, the world is not subject to a presiding intelligence. Let x = Matter is a necessary being. y = Gravitation is necessarily present. z = The world is not subject to a presiding intelligence. w = Motion exists. t = Gravitation is necessarily absent. v = A vacuum is necessary. ' The premises then are : (1) x c (y + t}, or x -y -t = 0. ^2) t z c -w, or t z w = 0. (3) y cv, or y -v = 0. (4) v c -x, or v x = 0. (5) x c z, or x -z = 0. And since gravitation cannot be both present and absent, (6) y t = 0. Combining these equations : x -y -t + t z iv + y -v + v x + x -z + y t = (7) From these premises, let it be required, first, to discover any conection between x, "Matter is a necessary being", and y, "Gravitation is necessarily present". For this purpose, it is sufficient to discover whether any one of the four, x y 0, x -y = 0, -x y = 0, or -x -y = 0, since these are the relations which state any implication which holds between x, or -x, and y, or -y. This can always be done by collecting the coefficients of x y, x -y, -x y, and -x -y, in the comprehensive expression of the data, such as equation (7), and finding which of them, if any, reduce to 1. But 32 See Boole, Laws of Thought, Chap. xiv. The premises assumed are supposed to be borrowed from Clarke's metaphysics. 218 A Survey of Symbolic Logic sometimes, as in the present case, this lengthy procedure is not necessary, because the inspection of the equation representing the data readily reveals such a relation. From. (7), [5-72] vx + -vy = 0. Hence [1-5] V x y + -v x y = (v + -v) xy = xy = Q, orxc -y, y c -x. If matter is a necessary being, then gravitation is not necessarily present; if gravitation is necessarily present, matter is not a necessary being. Next, let any connection between x and w be required. Here no such relation is easily to be discovered by inspection. Remembering that if a = 0, then a b = and a -b = ; From (7), (-y -t+tz + y-v + i) + -z + yt)wx + (tz + y -v + y t) w -x + (-y -t + y -v + v + -z + y f) -w x + (y-v + yf) -w -x = (8) Here the coefficient of w x reduces to 1, for [5-85], y -v + v = y + v, and t z + -z = t + -z and hence the coefficient is -y -t + y + t + v + -z + y t. But [5-96] (-y-t + y + t) + v + -z + yt = l+v + -z + yt = 1. Hence w x = 0, or w c -x, x c -w. -IV --y- --{ FIG. 26 -v -t Applications of the Boole-Schroder Algebra 219 None of the other coefficients in (8) reduces to 1. Hence the conclusion which connects x and w is: "If motion exists, matter is not a necessary being; if matter is a necessary being, motion does not exist". Further conclusions, relating other terms, might be derived from the same premises. All such conclusions are readily discoverable in the dia- gram of equation (7). In fact, the diagram is more convenient for such problems than the transformation of equations in the algebra. Another method for discovering the implications involved in given data is to state the data entirely in terms of the relation c , and, remembering that "If acb and bcc, then ace", as well as "acb is equivalent to -b c-a", to seek directly any connection thus revealed between the propo- sitions which are in question. Although by this method it is possible to overlook a connection which exists, the danger is relatively small. IV. THE APPLICATION TO RELATIONS The application of the algebra to relations is relatively unimportant, because the logic of relations is immensely more complex than the Boole- Schroder Algebra, and requires more extensive treatment in order to be of service. We shall, consequently, confine our discussion simply to the explanation of this interpretation of the algebra. A relation, taken in extension, is the class of all couples, triads, or tetrads, etc., which have the property of being so related. That is, the relation "father of" is the class of all those couples, (x;y], such that x is father of y: the dyadic relation R is the class of all couples (x; y) such that x has the relation R to y, x R y. The extension of a relation is the class of things which have the relation. We must distinguish between the class of couples (x; y) and the class of couples (y; x), since not all relations are symmetrical and x R y commonly differs from y R x. Since the properties of relations, so far as the laws of this algebra apply to them, are the same whether they are dyadic, triadic, or tetradic, etc., the discussion of dyadic relations w r ill be sufficient. The "product ", R x S, or R S, will represent the class of all those couples (x; y} such that x R y and x S y are both true. The "sum ", R + S, will be the class of all couples (x ; y} such that at least one of the two, x R y and x S y, holds. The negative of R, -R, will be the class of couples (x; y) for which x R y is false. The null-relation, 0, will be the null-class of couples. If the class of couples (t; u) for which t R u is true, is a class with no members, and the 220 A Survey of Symbolic Logic class of couples (v; w) for which v S w is true is also a class with no members, then R and S have the same extension. It is this extension which repre- sents. Thus R = signifies that there are no two things, i and u, such that i R u is true that nothing has the relation R to anything. Similarly, the universal-relation, 1, is the class of all couples (in the universe of dis- course). The inclusion, RcS, represents the assertion that every couple (x; y) for which x R y is true is also such that x S y is true; or, to put it otherwise, that the class of couples (x ; y) for which x R y is true is included in the class of couples (u; t>) for which u S v is true. Perhaps the most satisfactory reading of R cS is "The presence of the relation R implies the presence of the relation S". R = S, being equivalent to the pair, RcS and ScR, signifies that R and S have the same extension that the class of couples (x; y) for which x Ry is true is identically the class of couples (u; v) for which u S v is true. It is obvious that all the postulates, and hence all the propositions, of the Boole-Schroder Algebra hold for relations, so interpreted. 1-1 If R and x c\f/x~), is particularly important. In terms of it, the logical properties of relations including the properties treated in the last chapter but going beyond them can be established. This is the type of procedure used by Peirce and Schroder. The second method that of Principia Mathematica begins with the calculus of propositions, or calculus of material implication, in a form which is simpler and otherwise superior to the Two-Valued Algebra, then pro- ceeds from this to the calculus of prepositional functions and formal impli- cation, and upon this last bases not only the treatment of relations but also the "calculus of classes". It is especially important for the comprehension of the whole subject of symbolic logic that the agreement in results and the difference of method, of these two procedures, should be understood. Too often they appear to the student simply unrelated. I. THE TWO-VALUED ALGEBRA 1 If the elements a, b, ... p, q, etc., represent propositions, and a x b or a b represent the joint assertion of a and b, then the assumptions of the 1 See Schroder, Algebra der Logik: n, especially Fiinfzehnte Vorlesung. An excellent summary is contained in Schroder's Abriss (ed. M tiller), Teil u. 222 Systems Based on Material Implication 223 Boole-Schroder Algebra will all be found to hold for propositions, as was explained in the last chapter. 2 As was there made clear, p = will repre- sent "p is false", and p = 1, "p is true". Since and 1 are unique, it follows that any two propositions, p and q, such that p = and q = 0, or such that p = 1 and q = 1, are also such that p q. p = q, in the algebra, represents a relation of extension or "truth value", not an equiva- lence of content or meaning. -p symbolizes the contradictory or denial of p. The meaning of p + q is readily determined from its definition, p + q = -(-p -q) p + q is the denial of "p is false and q is false", or it is the proposition "At least one of the two, p and q, is true", p + q may be read loosely, "Either p is true or q is true". The possibility that both p and q should be true is not excluded. p c q is equivalent to p q = p and to p -q = 0. p c q is the relation of material implication. We shall consider its properties with care later in the section. For the present, we may note simply that p c q means exactly "It is false that p is true and q false". It may be read "If p is true, q is true", or " p (materially) implies q". With the interpretations here given, all the postulates of the Boole- Schroder Algebra are true for propositions. Hence all the theorems will also be true for propositions. But there is an additional law which holds for propositions: P = (P = 1) "The proposition, p, is equivalent to 'p is true'". It follows immediately from this that -p = (-p = 1) = (p = 0) 11 -p is equivalent to 'p is false'". It also follows that -p = -(p = 1), and hence -(p = 1) = (p = 0), and -(p = 0) = (p = 1) '"p = 1 is false ' is equivalent to p = 0", and '"p = is false ' is equivalent to p = 1". Thus the calculus of propositions is a two-valued algebra: every proposition is either = or = 1, either true or false. We may, then, proceed as follows: All the propositions of the Boole-Schroder Algebra 2 However, many of the theorems, especially those concerning functions, eliminations, and solutions, are of little or no importance in the calculus of propositions. 224 A Survey of Symbolic Logic which were given in Chapter II may be regarded as already established in the Two-Valued Algebra. We may, then, simply add another division of propositions the additional postulate of the Two-Valued Algebra and the additional theorems which result from it. Since the last division of the- orems in Chapter II was numbered 8-, we shall number the theorems of this section 9 . The additional postulate is: 9-01 For every proposition p, p = (p = 1). And for convenience we add the convention of notation : 9-02 -(p = q) is equivalent to p 4= x is by x\, x 2 , Xz, etc. This is not to presume that the number of such values of x in x s , etc., will be propositions; and 2/3), t(xz, y n ), t(x m , y n ), etc., are propositions. We shall now make a new use of the operators II and S, giving them a meaning similar to, but not identical with, the meaning which they had in Chapter II. To emphasize this difference in use, the operators are here set in a different style of type. We shall let 2 x Xi + x s , etc., are propositions, x represents the proposition "For all values of x, x n , 2 x x and Tl x x may not be finite. And any use of mathematical induction, or of theorems dependent upon that principle for proof, will then be invalid in this con- nection. Short of abandoning the proposed procedure, two alternatives are open to us: we can assume that the number of values of any variable in a prepositional function is always finite; or we can assume that any law of the algebra which holds whatever finite number of elements be involved holds for any number of elements whatever. The first of these assumptions would obviously be false. But the second is true, and we shall make it. This also resolves our difficulty concerning the possibility that the number of values of x in x 3 + . . ., or 2 x Xi x Xi+ x s + .... Def. 10-02 Hx}. Def. 10-05 -2 x Xi+ 10-1 states that "For some values of x, xi x tpxz x (px s x . . . c tpx\ and (f>Xi x Xi c x 2 would not hold generally. For example, let xc2x n ". Hxi x x s x ... =1) [9-01] And [5-971] Xi = 1, Xi, x n + P". And hence we have, by 10-23, TL x [x c Xi + tpX 2 + x s + . . . ) x P = ( .T is true for some x' implies P" is equivalent to "For every x, x implies P". It is easy to see that the second of these two expressions gives the first also : If Xi x xi + P) + (- x be " If a: is a hair of Mr. Blank's, x has fallen out". And let P be "Mr. Blank is bald". Then n^x cP will represent "If all of Mr. Blank's hairs have fallen out, then Mr. Blank is bald". And S z ( x + fa) be significant, xx2fa = 2 x (x xfa) cS^.r, and ^ f ((px xfa) cZfa Hence [5 34] 2 Z ( x + fa) fails to hold. "Either for every x, x is ugly, or for every x, x is beautiful ", is not equivalent to, " For every x, either x is ugly or x is beautiful ". Some x's may be ugly and others beauti- ful. But we have: Systems Based on Material Implication 243 10-53 [5-21] (px n c ( x n + fa n ), and ^.r n c ( x c n x (<^x + #r), and II i/^ c II I (^a; + fa) Hence [5 33] II x n , instead of writing fai c x x-fa), so that H. x (x + fa) = H x [9 3] X n C fa n = - X Cfa) C (x cfa) x (px n ] cfa n . [9-4, 10-61] If x cfa) c.