CORNELL UNIVERSITY LIBRARY GIFT OF The University of California. Cornell University Library BC135 .L67 + Survey of symbolic logic, by C.I. Lewis. 3 1924 028 923 451 olin Overs All books are subject to recall atter two weeks. Olin/Kroch Library DATE DUE Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924028923451 SEMICENTENNIAL PUBLICATIONS OF THE UNIVERSITY OF CALIFORNIA 1868-1918 A SURVEY OF SYMBOLIC LOGIC BY C. I. LEWIS UNIVERSITY OF CALIFORNIA PRESS BERKELEY 1918 >--,■■. A ^\o5SX\ PRESS OF THE NEW ERA PRINTING COMPANY LANCASTER, PA. C.f^ 0^ Uv^w.or ai; r 'w i9 I : r TABLE OF CONTENTS PREFACE . 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 Section II. Elementary Theorems Section HI. General Properties of Functions . . Section IV. Fundamental Laws of the Theory of Equations . . . Section V. Fundamental Laws of the Theory of Inequations . Section VI. Note on the Inverse Operations, "Subtraction" and "Division". ' 118 122 132 144 166 173 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 L The Two-Valued Algebra 222 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 Propositional Functions . 260 Section V. The Logic of Relations . . . . Section VI. The Logic of Principia Mathematica . 269 279U 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 ImpHcation. The Calculus of Consistencies and the Calculus of Ordinary Inference . 316 324 . Section V. The Meaning of " Implies " 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 TWO FRAGMENTS FROM LEIBNIZ. APPENDIX BIBLIOGRAPHY INDEX 373 389 407 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 ilber 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. f 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 sufBcient consideration of these questions is here given, though such statements as are made will be, I hope, accurate. Particularly in the last chapter, on "SymboHc 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 ~X 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 symboHc logic and the application of such methods as symbohc 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 Reptihlic?- Leibniz left frag- mentary developments of symboHc 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 MacCoU to Schroder whose Vorlesungen iiber 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 / Principii di Geometria. Frege's works, from the Begriffsschrift of 1879 to the Grundgesetze der Arithmetik (Vol. I, 1893; Vol. H, 1903) provide a comprehensive development of arithmetic by the logistic method. ' See the criticisms of contemporary mathematics and the program for the dialectic or philosophic development of mathematics in Bk. vi, Step. 510-11 and PMlebus, 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.^ 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' A history of the subject will not be attempted, if by history is rneant 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.* /^ II. Leibniz The history of symbolic logic and logistic properly begins with Leibniz.^ In the Neio Essays on the Human Understanding, Philalethes is made to say:^ "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. ' 1 Pp. 389-406. 5 Leibniz regards Raymond LuUy, Athanasius Kircher, John WOkins, and George Dalgarno (see Bibliography) as his predecessors in this field. But their writings contain little which is directly to the point. 1 Bk. IV, Chap, xvii, § 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 verj^ seriously to the more limited tasks which he might have accomplished unaided.^ 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. ^ 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."^ 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- 'The editor's introduction to "Scientia Generalis. Characteristic a" in Gerhardt's Philosophischen Schriften von Leibniz (Berlin, 1890), vii, gives an excellent account of Leibniz's correspondence upon this topic, together with other material of historic interest. (Work hereafter cited as G. Phil.) ' See Gerhardt, op. cit. especially iv and vii. 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, vii, 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.i" 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." " 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 symbohsm. 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. " ^^ 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. "^^ 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. " G. Phil, VII, 84. 12 G. Phil, VII, 21. " G. Phil., VII, 157. 8 A Survey of Symbolic Logic cepts is, indeed, real — vide the few primitives of Pmicipia MatJiematica. 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;^* 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 Scientias novce Generalis, he says that the utmost which we can hope to accomplish at present, toward the general art of invention, is a perfectly orderlj^ and finished reconstruction of existing science in terms of the ■ absolute primitives which analysis reveals.^' 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- " See G. Phil, vii, 194, footnote. 15 G. Phil, VII, 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. "^^ 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.'' 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" — with combinations which are absolute or relative, symmetrical or unsymmetrical, and so on.'" 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 Combinatoria,^^ and in the fragments which attempt to develop an algebra of logic .^^ Leibniz wrote De Arte Combinatoria when he was, in his own words, vix 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, VII, 31. 