*2 x (vx cfa). [10-22] 10-63 U [10-61] If H x (x cfa)*? tpx] c 2 fa. [9-4, 10-64] If xcfa) *TL x (fa cfcr)] cU x (ifXE cx). [10-61] If U x (x n cx n . Hence [10-23] n x (^arcfa;) Systems Based on Material Implication 245 This theorem states that formal implication is a transitive relation. It is another form of the syllogism in Barbara. For example let x cfar)]. [9-4, 10-65] 10-652 U r (\f/x cfar) c[U x (x = fa) = [H x (x n c x = fa;). Whatever value of x, x n may be, if x = \l/x) c[U x (\lfX = fa;) cTL x (x n = fan); TL x (x = H^a:); and H x ((x, y), 2 v n x (x, y) represents some relation of x and y, it does not neces- sarily represent any relation of the algebra, such as x cy or x = y; and it cannot represent relations which are not assertable. (xi, y),.U y (x, y), 2 y (xiy) + tt y (#, y} should be "For some x, every y is such that (x, y) should be "For every y, some x is such that (x, y} means "Either for x\ and every y, (x, y) expresses the propo- sition: "There is a proper fraction which is greater than any proper frac- tion ", which is false. In this example, if we should read SJTy "For some x and every y"; 11 See i, p. 161. Systems Based on Material Implication 249 IlySi "For every y and some x", we should make equivalent these two very different propositions. But cases where this caution is required are infrequent, as we shall see. Where both operators are II or both S, the two-dimensional array of propositions can be turned into a one-dimensional array, since every rela- tion throughout w r ill be in the one case x , in the other + , and both of these are associative and commutative. It follows from our discussion of the range of significance of a function of two variables that any such func- tion, V )(x, y) = Ii x , y y (x, y) with 2 X , v (x, y), is of little consequence for the theory of prepositional functions itself, but it will be of some importance in the theory of relations which is to be derived from the theory of functions of two or more variables. Having now somewhat tediously cleared the ground, we may proceed to the proof of theorems. Since 2 X) y (x, y) n - [10-2] 11-21 v(x,y} n c*L x , y (x, y), x r may be, U y (x r y)". And H y (x, y n ) . [11-26-27] 11-291 U x n y (x 2 yz) x + { ^(ar 3 yi) x (x, y} and H y 2 x (x, y) = [11-04] 2*2^(3-, y) = And [5-992] ? y (x, y) c $(x, y}} or H x H v [(x, y) CTJ/(X, y}]. By 11-06, these two are equivalent. We shall give the theorems only in the first of these forms. 11-4 U x , y [(x, y) c$(x, y}] *U X , v (x, y) c S x , v t(x, y)}. [10-64] 11-441 {II*, v [(x, y} cf(x, y)]}. [10-651] 11-452 n x , v [i(x,y)ct(x,y)] c {U x , y [ v [$(x, y} = t(x, y)] c |n s , v [(x t y, z) 254 A Survey of Symbolic Logic x . . ., and !!(, *)(x n y, 2) = n y ll z (x n y, z), etc., we shall be able to deduce n (l , y, *)>(#, y, 2) = n x n (lM z) (x, y, 2) And similarly for 2 (x , v , z) . This calls our attention to the fact that x + Il\f/y, etc., under the head of functions of one variable. The reason for this omission was that such expressions find their significant equivalents in propositions of the type n x lly(x x^y}, 2 x n y (x + -tyy), etc., and these are special cases of functions of two variables. We may also remind the reader of the difference between two such expressions as II Xl + ^l) X ( x . . . } + . . . Etc., etc. And for any such expression with two operators we have the same type of Systems Based on Material Implication 255 two-dimensional array as for a function of two variables in general. The only difference is that here the function itself has a special form, (1) [1-3] nXs x . . . ) x Ti-^y = (xx2ty = S^yxS^ar. (2) S #c x 2^y = (x + U\j/y = (x 3 x . . . ) + x (x + H\l/y will be 'Either- every number is odd or every number is even', but H x ll y ((px + \l/y} will be 'Every number is either odd or even'". The mistake of this supposed illustration lies in misreading Il x ll y (x + \l/y~). It is legitimate to choose, as in this case, tpx and \l/y such that their range is identical: but it is not legitimate to read U x lly(x). And by proof similar to (3) in 12-4, H\f/y + 2<#r = H v 2 x (x x^/), etc. We give, in summary form, the derivatives of (2), by way of illustration: Any one of c any one of x Hij/y Ii(f>x x ?,\l/y x ^y) n x Sj,( <& x ^y) x x + $y), etc., etc. This table summarizes one hundred fifty-six theorems, and these are only a portion of those to be got by such procedures. Functions of the type of (x c\f/y)-, and (4) S x S,,(^a; c^y). With the exception of the first, these relations are unfamiliar as "implications", though all of them could be illustrated from the field of mathematics. Nor are they par- ticularly useful: the results to be obtained by their use can always be got by means of material implications or formal implications. Perhaps Uxn y ((px c \l/y] is of sufficient interest for us to give its elementary properties. 12-7 n z n y ( x + H\l/y. (1) If UMvxcty), then [12-7] xcU\l/y, then [10-42] U v (2 sections. We may then further derive the calculus of logical classes, and a calculus of relations, by methods which are to be outlined in this section and the next. The present section will not develop the logic of classes, but will present the method of this development, and prove the possibility and adequacy of it. At the same time, certain differences will be pointed out between the calculus of classes as derived from that of prepositional functions and the Boole-Schroder Algebra considered as a logic of classes. In order to distinguish class-symbols from the variables, x, y, z, in prepositional func- tions, we shall here represent classes by a, /3, 7, etc. For the derivation of the logic of classes from that of propositional functions, a given class is conceived as the aggregate of individuals for which some propositional function is true. If z". And z(z), z(\j/z), etc., and 1, will be significant whenever x c^ar). [13-02] 2(^2) cz(^z) = II x [ar e 2(>2) ex e 2(^2)]. [13-01] ar n e z(x implies \f/x". [13-03] [2(>2) = 2(^2)] = II x [ar e z(2j == a: c^x), (/3 c 7) = n x (^a: c far), and (a c 7) = n x (

ar c far). The relation "is contained in" is transitive. 13-9 is the first form of the syllogism in Barbara. The second form is: 13-91 [(a c 8) x (ar n e a)] c (ar e fi). [13-6] (aCjS) = n x (x = far). The last three theorems illustrate particularly well the direct connection 266 A Survey of Symbolic Logic between formal implications and the relations of classes. 13-6 and 13-7 are alternative definitions of a c /3 and a = /3. Similar alternative defini- tions of the other relations would be : 16 We may give one theorem* especially to exemplify the way in which every proposition of the Two- Valued Algebra, since it gives, by 10-23, a formal implication or equivalence, gives a corresponding proposition con- cerning classes. We choose for this example the Law of Absorption. 13-92 [a+(aX/3)] = a. [13-04-05] [a+(aXj8)] = x{(xea) + [(xta) x(.re/3)]}. Hence [13-01] [x n e [a+ (a x/3)]} = { (* e a) + [(z e a) x(ar n ej8)]}. (1) But [13-03] {[a+(ax|8)] = a} = IIxC{(a;e a) + [() x(a?e/S)]) = (zea)]. (2) But [13-01] (x n e a) = x x x x #r) x ^] = [^.r x (^ 16 A more satisfactory derivation of these existence postulates is possible when the theory of prepositional functions is treated in greater detail. See Prindpia, I, pp. 217-18. 268 A Survey of Symbolic Logic But [1-04] (x n cipXn) is equivalent to [((x, y) n and v(x r y,} becomes important. For this allows us to treat R(x, y), or (xRy), as a function of one or of two variables, at will; and by 11-07, we can give our definition the alternative form: 15 -01 (x m y n }tzw(zRw) = x m R y n . Def. "The couple (x m y n ) belongs to the field, or extension, of the relation deter- mined by (2 R w) " means that x m R y n is true. 15-02 RcS = Il x , y [(x R y) c (x S y)]. Def. This definition is strictly parallel to 13-02, (a c /3) = Tl x (x e a C x e /3) because, by 15-01, (x R y} is (x, y) e R and (x S y) is (x, y) c S. A similar remark applies to the remaining definitions. 15-03 (R = S) = H x , v [(x Ry) = (xS y}]. Def. R and S are equivalent in extension when, for every x and every y, (x R y) and (x S y) are equivalent assertions. 18 See above, pp. 253 ff. Systems Based on Material Implication 271 15-04 RxS = xy[(xRy) x(xSy)]. Def. The logical product of two relations, R and S, is the class of couples (x, y. such that x has the relation R to y and x has the relation S to y. If R is "friend of", and S is "colleague of", R x S will be "friend and colleague of") 15-05 R + S = xy[(xRy) + (xSy)]. Def. The logical sum of two relations, R and S, is the class of couples (x, y) such that either x has the relation R to y or x has the relation S to y. R + S will be " Either R of or S of". 15-06 -R = xy-(xRy}. Def. -.R is the relation of x to i/ when a: does not have the relation R to y. It is important to note that R x S, R + S, and -R are relations : x(R x S)y, x(R+S)y, and x-Ry are significant assertions. The "universal-relation" and the "null-relation" are also definable after the analogy to classes. 15-07 1 = xy[t;(x,y)ct(x,y)}. Def. x has the universal-relation to y in case there is a function, f, such that (x, y} c(x, y}, i. e., in case x and y have any relation. 15-08 = -1. Def. Of course, 0, 1, + and x have different meanings for relations from their meanings for classes or for propositions. But these different meanings o 0, + , etc., are strictly analogous. As was pointed out in Section III of this chapter, for every theorem involving functions of one variable, there is a similar theorem involving functions of two variables, due to the fact that a function R). Instead we have (R c S) = (v/fl c wS) for (fi c S) = n x , [(* Ry)c(xS y)] = Ii x , y [(y *R . T ) c (y wfi )] = (RcS) "'Parent of implies 'ancestor of" is equivalent to '"Child of implies 'descendent of". The converses of compound relations is as folio W T S: for x(RxS)y = y(R*S)x = (y R x} x (y S x} = (x^Ry) x(xSy) = x(^R x * If x is employer and exploiter of y, the relation of y to x is "employee of and exploited by". Similarly v(R + S) = vR + S If x is either employer or benefactor of y, the relation of y to x is "either employee of or benefitted by". Other important properties of relations concern "relative sums" and Systems Based on Material Implication 275 "relative products". These must be distinguished from the non-relative sum and product of relations, symbolized by * and x . The non-relative product of "friend of" and "colleague of" is "friend and colleague of": their relative product is "friend of a colleague of". Their non-relative sum is "either friend of or colleague of": their relative sum is "friend of every non-colleague of ". We shall denote the relative product of R and S by R | S, their relative sum by R t S. 17-02 R\8 = xz{2 y [(xRy) x(ySz)]}. Def. R\S is the relation of the couple (x, z) when for some y, x has the relation R to y and y has the relation S to z. x is friend of a colleague of z when, for some y, x is friend of y and y is colleague of z. 17-03 R-tS = xz{U y [(xRy) + (ySz)]}. Def. R t S is the relation of x to z when, for every y, either x has the relation R to y or y has the relation S to z. x is friend of all non-colleagues of z when, for every y, either x is friend of y or y is colleague of z. It is noteworthy that neither relative products nor relative sums are commutative. "Friend of a colleague of" is not "colleague of a friend of". Nor is "friend of all non-colleagues of" the same as "colleague of all non- friends of". But both relations are associative. R\(S\T) = (R\S}\T for ? x {(wRx} x[x(S T)z}} = 2 x {(w Rx) x? y [(x S y} x(y T z)]} = 2 y 2 x {(wRx)x((xSy)x(yTz)}} = S,S{[(waOx(a:S30]x(3fr 2 )} = 2 y {2 x [(wRx)x(xSz)]x(y T z}} = 2 y {(w(RS)y]x(yTz)} "Friend of a (colleague of a neighbor of)" is "(friend of a colleague) of a neighbor of". Similarly, R t (S t T) = (R t S) t T "Friend of all (non-colleagues of all non-neighbors of)" is "(friend of all non-colleagues) of all non-neighbors of". De Morgan's Theorem holds for the negation of relative sums and prod- ucts. for -M(x R y) x (y S z)] } = U y -[(x Ry)x(yS z)} 276 A Survey of Symbolic Logic The negative of "friend of a colleague of" is "non-friend of all colleagues (non-non-colleagues) of ". Similarly, -(R t S) = -R\-S The negative of "friend of all non-colleagues of" is "non-friend of a non- colleague of ". Converses of relative sums and products are as follows: for x *(R | S)z = z(R \ S)x = S v [(z Ry)x(yS x)] = Z y ((ySx)x(zRy)} If x is employer of a benefactor of z, then the relation of z to x is "bene- fitted by an employee of". Similarly, (R t S) = *S t ~R If x is hater of all non-helpers of z, the relation of z to x is " helped by all who are not hated by". The relation of relative product is distributive with reference to non- relative addition. R\(S+ T} = (RS) + (R T) for x[R \(S+T)}z = Z v {(xRy)x [y(S + 7>] } Similarly, (R + S) T = (R \ T) + (S \ T) "Either friend or colleague of a teacher of" is the same as "either friend of a teacher of or colleague of a teacher of". A somewhat curious formula is the following: It holds since x[R \ (S x T)]z = 2 y {(x Ry)x [y(S x 7>] } = Z y {(xRy)x[(ySz)x(yTz)}\ and since a x (b xc) = (a x6) x (a xc), c 2,[( X Ry)x(yS z)} x 2,[(x Ry}x(yT z}] And this last expression is [x(R\S)z] x[x(R\ T)z]. Systems Based on Material Implication 277 If x is student of a friend and colleague of z, then x is student of a friend and student of a colleague of z. The converse implication does not hold, be- cause "student of a friend and colleague" requires that the friend and the colleague be identical, while "student of a friend and student of a col- league" does not. (Note the last step in the 'proof ', where S v is repeated, and observe that this step carries exactly that significance.) Similarly, (R x S) \ T c (R \ T) x (S \ T) The corresponding formulae with t instead of | are more complicated and seldom useful; they are omitted. The relative sum is of no particular importance, but the relative product is a very useful concept. In terms of this idea, "powers" of a relation are definable : R 2 = R R, R 3 = R* R, etc. A transitive relation, S, is distinguished by the fact that *S 2 c S, and hence S n c S. The predecessors of predecessors of predecessors ... of x are predecessors of x. This conception of the powers of a relation plays a prominent part in the analysis of serial order, and of the fundamental proper- ties of the number series. By use ofthis and certain other concepts, the method of "mathematical induction" can be demonstrated to be com- pletely deductive. 20 In the work of De Morgan and Peirce, "relative terms" were not given separate treatment. The letters by which relations were symbolized were also interpreted as relative terms by a sort of systematic ambiguity. Any relation symbol also stood for the class of entities which have that relation to something. But in the logistic development of mathematics, since that time, notably in Principia Mathematical relative terms are given the separate treatment which they really require. The "domain" of a given relation, R that is, the class of entities which have the relation R to some- thing or other may be symbolized by D'R, which can be defined as follows: 17-04 D'R = x[-2 y (xRy)}. Def. The domain of R is the class of .r's determined by the function "For some y, x has the relation R to y". If R be "employer of", D'R will be the class of employers. The "converse domain" of R that is, the class of things to each of 20 See Principia, i, Bk. n, Sect. E. 21 See i, ^33. The notation we use for domains and converse domains is that of Prin- cipia. 278 A Survey of Symbolic Logic which something or other has the relation R may be symbolized by Q.'R and similarly defined: 17-05 (Tfl = $[2 x (xRy)}. Def. The converse domain of R is the class of y's determined by the function "For some x, x has the relation R to y". If R be "employer of", Q'R will be the class of employees. The domain and converse domain of a relation, R, together constitute the "field "of R, C'R. 17-06 C'R = {2 y [(xRy) + (yRx)]}. The field of R will be the class of all terms which stand in either place in the relation. If R be "employer of", C'R is the class of all those who are either employers or employees. The elementary properties of such "relative terms" are all obvious: x n e T>'R = ? y (x n R y} y n *a'R = 2 x (xRy n ~) x n e C'R = -2y[(x n Ry) + (yR x n }} C'R = D'R + d'R However, for the logistic development of mathematics, these properties are of the highest importance. We quote from Principia Mathematical 22 " Let us ... suppose that R is the sort of relation that generates a series, say the relation of less to greater among integers. Then D'R = all integers that are less than some other integer = all integers, Q'R = all integers that are greater than some other integer = all integers except 0. In this case, C'.R = all integers that are either greater or less than some other integer = all integers .... Thus when R generates a series, C'R becomes important. ..." We have now surveyed the most fundamental and important characters of the logic of relations, and we could not well proceed further without elaboration of a kind which is here inadmissible. But the reader is warned that we have no more than scratched the surface of this important topic. About 1890, Schroder could write "What a pity! To have a highly developed instrument and nothing to do with it". And he proceeded to make a beginning in the bettering of this situation by applying the logic of relatives to the logistic development of certain portions of Dedekind's theory of number. Since that time, the significance of symbolic logic has been completely demonstrated in the development of Peano's Formulaire 22 1, p. 261. Systems Based on Material Implication 279 and of Principia Mathematica. And the very head and front of this develop- ment is a theory of relations far more extended and complete than any previously given. We can here adapt the prophetic words which Leibniz puts into the mouth of Philalethes : " I begin to get a very different opinion of logic from that which I formerly had. I had regarded it as a scholar's diversion, but I now see that, in the way you understand it, it is a kind of universal mathematics." \ VI. THE LOGIC OF Principia Mathematica We have now presented the extensions of the Boole-Schroder Algebra the Two-Valued Algebra, propositional functions and the propositions derived from them, and the application to these of the laws of the Two- Valued Algebra, giving the calculus of propositional functions. Beyond this, we have shown in outline how it is possible, beginning with the Two- Valued Algebra as a calculus of propositions, to derive the logic of classes in a form somewhat more satisfactory than the Boole-Schroder Algebra, and the logic of relations and relative terms. In so doing, we have presented as much of that development which begins with Boole and passes through the work of Peirce to Schroder as is likely to be permanently significant. But, our purpose here being expository rather than historical, we have not followed the exact forms which that development took. Instead, we have considerably modified it in the light of what symbolic logicians have learned since the publication of the work of Peirce and Schroder. Those who are interested to note in detail our divergence from the historical development will be able to do so by reference to Sections VII and VIII of Chapter I. But it seems best here to point out briefly what these alterations are that we have made. In the first place, we have interpreted 2x as the symbol of a sum, makes demonstration possible where otherwise a large number of assumptions must be made and, for further principles, a much more difficult and less obvious style of proof resorted to. 280 A Survey of Symbolic Logic In this part of their work, Peirce and Schroder can hardly be said to have formulated the assumptions or given the proofs. In the second place, the Boole-Schroder Algebra the general outline of which is already present in Peirce 's work probably seemed to Peirce and Schroder an adequate calculus of classes (though there are indications in the paper of 1880 that Peirce felt its defects). With this system before them, they neglected the possibility of a better procedure, by beginning with the calculus of propositions and deriving the logic of classes from the laws which govern prepositional functions. And although the principles which they formulate for prepositional functions are as applicable to func- tions of one as of two variables, and are given for one as well as for two, their interest was almost entirely in functions of two and the calculus of relatives which may be derived from such functions. The logic of classes which we have outlined is, then, something which they laid the foundation for, but did not develop. The main purposes of our exposition thus far in the chapter have been two: first, to make clear the relation of this earlier treatment of symbolic logic with the later and better treatment to be discussed in this section; and second, to present the logic of prepositional functions and their deriva- tives in a form somewhat simpler and more easily intelligible than it might otherwise be. The theoretically sounder and more adequate logic of Prin- cipia Mathematica is given a form which so far as prepositional functions and their derivatives is concerned seems to us to obscure, by its notation, the obvious and helpful mathematical analogies, and requires a style of proof which is much less obvious. With regard to this second purpose, we disclaim any idea that the development we have given is theoretically adequate; its chief value should be that of an introductory study, prepara- tory to the more complex and difficult treatment which obviates the the- oretical shortcomings. Incidentally, the exposition which has been given will serve to indicate how much we are indebted, for the recent development of our subject, to the earlier work of Peirce and Schroder. The Peirce-Schroder symbolic logic is closely related to the logic of Peano's Formulaire de Mathematiques and of Principia Mathematica. This connection is easily overlooked by the student, with the result that the sub- ject of his first studies the Boole-Schroder Algebra and its applications is likely to seem quite unrelated to the topic which later interests him the logistic development of mathematics. Both the connections of these two Systems Based on Material Implication 281 and their differences are important. We shall attempt to point out both. And because, for one reason, clearness requires that we stick to a single illustration, our comparison will be between the content of preceding sections of this chapter and the mathematical logic of Book I, Prindpia Mathematical The Two- Valued Algebra is a calculus produced by adding to and re- interpreting an algebra intended primarily to deal with the relations of classes. And it has several defects which reflect this origin. In the first place, the same logical relation is expressed, in this system, in two different ways. We have, for example, the proposition "If p c q and q c r, then per", where p, q, and r are propositions. But "if . . . , then ..." is supposed to be the same relation which is expressed by c in p c q, q c r, and per. Also, "and" in "peg and q c r " is the relation which is other- wise expressed by x and so on, for the other logical relations. The system involves the use of "if . . . , then . . .", "... and . . .", "either . . . or . . .", ". . .is equivalent to . . .", and "... is not equivalent to . . .", just as any mathematical system may; yet these are exactly the relations c , x , + , = , and =1= whose properties are supposed to be investigated in the system. Thus the system takes the laws of the logical relations of propositions for granted in order to prove them. Nor is this paradox removed by the fact that we can demonstrate the interchange- ability of "if . . . , then ..." and c, of ". . . and ..." and x, etc. For the very demonstration of this interchangeability takes for granted the logic of propositions; and furthermore, in the system as developed, it is impossible in most cases to give a law the completely symbolic form until it has first been proved in the form which involves the non-symbolic expression of relations. So that there is no way in which the circularity in the demonstration of the laws of propositions can be removed in this system. Another defect of the Two-Valued Algebra is the redundance of forms. The proposition p or "p is true" is symbolized by p, by p = 1, by p =(= 0, 23 Logically, as well as historically, the method of Peano's Formulaire is a sort of intermediary between the Peirce-Schroder mode of procedure and Prindpia. The general method of analysis and much of the notation follows that of the Formulaire. But the Formulaire is somewhat less concerned with the extreme of logical rigor, and somewhat more concerned with the detail of the various branches of mathematics. Perhaps for this reason, it lacks that detailed examination and analysis of fundamentals which is the dis- tinguishing characteristic of Prindpia. For example, the Formulaire retains the ambiguity of the relation a (in our notation, c ): p Dr two dots, indicating a product is always inferior to a stop indicated by the same number of dots but not indicating a product. The postulates of the system in question are as follows: #1-1 Anything implied by a true elementary proposition is true. Pp. ("Pp." stands for "Primitive proposition".) *1 11 When r.D:pvg.D.pvr. Pp. In our notation, (qcr) c [(p + q) c(p + r)]. Note that the sign of assertion in each of the above is followed by a 284 A Survey of Symbolic Logic sufficient number of dots to indicate that the whole of what follows is asserted. #1-7 If p is an elementary proposition, ~p is an elementary proposition. Pp. #1-71 If p and q are elementary propositions, pvq is an elementary proposition. Pp. #1-72 If ~;p .3- ~;p r ~P~I Dem. Taut \-i ~p v ~p . 3 . ~p (1) [(!).(*! 01)] hipD-p.D.