1' See New Essays on the Human Understanding, Bk. iv. Chap, xvii, §§ 9-13. 18 See G. Phil, vii, 31, 198 #., and 204. 15 G. Phil, IV, 35-104. Also Gerhardt, Leibnizens mathematische Schriften (1859), v, 1-79. ^0 Seientia Generalis. Characteristica, xv-xx, G. Phil., vii. 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; ^^ 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. ^^ 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.^' 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.^* 21 Leibniz tells us that he takes this problem from the Ars Magna of Raymond LuUy. See G. Phil, v, 62. 22 See letter to Tscliirnhaus, 1678, Gerhardt, Math., iv, 451-63. Cf. Cantor, Geschichte d. Math., HI, 39 ff. 23 See the Synopsis, G. Phil, iv, 30-31. ^ See Couturat, op. eit., 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 sen contiguum, 5. dis- situm sen 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 tCiv 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 rfis 11. §. 3. Continuum est A ad B, si rou A ^ 9 est 4 et 7 rci 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 y 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.^^ 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 accompHshed 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 26 Dated April, 1679. Couturat (op. cit., p. 326, footnote) gives the titles of these as follows: "Elemenia Characteristicae Universalis (Collected manuscripts of Leibniz in the Hanover Library, PHIL., v, 8 b); Calculi universalis Elementa (PHIL., v, 8 c); Calculi universalis investigaiiones (PHIL., v, 8 d); Modus examinandi consequentias per numeros (PHIL., V, 8 e); Regulae ex quihus de bonitate consequeniiarum formisque et modis syllogis- morum calegoricum 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 ov S = Py (where yisa 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 =)= Py- And since the universal affirmative is iS = Py, the particular negative should be S 4= 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 =t= Py, where x and y are arbitrary, will never be true. Such difficulties led Leibniz to compHcate 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.^^ Leibniz cherished the 26 G. Phil., VII, xvi-xx. Of these, xvi, without title, states rules for inference in terms of inclusion and exclusion; Difficultales quaedam logicae treats of subalternation and conversion and of the symbolic expression for various types of propositions; xviii, 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 ov B to be included in A is to affirm the predicate B universally of the subject ^"." However, Leibniz evidently considers the calculus to have many appHcations, 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 indifPerently 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?^ 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 ahstractis. demonstrandi in ahstractis 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 {O. Phil., vii, 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 oodenda. "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 sapientum. "10. De Arte Inveniendi. "11. De Synthesi seu Arte combinatoria. "12. De Analysi. "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. 2' G. Phil., VII, 208. 2* "Generates Inquisitiones" (1686): see Couturat, OpzisCMtes e(c., 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.^' In DifficuUates 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: ^^ Univ. aff.; All A is B: AB = A, or A non-B does not exist. Part, neg.; Some A is not B; AB =1= ^, or ^ non-B exists. Univ. neg.; Xo A is B; AB does not exist. Part, aif.; Some ^ is 5; A B 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 symbohc 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 O. Phil., vii, 231, foot- note; and "Specimen certitudinis seu de conditionibus," Dutens, Leibnitii Opera, iv, Part III, pp. 92 ff., also Couturat, La Logique de Leibniz, p. 348, footnote, and p. 353, footnote. '»(?. Phil, VII, 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: ^i All .4 is 5: AB = A. Some A is not B: AB =|= A. Xo AhB: AB does not exist, or AB + AB Ens. Some .4 is £ : 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 + 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 ii AB = A (All A is B), YAB = YA (Some AisB); but if AB f A, it does not follow that YAB =|= YA, for if Y = B, YAB = YA. Again, if AB #= AB Ens (No A is 5), YAB =t= F^5 Ens (It is false that some A is B); but if AB = AB Ens (Some A is i?), 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 foohsh men are foolish wise persons, because "foolish" is incompatible with "wise".^^ 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 : ^'^ 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. '2 G. Phil., VII, 215: the illustration is mine. 33 G. Phil, VII, 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 6 and [all] a is c and [all] a is d. " "* 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- aderistica. The first of these, with the title Non inelegans specimen devion- 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 ot B (or both). Thus the inclusion relation, ^ 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 Httle 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 ol 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 = Lis 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 i is a hne, ^1 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). It L — A = N, A and iV must be mutually exclusive (incominunicantia) . 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"."