-p "Taut" is the abbreviation for the Principle of Tautology, *l-2 above. ~plp indicates that ~p is substituted in this postulate for p, giving (1). This operation is valid by *l-7. Then by the definition #1-01, above, p D ~p is substituted for its defined equivalent, ~p v ~p, and the proof is complete. *2-05 H q^r .2: poq .D. p^r Dem. Sum \ \-z . qor .0 ' ~p vo . D . ~p vr (1) L p J [(1) . (-&1-01)] \-l . q^r .D: poq ,D. por Here "Sum" refers to ^1-6, above. And (1) is what ^1-6 becomes when ~p is substituted for p. Then, by *1-01, p^>q and p^r are substituted for their defined equivalents, ~pvg and ~pvr, in (1), and the resulting expression is the theorem to be proved. The next proof illustrates the use of *1 1 and *1 11. &2-06 h: p^q -3! q^r .3. por Dem. p r ' ?3? ' "-'I , q, r J 3r.3:j>Dg'.D.p3r!.3:.;pDqr.D:gor.3.p3r (1) [#2-05] h! q^r. ?:p^q. o m por (2) [(1) . (2) . #1-11] \-: . poq .0: qor .o.p^r 286 A Survey of Symbolic Logic " Comm " is #2 04, previously proved, which is p.o.q^riosqo.por. When, in this theorem, q 3 r is substituted for p, p 3 q for q, and p 3 r for r, it becomes the long expression (1). Such substitutions are valid by *l-7, #1-71, and the definition *1-01: if p is a proposition, ~p is a proposition; if ~p and q are propositions, ~p v q is a proposition ; and p 3 q is the defined equivalent of ~p v q. Thus poq can be substituted for p. If we replace the dots by parentheses, etc., (1) becomes h { (g 3 r) 3 [(p 3 g) 3 (p 3 r)] } 3 { (p 3 g) 3 [(q 3 r) 3 (p 3 r)} } But, as (2) states, what here precedes the main implication sign is identical with a previous theorem, ^2-05. Hence, by *1-11, what follows this main implication sign the theorem to be proved can be asserted. Further proofs would, naturally, be more complicated, but they involve no principle not exemplified in the above. These three operations sub- stitutions according to *l-7, *1-71, and *l-72; substitution of defined equivalents; and "inference" according to #1-1 and *1-11 are the only processes which ever enter into any demonstration in the logic of Principia. The result is that this development avoids the paradox of taking the logic of propositions for granted in order to prove it. Nothing of the sort is assumed except these explicitly stated postulates whose use we have ob- served. And it results from this mode of development that the system is completely symbolic, except for a few postulates, *1 1, *1 7, etc., involving no further use of "if . . . , then . . .", "either ... or ...","... and " etc j wlfVft We have now seen that the calculus of propositions in Principia Mathe- matica avoids both the defects of the Two-Valued Algebra. The further comparison of the two systems can be made in a sentence : Except for the absence, in the logic of Principia, of the redundance of forms, p, p = 1, p =t= 0, etc., etc., and the absence of the entities and 1, the two systems are identical. Any theorem of this part of Principia can be translated into a valid theorem of the Two-Valued Algebra, and any theorem of the Two-Valued Algebra not involving and 1 otherwise than as { = } or { = 1 } can be translated into a valid theorem of Principia. In fact, the qualification is not particularly significant, because any use of and 1 in the Two- Valued Algebra reduces to their use as { = } and { = 1 } . For as a term of a sum, and 1 as a factor, immediately disappear, while the presence of as a factor and the presence of 1 in a sum can always be other- wise expressed. But p = is -p, and p I is p. Hence the two systems Systems Based on Material Implication 287 are simply identical so far as the logical significance of the propositions they contain is concerned. 25 The comparison of our treatment of propositional functions with the same topic in Principia is not quite so simple. 26 In the first place, there is, in Principia, the "theory of types," which concerns the range of significance of functions. But we shall omit con- sideration of this. Then, there are the differences of notation. Where we write TLK4'24 in Principia. 1-3 is ^-4-3 in Principia. 1 -4 is ^2-3 in Principia. 1 -5 is equivalent to "If a; = 0, then a x = 0", hence to -x c -(a x), which is a consequence of *3-27 in Principia, by *2-16. 1-61, in the form -(x -a) c (x a = x), is a consequence of ^-4 -71 and ^4-61 in Principia, by *4-01 and *3-26. 1-62, in the form [(y a = y}(y -a = y)]c-y, is a consequence of ^-4-71, ^-5-16, and ^2-21 in Principia. 1-7 is equivalent to [(x = \)(y = 0)] c (x = -y), hence to (x -y} c (x = -y), which is an immediate consequence of ^5-1 in Principia. 1 8 is ^4 57 in Principia. 1-9 is ^4-71 in Principia. 9-01 is equivalent to (q = 1) c [p = (p = q)], hence to q c [p = (p = q)], which is an immediate consequence of ^5-501 in Principia. 26 See Principia, i, 15-21. 288 A Survey of Symbolic Logic In this last, note the difference in the scope of the "quantifier" (x) on the two sides. If the dots be replaced by parentheses, *9 03 will be {[(*) - (px] vp} = {(x) ,[.r + P), which is 10-3. *9-06 is P+ Z(px = S Z (P+ (px), which is 10-31. *9-07 is Hx 2 , etc. (or Xz, etc. These are simply assumed as new primitive ideas, (x) . x + P), etc., not making any use, after 10-23, of the properties of n y "ft is impossible that p and q both be true" would be "p and q are inconsistent". Hence -~(pq), "It is possible that p and q both be true", represents " p and q are consistent". 1-02 Strict Implication. p-*q = ~(p-q)- Def. 1-03 Material Implication. pcq = -(p-q). Def. 1-04 Strict Logical Sum. p*q = ~(-p~~q)' Def. 1-05 Material Logical Sum. p+q = ~(~p~q)- Def. 1-06 Strict Equivalence, (p = q) = (p -iq)(q-* P) Def. We here define the defining relation itself, because by this procedure we. establish the connection between strict equivalence and strict implication. Also, this definition makes it possible to deduce expressions of the type, p = q something which could not otherwise be done. 5 But p = q re- mains a primitive idea as the idea that one set of symbols may be replaced by another. 1 07 Material Equivalence, (p = q) = (p c q) (qcp*). Def. These eight relations the seven defined above and the primitive rela- tion, p q divide into two sets, p q, p c q, p + q, and p = q are the relations which figure in any calculus of Material Implication. We shall refer to them as the "material relations", p o q, p-lq, p*q, and p = q involve the idea of impossibility, and do not belong to systems of Material Impli- cation. These may be called the "strict relations". We may anticipate a little and exhibit the analogy of these two sets, which results from the theorem ~(p q} = -(P 1} shortly to be proved. Strict relations : Material relations : p -J q = -(p O -q) p c q = -(p -q) p A q = -(-p o - (f>x H \f/x is the proposition ~(x c \l/x) "If it is impossible that x formally implies fa". This connects the novel theorems of this theory of prepo- sitional functions with the better known propositions which result from the extension of Material Implication. Similarly we shall have ( x + fa] and S.r(

A ^.r) = ~(x x-^a;). a -J /3 may be correctly interpreted "The class-concept of a, that is, , contains or implies the class-concept of /3, that is, \l/". That this should be true may not be at once clear to the reader, but it will become so if he study the properties of a * j8, and of xcfa) will be true if H x denote only actual x's; false if it denote all possible x's. One illustration is as good as a hundred; if H x (xcfa) be confined to actual x's, then it signifies a relation of extension, "The class of things of which ipx is true is contained in the class of things of which fa is true". 330 A Survey of Symbolic Logic It might be thought that the meaning of H x (X C $X) or by some equivalent definition. If U x (x c\f/x) to mean "For all possible x's, (pxc^f/x, then two courses are open: (1) we can maintain that whatever is true of all existent things is true of all possible thus abrogating a useful and probably indispensable logical distinction; or (2) we can allow that what is true or false of the possible depends upon its nature as conceived or defined. If we make the second choice here, the consequence is that a c 13, or z((pz) c z(\fsz), defined by z(z) c z($z) = H x (x -J \l/x means "It is impossible that 1-2 aeNo.3. a + eN 1 3 s e Cls Oes:aes.3 f ,.a + es:D.N es 1-4 a, & e NO . a + =b + mO.a = b l-5aNo3a + -=0 The symbol D here represents ambiguously "implies" or "is contained in" the relation c of the Boole-Schroder Algebra. This and the idea of a class, "Cls", and the e-relation, are defined and their properties demon- strated in the "mathematical logic". In terms of these, the above propo- sitions may be read: 4 All our references will be to the fifth edition, which is written in the proposed inter- national language, Interlingua, and entitled Formulario Mathemalico, Editio v (Tomo v de Formulario complete). 5 The independence of various branches in the Formulaire is somewhat greater than a superficial examination reveals. Not only are there primitive propositions for arithmetic and geometry, but many propositions are assumed as "definitions" which define in that discursive fashion in which postulates define, and which might as well be called postulates. Observe, for example, the definitions of + and X, to be quoted shortly. 6 Section n, 1, p. 27. 7 Ibid. Symbolic Logic, Logistic, and Mathematical Method 345 1-0 NO is a class, or 'number' is a common name. I 1 is a number. 1-2 If a is a number, then the successor of a is a number. 1-3 If s is a class, and if is contained in s, and if, for every a, 'a is contained in s' implies 'the successor of a is contained in s', then N is contained in s (every number is a member of the class s}. (1-3 is the principle of "mathematical induction".) 1-4 If a and b are numbers, and if the successor of a = the successor of b, then a = b. 1-5 If a is a number, then the successor of a =}= 0. The numbers are then defined in the obvious way: 1=0+, 2 = 1+, 3 = 2+, etc. 8 The relation +, which differs from the primitive idea, a +, is then defined by the assumptions: 9 3-laeN .3.a + = a (If a is a number, then a + = a.) 3-2 a, 6 e N . 3. a + (b +) = (a + 6) + (If a and b are numbers, then a + 'the successor of b' = 'the successor of a + 6'.) The relation X is defined by: 10 1-0 a, b, c e No. a. a X = 1-01 a, b, c No . 3 . a X (b + 1) = (a X 6) + a It will be clear that, except for the expression of logical relations, such as f and o , in ideographic symbols, these postulates and definitions are of the same general type as any set of postulates for abstract arithmetic. A class, NO, of members a, b, c, etc., is assumed, and the idea of a +, "suc- cessor of a". The substantive notions, "number" and "zero", the de- scriptive function, "successor of," the relations + and X, are not analysed but are taken as simple notions. 11 However, the properties which numbers have by ^virtue of being members of a class, N , are not taken for granted, as would necessarily be done in a non-logistic treatise they are specifically set forth in propositions of the "mathematical logic" which precedes. And the other principles by which proof is accomplished are similarly demonstrated. Of the specific differences of method to which this explicit- ness of the logic leads, we shall speak shortly. 8 See ibid., p. 29. 9 Ibid. 10 See ibid., 2, p. 32. II Peano does not suppose them to be unanalyzable. He says (p. 27): "Quaesitione si nos pote defini No, significa si nos pote scribe aequalitate de forma, No = expressione composito per signos noto ~ ~ 7 ... -, quod non est facile". (This was written after the publication of Russell's Principles of Mathematics, but before Principia Mathematical 346 A Survey of Symbolic Logic In Principia Mathematica, there are no separate assumptions of arith- metic, except definitions which express equivalences of notation and make possible the substitution of a single symbol for a complex of symbols. There are no postulates, except those of the logic, in the whole work. In other words, all the properties of numbers, of sums, products, powers, etc., are here proved to be what they are, solely on account of what number is, what the relations + and X are, etc. Postulates of arithmetic can be' dispensed with because the ideas of arithmetic are thoroughly analysed. The lengths to which such analysis must go in order to derive all the proper- ties of number solely from definitions is naturally considerable. We should be quite unable, within reasonable space, to give a satisfactory account of the entities of arithmetic in this manner. In fact, the latter half of Volume I and the first half of Volume II of Principia Mathematica may be said to do nothing but just this. However, we may, as an illustration, follow out the analysis of the idea of "cardinal number". This will be tedious but, with patience, it is highly instructive. We shall first collect the definitions which are involved, beginning with the definition of cardinal number and proceeding backward to the definition of the entities in terms of which cardinal number is defined, and then to the entities in terms of which these are defined, and so on. 12 *100-02 NC = D'Nc. Df "Cardinal number" is the defined equivalent of "the domain of (the rela- tion) Nc". *33-01 D = aR[a = x{(3.y} .xRy]. Df " D " is the relation of (a class) a to (a relation) R, when a and R are such that a is (the class) x which has the relation R to (something or other) y. That is, "D" is the relation of a class of x's, each of which has the relation R to something or other, to that relation R itself. *30-01 R'y = (TX)(xRy}. Df " R'y" means "the x which has the relation R to y". Putting together this definition of the use of the symbol ' and the definition of "D", we see that "D'Nc" is "the x which has the relation D to Nc", and this is a class a such that every member of a has the rela- 12 The place of any definition quoted, in Principia, is indicated by the reference number. The " translations" of these definitions are necessarily ambiguous and sometimes inaccurate, and, of course, any "translation" must anticipate what here follows but in Principia precedes. Symbolic Logic, Logistic, and Mathematical Method 347 tion Nc to something or other. "D'R" is "the domain of the relation R". If "R" be "precedes", then "D'R" will be "the class of all those things which precede anything". "Cardinal number", "NC," is defined as "D'Nc", "the domain of the relation Nc". We now turn to the meaning of "Nc". *100-01 Nc = sm. Df " Nc " is the relation of the class of referents of " sm " to " sm " itself. First, let us see the meaning of the arrow over "sm". *32-01 R = ay{a = x(xRy)}. Df " R" is "the relation of a to y, where a and y are such that a is the class of x's, each of which has the relation R to y". If " R" be "precedes", " R" will be the relation of the class "predecessors of y" to y itself. Now for "sm". We shall best not study its definition but a somewhat simpler proposition. *73-l asm/3. = . (3#) .flel -1 -a = D'R. (3 = d'R " a sm j8" is equivalent to "For some relation R, R is a one-to-one relation, while a is the domain of R and /3 is the converse-domain of R". We have here anticipated the meaning of "d'R" and of "1 > 1". *33-02 a = $R[p = ${(3.x).xRy}]. Df "G" is "the relation of (a class) /3 to (a relation) R, when /3 and R are such that /3 is the class of y's, for each of which (something or other) x has the relation R to y". Comparing this with the definition of "D" and of "D'R" above, we see that "Q'R", the converse-domain of R, is the class of those things to which something or other has the relation R. If " R" be "precedes", "Q.'R" will be the class of those things which are preceded by something or other. *71-03 1 -1 = R(R"a'Rcl.R"D'Rcl). Df This involves the meaning of "R", of ", and of "1 ". *32-02 # = $x{(3 = y(xRy}}. Df " R" signifies "the relation of /3 to x, when ft and x are such that j8 is the class of y's to which x has the relation R. *37-01 R"p = {(30) .yt-p.xRy}. Df " R"P" is "the class of x's such that, for some y, y is a member of /3, and x has the relation R to y. In other words, " R"P" (the R's of the /3's) is the 348 A Survey of Symbolic Logic class of things which have the relation R to some member or other of the class j8. If "R" be "precedes", "jR"/3" will be the class of predecessors of all (any) members of 0. With the help of this last and of preceding definitions, we can now read *71-03. "1 - 1" is "the class (of relations) R, such that whatever has the relation R to any member of the class of things-to-which-anything-has- the-relation-^, is contained in 1; and whatever is such that any member of the class of those-things-which-have-the-relation-.R-to-anything has the relation R to it, is contained in 1 . " Or more freely and intelligibly : " 1 1 " is the class of relations, R, such that if a R ft is true, then a is a class of one member and /3 is a class of one member: "1 > 1" is the class of all one-to-one correspondences. Hence " a sm /3 " means " There is a one-to- one correspondence of the members of a with the members of /?. "sm" is the relation of classes which are (cardinally) similar. The analysis of the idea of cardinal number has now been carried out until the undefined symbols, except "1", are all of them logical symbols; of relations, R; of classes, a, (3, etc.; of individuals, x, y, etc.; of prepo- sitional functions such as xRy [which is a special case of x for some x", (3 a:) . z, etc. and the idea of "negation", indi- cated by writing ~ immediately before the proposition. And in part, the rules of operation are contained in certain postulates, distinguished by their non-symbolic form: "If p is an elementary proposition, ~p is an elementary proposition", "If p and q are elementary propositions, pvq is an elementary proposition", and "If pp and $p are elementary propo- sitional functions which take elementary propositions as arguments, B. 3 We write A 4= B where the text has A non o B. 373 374 A Survey of Symbolic Logic coincide, and that one is called the container. And conversely, if any term be contained in another, then it will be one of a plurality which taken together coincide with that other. For example, if A and B taken together coincide with L, then A, or B, will be called the inexislent (inexistens) or the contained; and L will be called the container. However, it can happen that the container and the contained coincide, as for example, if (A and B) = L, and A and L coincide, for in that case B will contain nothing which is different from A. . . .* Scholium. Not every inexistent thing is a part, nor is every container a whole e. g., an inscribed square and a diameter are both in a circle, and the square, to be sure, is a certain part of the circle, but the diameter is not a part of it. We must, then, add something for the accurate explanation of the concept of whole and part, but this is not the place for it. And not only can those things which are not parts be contained in, but also they can be subtracted (or "abstracted", detrahi); e. g., the center can be subtracted from a circle so that all points except the center shall be in the remainder; for this remainder is the locus of all points within the circle whose distance from the circumference is less than the radius, and the difference of this locus from the circle is a point, namely the center. Similarly the locus of all points which are moved, in a sphere in which two distinct points on a diameter remain unmoved, is as if you should subtract from the sphere the axis or diameter passing through the two unmoved points. On the same supposition [that A and B together coincide with L], A and B taken together are called constituents (constituentia), and L is called that which is constituted (constitutum). Charact. 3. A + B = L signifies that A is in or is contained in L. Scholium. Although A and B may have something in common, so that the two taken together are greater than L itself, nevertheless what we have here stated, or now state, will still hold. It will be well to make this clear by an example: Let L denote the straight line RX, and A denote a part of it, say the line RS, and B denote another part, say the line XY. Let either of these parts, RS or R S X XY, be greater than half the whole line, RX; then certainly it cannot be said that A + B equals L, or RS + XY equals RX. For inasmuch as YS is a common part of RS and XY, RS + XY will be equal to RX + SY. And yet it can truly be said that the lines RS and XY together coincide with the line RS. S p M N 'R ^ 'Y NS f 's \ v- Def. 4. If some term M is in A and also in B, it is said to be common to them, and they are said to be communicating (communicantia) . 6 But if they have s'~~" '' nothing in common, as A and N (the lines RS and B /' XS, for example), they are said to be non-communi- "^ eating (incommunicantia). Def. 5. If A is in L in such wise that there is another term, N, in which belongs everything in L except what is in A, and of this last nothing belongs in N, then A is said to be subtracted (delrahi) or taken away (removeri), and N is called the remainder (residuum). Charact. 4. L A = N signifies that L is the container from which if A be sub- tracted the remainder is N. Def. 6. If some one term is supposed to coincide with a plurality of terms which are added (positis) or subtracted (remolis), then the plurality of terms are called the con- stituents, and the one term is called the thing constituted. 7 4 Lacuna in the text, followed by "significet A, significabit Nihil". 5 Italics ours. 6 The text here has "communicatia", clearly a misprint. 7 Leibniz's idea seems to be that if A + N = L then L is "constituted" by A and N, and also if L A = N then L and A "constitute" N. But it may mean that if L A = N, then A and N " constitute " L. Two Fragments from Leibniz 375 Scholium. Thus all terms which are in anything are constituents, but the reverse does not hold; for example, L A = N, in which case L is not in A. Def. 7. Constitution (that is, addition or subtraction) is either tacit or expressed, N or M the tacit constitution of M itself, as A or A in which N is. The expressed constitution of N is obvious. 8 Def. 8. Compensation is the operation of adding and subtracting the same thing in the same expression, both the addition and the subtraction being expressed [as A + M M]. Destruction is the operation of dropping something on account of compensation, so that it is no longer expressed, and for M M putting Nothing. Axiom 1. If a term be added to itself, nothing new is constituted or A + A = A. Scholium. With numbers, to be sure, 2 + 2 makes 4, or two coins added to two coins make four coins, but in that case the two added are not identical with the former two ; if they were, nothing new would arise, and it would be as if we should attempt in jest to make six eggs out of three by first counting 3 eggs, then taking away one and counting the remaining 2, and then taking away one more and counting the remaining 1. Axiom 2. If the same thing be added and subtracted, then however it enter into the constitution of another term, the result coincides with Nothing. Or A (however many times it is added in constituting any expression) A (however many times it is subtracted from that same expression) = Nothing. Scholium. Hence A A or (A + A ) A or A (A + A), ete. = Nothing. For by axiom 1, the expression in each case reduces to A A. Postulate 1. Any plurality of terms whatever can be added to constitute a single term; as for example, if we have A and B, we can write A + B, and call this L. Post. 2. Any term, A, can be subtracted from that in which it is, namely A + B or L, if the remainder be given as B, which added to A constitutes the container L that is, on this supposition [that A + B = L] the remainder L A can be found. Scholium. In accordance with this postulate, we shall give, later on, a method for finding the difference between two terms, one of which, A, is contained in the other, 'L, even though the remainder, which together with A constitutes L, should not be given that is, a method for finding L A, or A + B A, although A and L only are given, and B is not. THEOREM 1 Terms which are the same with a third, are the same with each other. If A = B and B = C, then A = C. For if in the proposition A = B (true by hyp.) C be substituted for B (which can be done by def. 1, since, by hyp., B = C), the result is A = C. Q.E.D. THEOREM 2 // one of two terms which are the same be different from a third term, then the other of the two will be different from it also. If A = B and B =J= C, then A 4= C. For if in the proposition B =f= C (true by hyp.) A be substituted for B (which can be done by def. 1, since, by hyp., A = B), the result is A * C. Q.E.D. [Theorem in the margin of the manuscript.] Here might be inserted the following theorem: Whatever is in one of two coincident terms, is in the other also. If A is in B and B = C, then also A is in C. For in the proposition A is in B (true by hyp.) let C be substituted for B. THEOREM 3 // terms which coincide be added to the same term, the results will coincide. If A = B, then A + C = B + C. For if in the proposition A + C = A + C (true 8 This translation is literal: the meaning is obscure, but see the diagram above. 376 A Survey of Symbolic Logic per se) you substitute B for A in one place (which can be done by def. 1, since A = B), it gives A + C = B + C. Q.E.D. COROLLARY. // terms which coincide be added to terms which coincide, the results will coincide. If A = B and L = M, then A -f L = B + M . For (by the present theorem) since L = M, A + L A + M, and in this assertion putting B for A in one place (since by hyp. A = B) gives A + L = B + M. Q.E.D. THEOREM 4 A container of the container is a container of the contained', or if that in which some- thing is, be itself in a third thing, then that which is in it will be in that same third thing that is, if A is in B and B is in C, then also A is in C. For A is in B (by hyp.), hence (by def. 3 or charact. 3) there is some term, which we may call L, such that A + L = B. Similarly, since B is in C (by hyp.), B + M = C, and in this assertion putting A + L f or B (since we show that these coincide) we have A + L + M = C. But putting N f or L + M (by post. 1) we have A + N = C. Hence (by def. 3) A is in C. Q.E.D. THEOREM 5 Whatever contains terms individually contains also that which is constituted of them. If A is in C and B is in C, then A + B (constituted of A and B, def. 4) is in C. For since A is in C, there will be some term M such that A + M = C (by def. 3). Similarly, since B is in C, B + N = C. Putting these together (by the corollary to th. 3) , we have A + M + B + N = C + C. But C + C = C (by ax. 1), hence A+M+B + N = C. And therefore (by def. 3) A + B is in C. Q.E.D. 9 THEOREM 6 Whatever is constituted of terms which are contained, is in that which is constituted of the containers. If A is in M and B is in N, then A + B is in M + N. For A is in M (by hyp.) and M is in M + N (by def. 3), hence A is in M + N (by th. 4). Similarly, B is in N (by hyp.) and N is in M + N (by def. 3), hence B is in M + N (by th. 4). But if A is in M + N and B is in M + N, then also (by th. 5) A + B is in M + N. Q.E.D. THEOREM 7 // any term be added to that in which it is, then nothing new is constituted; or if B is in A, then A + B = A. For if B is in A, then [for some C] B + C = A (def. 3). Hence (by th. 3) A + B = B + C + B = B + C (by ax. 1) = A (by the above). Q.E.D. CONVERSE OF THE PRECEDING THEOREM // by the addition of any term to another nothing new is constituted, then the term added is in the other. If A + B = A, then B is in A; for B is in A + B (def. 3), and A + B = A (by hyp.). Hence B is in A (by the principle which is inserted between ths. 2 and 3). Q.E.D. THEOREM 8 // terms which coincide be subtracted from terms which coincide, the remainders will coincide. If A = L and B = M , then A B = L - M. For A - B = A B (true per se), 9 In the margin of the manuscript at this point Leibniz has an untranslatable note, the sense of which is to remind him that he must insert illustrations of these propositions in common language. Two Fragments from Leibniz 377 and the substitution, on one or the other side, of L for A and M for B, gives A B = L - M. Q.E.D. [Note in the margin of the manuscript.] In dealing with concepts, subtraction (de- tractio) is one thing, negation another. For example, "non-rational man" is absurd or impossible. But we may say; An ape is a man except that it is not rational. [They are] men except in those respects in which man differs from the beasts, as in the case of Grotius's Jumbo 10 (Homines nisi qua bestiis differt homo, ut in Jambo Grotii). "Man" "rational" is something different from "non-rational man". For "man" "rational" = "brute". But "non-rational man" is impossible. "Man" "animal" "rational" is Nothing. Thus subtractions can give Nothing or simple non-existence even less than nothing but negations can give the impossible. 11 THEOREM 9 (1) From an expressed compensation, the destruction of the term compensated follows, provided nothing be destroyed in the compensation which, being tacitly repeated, enters into a constitution outside the compensation [that is, + N N appearing in an expression may be dropped, unless N be tacitly involved in some other term of the expression]; (2) The same holds true if whatever is thus repeated occur both in what is added and in what is subtracted outside the compensation; (3) If neither of these two obtain, then the substitution of destruction for compensa- tion [that is, the dropping of the expression of the form -f- N N] is impossible. Case 1. If A + N M N = A M, and A, N, and M be non-communicating. For here there is nothing in the compensation to be destroyed, + N N, which is also outside it in A or M that is, whatever is added in + N., however many times it is added, is in + N, and whatever is subtracted in N, however many times it is subtracted, is in N. Therefore (by ax. 2) for + N N we can put Nothing. Case 2. If A + B B G = F, and whatever is common both to A + B [i. e., to A and B] and to G and B, is M, then F = A G. In the first place, let us suppose that whatever A and G have in common, if they have anything in common, is E, so that if they have nothing in common, then E = Nothing. Thus [to exhibit the hypothesis of the case more fully] A = E + Q + M, B = N + M, and G = E + H + M, so that F=E + Q + M + N-N-M-H-M, where all the terms E, Q, M, N, and H are non-communicating. Hence (by the preceding case) F = Q H=E+Q+M E -H - M = A -G. Case 3. If A -f- B B D = C, and that which is common to A and B does not coincide with that which is common to B + D [i. e., to B and D], then we shall not have C = A - D. FoT\etB=E + F + G, and A = H + E, and D = K + F, so that these constituents are no longer communicating and there is no need for further resolution. Then C = H+E + F + G~E-F-G-K-F, that is (by case 1) C = H - K, which is not = A D (since A D = H + E K F), unless we suppose, contrary to hypothesis, that E = F that is that B and A have something in common which is also common to B and D. This same demonstration would hold even if A and D had something in common. 10 Apparently an allusion to some description of an ape by Grotius. 11 This is not an unnecessary and hair-splitting distinction, but on the contrary, per- haps the best evidence of Leibniz's accurate comprehension of the logical calculus which appears in the manuscripts. It has been generally misjudged by the commentators, because the commentators have not understood the logic of intension. The distinction of the merely non-existent and the impossible (self-contradictory or absurd) is absolutely essential to any calculus of relations in intension. And this distinction of subtraction (or in the more usual notation, division) from negation, is equally necessary. It is by the confusion of these two that the calculuses of Lambert and Castillon break down. 378 A Survey of Symbolic Logic THEOREM 10 A subtracted term and the remainder are non-communicating. If L A = N, 1 affirm that A and 2V have nothing in common. For by the definition of "subtraction" and of "remainder", everything in L remains in N except that which is in A, and of this last nothing remains in N. THEOREM 11 Of that which is in two communicating terms, whatever part is common to both and the two exclusive parts are three non-communicating terms. If A and B be communicating terms, and A = P + M and B = N + M , so that whatever is in A and B both is in M, and nothing of that is in P or N, then P, M, and N are non-communicating. For P, as well as N, is non-communicating with M, since what- ever is in M is in A and B both, and nothing of this description is in P or N. Then P and N are non-communicating, otherwise what is common to them would penetrate into A and B both. PROBLEM To add non-coincident terms to given coincident terms so that the resulting terms shall coincide. If A = A, I affirm that it is possible to find two terms, B and N, such that B =J= N and yet A + B = A + N. Solution. Choose some term M which shall be contained in A and such that, N being chosen arbitrarily, M is not contained in N nor N in M, and let B = M + N. And this will satisfy the requirements. Because B = M + N (by hyp.) and M and N are neither of them contained in the other (by hyp.), and yet A + B = A + N, since A + B = A + M + N and (by th. 7, since, by hyp., M is in A) this is = A + N. THEOREM 12 Where non-communicating terms only are involved, whatever terms added to coincident terms give coincident terms will be themselves coincident. That is, if A + B = C + D and A = C, then B = D, provided that A and B, as well as C and D, are non-communicating. For A + B C C -{ D C (by th. 8) ; but A + B - C = A + B - A (by hyp. that A = C), and A + B - A = B (by th. 9, case 1, since A and B are non-communicating), and (for the same reason) C + D C = D. Hence B = D. Q.E.D. THEOREM 13 In general; if other terms added to coincident terms give coincident terms, then the terms added are communicating. If A and A coincide or are the same, and A + B = A + N, I affirm that B and N are communicating. For if A and B are non-communicating, and A and N also, then B N (by the preceding theorem). Hence B and N are communicating. But if A and B are communicating, let A P + M and B = Q + M, putting M for that which is common to A and B and nothing of this description in P or Q. Then (by ax. 1) A + B = P + Q + M = P + M + N. But P, Q, and M are non-communicating (by th. 11). Therefore, if A?" is non-communicating with A that is, with P + M then (by the preceding theorem) it results from P + Q+M = P + M + N that Q = N. Hence N is in B; hence N and B are communicating. But if, on the same assumption (namely, that P + Q + M = P + M + N, or A is communicating with B) N also be communicating with P + M or A, then either N will be communicating with M, from which it follows that it will be communicating with B (which contains M) and the theorem will hold, or, N will be com- municating with P, and in that case we shall in similar fashion let P = G + H and N = F + H, so that G, F, and H are non-communicating (according to th. 11), and from P + Q Two Fragments from Leibniz 379 + M = P + M + N we get G + H+Q+M = G + H + M + F + H. Hence (by the preceding theorem) Q = F. Hence N ( = F + H) and B ( = Q + M) have something in common. Q.E.D. Corollary. From this demonstration we learn the following: If any terms be added to the same or coincident terms, and the results coincide, and if the terms added are each non-communicating with that to which it is added, then the terms added [to the same or coincidents] coincide with each other (as appears also from th. 12). But if one of the terms added be communicating with that to which it is added, and the other not, then [of these two added terms] the non-communicating one will be contained in the communicating one. Finally, if each of the terms is communicating with that to which it is added, then at least they will be communicating with each other (although in another connection it would not follow that terms which communicate with a third communicate with each other). To put it in symbols: A + B = A + N. If A and B are non-communicating, and A and N likewise, then B = N. HA and B are communicating but A and N are non-com- municating, then A" is in B. And finally, if B communicates with A, and likewise N com- municates with A, then B and 2V at least communicate with each other. XX Def. 1. Terms which can be substituted for one another wherever we please without altering the truth of any statement (salva veritate), are the same (eadem) or coincident (coincidentia). For example, "triangle" and "trilateral", for in every proposition demon- strated by Euclid concerning "triangle", "trilateral" can be substituted without loss of truth. A = B 12 signifies that A and B are the same, or as we say of the straight line XY and the straight line YX, XY = YX, or the shortest path of a [point] moving from X to Y coincides with that from Y to X. Def. 2. Terms which are not the same, that is, terms which cannot always be sub- stituted for one another, are different (diver sa). Such are "circle" and "triangle", or "square" (supposed perfect, as it always is in Geometry) and "equilateral quadrangle", for we can predicate this last of a rhombus, of which "square" cannot be predicated. A =j= B 13 signifies that A and B are different, as for example, R __ Y _ *? _ ^ the straight lines XY and RS. Prop. 1. If A = B, then also B = A. If anything be the same with another, then that other will be the same with it. For since A = B (by hyp.), it follows (by def. 1) that in the statement A = B (true by hyp.) B can be substituted for A and A for B; hence we have B = A. Prop. 2. If A =f= B, then also B ^ A. If any term be different from another, then that other will be different from it. Otherwise we should have B A, and in consequence (by the preceding prop.) A = B, which is contrary to hypothesis. Prop. 3. If A = B and B = C, then A = C. Terms which coincide with a third term coincide with each other. For if in the statement A = B (true by hyp.) C be substituted for B (by def. 1, since A = B), the resulting proposition will be true. Coroll. HA=B and B = C and C = D, then A = D; and so on. For A = B = C, hence A = C (by the above prop.). Again, A = C = D; hence (by the above prop.) A = D. Thus since equal things are the same in magnitude, the consequence is that things equal to a third are equal to each other. The Euclidean construction of an equilateral triangle makes each side equal to the base, whence it results that they are equal to each 12 A = B f or A oo B, as before. 13 A 4= B for A non oo B, as before. 380 A Survey of Symbolic Logic other. If anything be moved in a circle, it is sufficient to show that the paths of any two successive periods, or returns to the same point, coincide, from which it is concluded that the paths of any two periods whatever coincide. Prop. 4. If A = B and B =J= C, then A ^ C. If of two things which are the same with each other, one differ from a third, then the other also will differ from that third. For if in the proposition B 4= C (true by hyp.) A be substituted for B, we have (by def. 1, since A = B) the true proposition A 4= C. Def. 3. A is in L, or L contains A, is the same as to say that L can be made to coin- cide with a plurality of terms, taken together, of which A is one. Def. 4. Moreover, all those terms such that whatever is in them is in L, are together called components (componentia) with respect to the L thus composed or constituted. B N = L signifies that B is in L; and that B and N together compose or constitute L. 14 The same thing holds for a larger number of terms. Def. 5. I call terms one of which is in the other subalternates (suballernantia), as A and B if either A is in B or B is in A. Def. 6. Terms neither of which is in the other [I call] disparate (disparata). Axiom 1. B N = N B, or transposition here alters nothing. Post. 2. Any plurality of terms, as A and B, can be added to compose a single term, A B or L. Axiom 2. A A = A. If nothing new be added, then nothing new results, or repetition here alters nothing. (For 4 coins and 4 other coins are 8 coins, but not 4 coins and the same 4 coins already counted). Prop. 5. If A is in B and A = C, then C is in B. That which coincides with the in- existent, is inexistent. For in the proposition, A is in B (true by hyp.), the substitution of C for A (by def. 1 of coincident terms, since, by hyp., A = C) gives, C is in B. Prop. 6. // C is in B and A = B, then C is in A. Whatever is in one of two coincident terms, is in the other also. For in the proposition, C is in B, the substitution of A for C (since A = C) gives, A is in B. (This is the converse of the preceding.) Prop. 7. A is in A. Any term whatever is contained in itself. For A is in A A (by def. of "inexistent", that is, by def. 3) and A A = A (by ax. 2). Therefore (by prop. 6), A is in A. Prop. 8. If A = B, then A is in B. Of terms which coincide, the one is in the other. This is obvious from the preceding. For (by the preceding) A is in A that is (by hyp.), in B. Prop. 9. If A = B, then A ffi C = B C. If terms which coincide be added to the same term, the results will coincide. For if in the proposition, A C = A C (true per se), for A in one place be substituted B which coincides with it (by def. 1), we have A C = B C. A_C / ,_A_^ \ A. "triangle" "I /,''' "^>- N C \ /"coincide // \ ,'' N\ B "trilateral" } i* A & /^ t{ a/tTiiloi-iai-ol -f T-ionrrl^i " '}' ^^ ^' A C" equilateral triangle" \ x ^ -'' C '' 5 C "equilateral trilateral" /-coincide "" V T> S "^ /' ~B~9~C 14 In this fragment, as distinguished from XIX, the logical or "real" sum is repre- sented by . Leibniz has carelessly omitted the circle in many places, but we write wherever this relation is intended. Two Fragments from Leibniz 381 Scholium. This proposition cannot be converted much less, the two which follow. A method for finding an illustration of this fact will be exhibited below, in the problem which is prop. 23. Prop. 10. // A = L and B = M, then A B = L M . If terms which coincide be added to terms which coincide, the results will coincide. For since B = M, A B = A M (by the preceding), and putting L for the second A (since, by hyp., A = L) we have A B = L M. A "triangle", and L "trilateral" coin- / cide. B ""regular" coincides with M "most J/ capacious of equally-many-sided figures with R{ equal perimeters". " Regular triangle " coin- V s cides with "most capacious of trilateral mak- ing equal peripheries out of three sides". Scholium. This proposition cannot be converted, for if A B = L M and A = L, still it does not follow that B = M, and much less can the following be converted. Prop. 11. // A = L and B = M and C = N, then ABC=LMN. And so on. // there be any number of terms under consideration, and an equal number of them coincide with an equal number of others, term for term, then that which is composed of the former coincides with that which is composed of the latter. For (by the preceding, since A = L and B = M) we have A B L M. Hence, since C = N, we have (again by the preceding) A@B@C = LMN. Prop. 12. // B is in L, then A B will be in A L. If the same term be added to what is contained and to what contains it, the former result is contained in the latter. For L = B N (by def. of "inexistent"), and A B is in B N A (by the same), that is, A B is in L A. B "equilateral", L "regular", A "quad- //' v \ \ rilateral". "Equilateral" is in or is attribute v \ \ of "regular". Hence "equilateral quadrilat- (^ Y _ r- _ V^ W-' eral" is in "regular quadrilateral" or "perfect \ \ N / / / square". YS is in RX. Hence RT YS, Vx --__JSXl ^'' / or RS, is in RT RX, or in RX. A X~# ^'' "~r" Scholium. This proposition cannot be converted; for if A B is in A L, it does not follow that B is in L. Prop. 13. // L B = L, then B is in L. If the addition of any term to another does not alter that other, then the term added is in the other. For B is in L B (by def. of "in- existent") and L B = L (by hyp.), hence (by prop. 6) B is in L. RY RX = RX. Hence RY is in RX. R ! _ T _ ^ RY is in RX. Hence RY RX = RX. \T ' ! B~L 382 A Survey of Symbolic Logic Let L be "parallelogram" (every side of which is parallel to some side), 15 B be "quadri- lateral". "Quadrilateral parallelogram" is in the same as "parallelogram". Therefore to be quadrilateral is in [the intension of] "parallelogram". Reversing the reasoning, to be quadrilateral is in "parallelogram". Therefore, "quadrilateral parallelogram" is the same as "parallelogram". Prop. 14. If B is in L, then L B = L. Subalternates compose nothing new; or if any term which is in another be added to it, it will produce nothing different from that other. (Converse of the preceding.) If B is in L, then (by def. of "inexistent") L = B P. Hence (by prop. 9)L@B=BPB, which (by ax. 2) is = B P, which (by hyp.) is = L. Prop. 15. If A is in B and B is in C, then also A is in C. What is contained in the contained, is contained in the container. For A is in B (by hyp.), hence A L = B (by def. of "inexistent"). Similarly, since B is in C, B M = C, and putting A L for B in this statement (since we have shown that these coincide), we have A L M = C. Therefore (by def. of "inexistent") A is in C. R T S X RT is in RS, and RS in RX. V "A / ~~7 Hence RT is in RX. / A "quadrilateral", B "parallelogram", y c C "rectangle". To be quadrilateral is in [the intension of] "parallelogram", and to be parallelogram is in "rectangle" (that is, a figure every angle of which is a right angle). If instead of concepts per se, we consider individual things comprehended by the concept, and put A for "rectangle", B for "parallelogram", C for "quadrilateral", the relations of these can be inverted. For all rectangles are comprehended in the number of the parallelograms, and all parallelograms in the number of the quadrilaterals. Hence also, all rectangles are contained amongst (in) the quadrilaterals. In the same way, all men are contained amongst (in) all the animals, and all animals amongst all the material substances, hence all men are contained amongst the material substances. And conversely, the concept of material substance is in the concept of animal, and the concept of animal is in the concept of man. For to be man contains [or implies] being animal. Scholium. This proposition cannot be converted, and much less can the following. Coroll. If A N is in B, N also is in B. For N is in A N (by def. of " inexistent "). Prop. 16. // A is in B and B is in C and C is in D, then also A is in D. And so on. That which is contained in what is contained by the contained, is in the container. For if A is in B and B is in C, A also is in C (by the preceding). Whence if C is in D, then also (again by the preceding) A is in D. Prop. 17. // A is in B and B is in A, then A = B. Terms which contain each other coincide. For if A is in B, then A N = B (by def. of "inexistent"). But B is in A (by hyp.), hence A N is in A (by prop. 5). Hence (by coroll. prop. 15) N also is in A. Hence (by prop. 14) A = A N, that is, A = B. B_ RT, N; RS, A; SR RT, B. ,''' ^^ To be trilateral is in [the intension of] //'' \ "triangle", and to be triangle is in "trilat- eral". Hence "triangle" and "trilateral" coincide. Similarly, to be omniscient is to be \ " " / omnipotent. \ x ,'' 15 Leibniz uses "parallelogram" in its current meaning, though his language^may suggest a wider use. Two Fragments from Leibniz 383 Prop. 18. // A is in L and B is in L, then also A B is in L. What is composed of two, each contained in a third, is itself contained in that third. For since A is in L (by hyp.), it can be seen that A M = L (by def. of "inexistent"). Similarly, since B is in L, it can be seen that B N = L. Putting these together, we have (by prop. 10) A M BN = LL. Hence (by ax. 2) 18 AM@BN = L, Hence (by def. of 16 The number of the axiom is given in the text as 5, a misprint, "inexistent") A B is in L. RYS is in RX. / \ YSTisinRX. Bfr f Hence BT is in BX. *\ \ / J A "equiangular", B "equilateral", A B "equiangular equilateral" or "regular", L "square". "Equiangular" is in [the intension of] "square", and "equilateral" is in "square". Hence "regular" is in "square". Prop. 19. // A is in L and B is in L and C is in L, then A B C is in L. And so on. Or in general, whatever contains terms individually, contains also what is composed of them. For A B is in L (by the preceding). But also C is in L (by hyp.), hence (once more by the preceding) A B C is in L. Scholium. It is obvious that these two propositions and similar ones can be con- verted. For if A B = L, it is clear from the definition of "inexistent" that A is in L, and B is in L. Likewise, if A B C = L, it is clear that A is in L, and B is in L, and C is in L. 17 Also that A B is in L, and A C is in L, and B C is in L. And so on. Prop. 20. // A is in M and B is in N, then A B is in M N. If the former of one pair be in the latter and the former of another pair be in the latter, then what is composed of the former in the two cases is in what is composed of the latter in the two cases. For A is in M (by hyp.) and M is in M N (by def. of "inexistent"). Hence (by prop. 15) A is in M N Similarly, since B is in N and N is in M N, then also (by prop. 18) A B is in M N' RT is in RY and ST is in SX, hence RT ,''" ^^^^ ST, or RY, is in RY SX, or in RX. 18 ^'"^"X^'^N If A be "quadrilateral" and B "equi- fr / \ N >^ angular", A 5 will be "rectangle". If M f I \ ^ be "parallelogram" and N "regular", M ;- - ^ - f - J - IT N will be "square". Now "quadrilateral" V \ /' /I ib in [the intension of] "parallelogram", and v \^^-ii--^--_B.-''/ "equiangular" is in "regular", hence "rec- NX X^ ^' tangle" (or "equiangular quadrilateral") is m "regular parallelogram or square". Scholium. This proposition cannot be converted. Suppose that A is in M and A B is in M N, still it does not follow that B is in N; for it might happen that B as well as A is in M, and whatever is in B is in M, and something different in N. Much less, therefore, can the following similar proposition be converted. Prop. 21. IfAisinMandBisinNandCisinP,thenA B CisinM N @ P. 17 To be consistent, Leibniz should have written "A B is in L" instead of "A 5 = L", and "A 5 C is in L" instead of "A . C = L" but note the method of the proof. to 18 The text has RY here instead of BX: the correction is obvious. 384 A Survey of Symbolic Logic And so on. Whatever is composed of terms which are contained, is in what is composed of the containers. For since A is in M and B is in N, (by the preceding), A B is in M N. But C is in P, hence (again by the preceding) ABC'is'v\MN@P. Prop. 22. Two disparate terms, A and B, being given, to find a third term, C, different from them and such that with them it composes subalternates A C and B C that is, such that although A and B are neither of them contained in the other, still A @ C and B C shall one of them be contained in the other. Solution. If we wish that A C be contained in B C, but A be not contained in B, this can be accomplished in the following manner: Assume (by post. 1) some term, D, such that it is not contained in A, and (by post. 2) let A D = C, and the requirements are satisfied. - .. /,'' A C "-s x \ For A C = A A D (by construc- f \ \ tion) = A D (by ax. 2). Similarly, B C I _ S _ Vr _ \j - B A D (by construction). But A \~ /\ ,/\ f D is in 5 A D (by def. 3). Hence ^-*' X "-j'/' / N *"~R"' A CisinB C. Which was to be done. 19 L> SY and YX are disparate. If RS SY = YR, then SY YR will be in ZF Ffl. Let A be "equilateral", B "parallelogram", D "equiangular", and C "equiangular equilateral" or "regular", where it is obvious that although "equilateral" and "parallelo- gram" are disparate, so that neither is in the other, yet "regular equilateral" is in "regular parallelogram " or "square ". But, you ask, will this construction prescribed in the problem succeed in all cases? For example, let A be "trilateral", and B "quadrilateral"; is it not then impossible to find a concept which shall contain A and B both, and hence to find B C such that it shall contain A C, since A and B are incompatible? I reply that our general construction depends upon the second postulate, in which is contained the assumption that any term and any other term can be put together as components. Thus God, soul, body, point, and heat compose an aggregate of these five things. And in this fashion also quadrilateral and trilateral can be put together as components. For assume D to be anything you please which is not contained in "trilateral", as "circle". Then A D is "trilateral and circle", 20 which may be called C. But C A is nothing but "trilateral and circle" again. Consequently, whatever is in C B is also in "tri- lateral", in "circle", and in "quadrilateral". But if anyone wish to apply this general calculus of compositions of whatever sort to a special mode of composition; for example if one wish to unite "trilateral" and "circle" and "quadrilateral" not only to compose an aggregate but so that each of these concepts shall belong to the same subject, then it is necessary to observe whether they are compatible. Thus immovable straight lines at a distance from one another can be added to compose an aggregate but not to compose a continuum. Prop. 23. Two disparate terms, A and B, being given, to find a third, C, different from them [and such that A B = A C]. 21 Solution. Assume (by post. 2) C = A B, and this satisfies the requirements. For since A and B are disparate (by hyp.) that is (by def. 6), neither is in the other 19 Leibniz has carelessly substituted L in the proof where he has D in the proposition and in the figure. We read D throughout. 20 Leibniz is still sticking to intensions in this example, however much the language may suggest extension. 21 The proof, as well as the reference in the scholium to prop. 9, indicate that the statement of the theorem in the text is incomplete. We have chosen the most conservative emendation. Two Fragments from Leibniz 385 therefore (by prop. 13) it is impossible that C = A or C = B. Hence these three are differ- ent, as the problem requires. Thus AC=AAB (by construction), which (by ax. 2) is = A B. Therefore A C = A B. Which was to be done. Prop. 24. To find a set of terms, of any desired number, which differ each from each and are so related that from them nothing can be composed which is new, or different from every one of them [i. e., such that they form a group with respect to the operation ]. Solution. Assume (by post. 1) any terms, of any desired number, which shall be different from each other, A, B, C, and D, and from these let A B = M, M C = N, and N D P. Then A, B, M, N, and P are the terms required. For (by construction) M is made from A and B, and hence A, or B, is in M, and M in N, and N in P. Hence (by prop. 16) any term which here precedes is in any which follows. But if two such are united as components, nothing new arises; for if a term be united with itself, nothing new arises; L L = L (by ax. 2). 22 If one term be united with another as components, a term which precedes will be united with one which follows; hence a term which is contained with one which contains it, as L N, but L N = N (by prop. 14) , 23 And if three are united, as L N P, then a couple, L N, will be joined with one, P. But the couple, L N, by themselves will not compose anything new, but one of themselves, namely the latter, N, as we have shown; hence to unite a couple, L N, with one, P, is the same as to unite one, N, with one, P, which we have just demonstrated to compose nothing new. And so on, for any larger number of terms. Q.E.D. Scholium. It would have been sufficient to add each term to the next, which contains it, as M, N, P, etc., and indeed this will be the situation, if in our construction we put A = Nothing and let B = M . But it is clear that the solution which has been given is of somewhat wider application, and of course these problems can be solved in more than one way; but to exhibit all their possible solutions would be to demonstrate that no other ways are possible, and for this a large number of propositions would need to be proved first. But to give an example: five things, A, B, C, D, and E, can be so related that they will not compose anything new only in some one of the following ways: first, if A is in B and B in C and C in D and D in E; second, if A B = C and C is in D and D in E; third, if A B = C and A is in D and B D = E. The five concepts which follow are related in the last, or third, way; A "equiangular", B "equilateral", C "regular", D "rectangle", E "square", from which nothing can be composed which does not coincide with them, since "equiangular equilateral" coincides with "regular", and "equiangular" is in [the intension of] "rectangle", and "equilateral rectangle" coincides with "square". Thus "regular equiangular" figure is the same as that which is at once "regular" and "regular equi- lateral", and " equiangular rectangle " is "rectangle", and " regular rectangle " is "square". Scholium to defs. 3, 4, 5, and 6. We say that the concept of the genus is in the concept of the spe :ies; the individuals of the species amongst (in) the individuals of the genus; a part in the whole; and indeed the ultimate and indivisible in the continuous, as a point is in a line, although a point is not a part of the line. Likewise the concept of the attribute or predicate is in the concept of the subject. And in general this conception is of the widest application. We also speak of that which is in something as contained in that in which it is. We are not here concerned with the notion of "contained" in general with the manner in which those things which are "in" are related to one another and to that which contains them. Thus our demonstrations cover also those things which compose something in the distributive sense, as all the species together compose the genus. Hence all the inexistent things which suffice to constitute a container, or in which are all things which are in the container, are said to compose that container; as for example, A B are said to compose L, if A, B, and L denote the straight lines RS, YX, and RX, for RS YX = RX. And such parts which complete the whole, I am accustomed to call "cointe- grants", especially if they have no common part; if they have a common part, they are 22 The number of the axiom is omitted in the text. 23 The number of the prop, is omitted in the text. 26 386 A Survey of Symbolic Logic called "co-members", as RS and RX. Whence it is clear that the same thing can be composed in many different ways if the things of which it is composed are themselves composite. Indeed if the resolution could finally be carried to infinity, the variations of composition would be infinite. Thus all synthesis and analysis depends upon the principles here laid down. And if those things which are contained are homogeneous with that in which they are contained, they are called parts and the container is called the whole. If two parts, however chosen, are such that a third can be found having a part of one and a part of the other in common, then that which is composed of them is continuous. Which illustrates by what small and simple additions one concept arises from another. And I call by the name "subalter nates" those things one of which is in the other, as the species in the genus, the straight line RS in the straight line RX; "disparates" where the opposite is the case, as the straight lines RS and YX, two species of the same genus, perfect metal and imperfect metal and particularly, members of the different divisions of the same whole, which (members) have something in common, as for example, if you divide "metal" into "perfect" and "imperfect", and again into "soluble in aquafortis" and "insoluble", it is clear that "metal which is insoluble in aqua fortis" and "perfect metal" are two dispa- rate things, and there is metal which is perfect, or is always capable of being fulminated in a cupel, 24 and yet is soluble in aquafortis, as silver, and on the other hand, there is imperfect metal which is insoluble in aqua fortis, as tin. Scholium to axioms 1 and 2. Since the ideal form of the general [or ideal form in general, speciosa generalis] is nothing but the representation of combinations by means of symbols, and their manipulation, and the discoverable laws of combination are various, 25 it results from this that various modes of computation arise. In this place, however, we have nothing to do with the theory of the variations which consist simply in changes of order [i. e., the theory of permutations], and AB [more consistently, A B] is for us the same as BA [or B A], And also we here take no account of repetition that is A A [more consistently, A A] is for us the same as A. Thus wherever these laws just mentioned can be used, the present calculus can be applied. It is obvious that it can also be used in the composition of absolute concepts, where neither laws of order nor of repetition obtain; thus to say "warm and light" is the same as to say "light and warm", and to say "warm fire" or "white mi k", after the fashion of the poets, is pleonasm; white milk is nothing different from milk, and rational man that is, rational animal which is rational is nothing different from rational animal. The same thing is true when certain given things are said to be contained in (inexistere) certain things. For the real addition of the same is a useless repetition. When two and two are said to make four, the latter two must be different from the former. If they were the same, nothing new would arise, and it would be as if one should in jest attempt to make six eggs out of three by first counting 3 eggs, then taking away one and counting the remaining 2, and then taking away one more and counting the remaining 1. But in the calculus of numbers and magnitudes, A or B or any other symbol does not signify a certain object but anything you please with that number of congruent parts, for any two feet whatever are denoted by 2; if foot is the unit or measure, then 2 + 2 makes the new thing 4, and 3 times 3 the new thing 9, for it is presupposed that the things added are always different (although of the same magnitude) ; but the opposite is the case with certain things, as with lines. Suppose we describe by a moving [point] the straight line, RY YX = RYX or P B = L, going from R to X. If we suppose this same [point] then to return from X to Y and stop there, although it does indeed describe YX or B a second time, it produces nothing different than if it had described YX once. Thus L B is the same as L that is, P B B or RY YX XY is the same as RY YX. This caution is of much importance in making judgments, by means of the magnitude and motion of those things which generate 26 or describe, concerning the 24 The text here has". . . fulminabile persistens in capella": the correction is obvious. 25 ". . . variaeque sint combinandi leges excogitabiles, . . ." "Excogitabiles", "discoverable by imagination or invention", is here significant of Leibniz's theory of the relation between the "universal calculus" and the progress of science. 28 Reading "generant" for "generantur" a correction which is not absolutely neces- Two Fragments from Leibniz 387 magnitude of those things which are generated or described. For care must be taken either that one [step in the process] shall not choose the track of another as its own that is, one part of the describing operation follow in the path of another or else [if this should happen] this [reduplication] must be subtracted so that the same thing shall not be taken too many times. It is clear also from this that "components", according to the concept which we here use, can compose by their magnitudes a magnitude greater than the magnitude of the thing which they compose. 27 Whence the composition of things differs widely from the composition of magnitudes. For example, if there are two parts, A or RS and B or RX, of the whole line L or RX, and each of these is greater than half of RX itself if, for example, RX is 5 feet and RS 4 feet and YX 3 feet obviously the magnitudes of the parts compose a magnitude of 7 feet, which is greater than that of the whole; and yet the lines RS and YX themselves compose nothing different from RX, that is, RS YX = RX. Accord- ingly I here denote this real addition by , as the addition of magnitudes is denoted by +. And finally, although it is of much importance, when it is a question of the actual generation of things, what their order is (for the foundations are laid before the house is built), still in the mental construction of things the result is the same whichever ingredient we consider first (although one order may be more convenient than another), hence the order does not here alter the thing developed. This matter is to be considered in its own time and proper place. For the present, however, RY YS SX is the same as YS RY SX. Scholium to prop. 24. If RS and YX are different, indeed disparate, so that neither is in the other, then let RS YX = RX, and RS RX will be the same as YX RX For the straight line RX is always composed by a process of conception (in notionibus). If A is "parallelogram" and B "equiangular" which are disparate terms let C be A B, that is, "rectangle". Then "rectangular parallelogram" is the same as "equiangular rectangle ", for either of these is nothing differ- ent from "rectangle". In general, if Maevius ^ is A and Titius B, the pair composed of the j^f y= K. "jP^ two men C, then Maevius together with this \\ x / N