^ In intension, the relation seems to indicate an abstraction, not a negative qualification. But there are difiiculties, due to the overlapping of concepts. Say that "man" + "woodworking" = "carpenter" and "man" + "white-skinned" '= 0. Phil., VII, 231, footnote. Couturat in commenting on this (op. cit., pp. 377-78) says: ''Ailleurs Leibniz essaie de pr^ciser cette opposition en disant: 'A — A est Nihilum. Sed A non-A est Absurdum.' "Mais il oublie que le n^ant (non-Ens) n'est pas autre chose que ce qu'il appelle I'absurde ou I'impossible, c'est-^-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 no^-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 fund^-mental 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 .^'^ Venn makes it clear that Plouc- '" 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 1& quet's calculus was a calculus of intension and that it involved the quanti- fication of the predicate. Lambert^' wrote voluminously on the subject of logic, but his most important contribution to symbolic procedure is contained in the Seeks- Versiiche einer Zeichenkimst 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. Materi^al 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 {Gleichgultigkeit) = addition {Zusetzung) + abstraction (Absonderung) — opposition (des Gegeniheils) X universality > particularity < copula ~ given concepts (Begriffe) a, b, c, d, etc. undetermined concepts n, m, I, etc. unknowns x, y, z. the genus 7 the difference S 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, a5 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+aS = a{y + 5) = a 3' Johann Heinrich Lambert (1728-77), German physicist, mathematician, and astrono- mer. He is remembered chiefly for his development of the equation x»+px = g in an infinite series, and his proof, in 1761, of the irrationality of t. 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 {ErMdrung) of a. As immediate consequences of (1), we have also (2) ay = a — a5 (3) a5 = 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 h, have any part of their connotation in common, then {a + h) — 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 anddifference may be supposed to be mutually exclusive. We shall return to this topic later. More complex laws of genus and difference may be ehcited 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 + by = ay^ ^^ ayb + a8y + ab'^ Proof: ay = ayy + ayb and a6 = ady + a85 But a = ay + ad. Hence Q.E.D. That is to say: if one wish 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 prehminary step. Suppose the explanation carried one degree further and the resulting terms arranged as follows: a = a{y^ + 775 + yS8 + 5') + 7S7 + 57S + 577 + 557 The three possible arrangements of two 7's and one S might be summarized The Development of Symbolic Logic 21 by 37^5; the three arrangements of two 5's and one y by 3yS\ With this convention, the formula for an explanation carried to any degree, n, is: (5) a = air'* nT'-S + ""^^ y^-.,. ^ njn - l){n - 2) ^^_^^^ ^ ^^^ This "Newtonian formula" is a rather pleasant mathematical conceit. Two further interesting laws are given: (6) a = aS + ayS + ay^5 + ay^5 + . . . etc. Proof: a = ay + aS But ay = ay^ + ay8 and ay^ = ay^ + ay^8 ay^ = ay^ + ay^d, etc. etc. (7) a = ay + aSy + aS^y + a5^y + . . . etc. Proof: a = ay + a5 But a5 = aSy + aS^ and ad^ = ad'^y + a5^ a8^ = ab^y + a5^ etc. etc. Just as the genus of a is represented by ay, the genus of the genus of a by ay'^, etc., so a species of which a is genus may be represented by ay~^, and a species of which a is genus of the genus by ay-', etc. In general, as a7" represents a genus above a, so a species below a may be represented by a 07"" or — Similarly ainy concept of which a is difference of the difference of the differ- ence . . . etc., may be represented by a ad-" or Also, just as a = a(y + 8)", where o is a concept and a(y + 5)" its "explana- tion", so -. -^ = a, where ; — - is 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.^" '' Sechs Versuche, p. 5. " Ibid., pp. 9-10. 22 A Survey of Symbolic Logic The concept ay'^ + a8y is very different from the concept (ay + a8)y, because (8) (ay + aS)y — a{y + 6)7 = ay{y + S) = ay while ay^ + 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 7 7 species x of which a is the genus, i. e., a (9) -y = a 7 But ay/y 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.^^ 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 o | & that part of a which is different from b,"^^ then (10) a\b + b\a+ ab + ab = a+b Also X — x\a = ay, or x = ay + x\a ax = a5\ %,,,}y>^ ^j^ a — ax = ad a = ax + a5 ax = a — aS = ay = xy ■" Ibid., p. 10. ^ Lambert sometimes uses o | 6 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 hy y + x\y = X. The manner in which Lambert deduces the second of these expressions from the first is interesting."' If 2/ 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 = ci- He has subtracted y from both sides, in the last step, and we observe that '^y — 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 rehance 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 oi 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 *' Sechs Versuche, p. 12. -i A Suney of Symbolic Logic X — b represents the abstraction of the whole of b from .r. 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," 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 = a5, a — a5 = 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 — 6 becomes wholly doubtful unless [ — ] be treated as a wholly con- ventional inverse of [+ ], 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 hy the quantification of the predicate. ^^ The proposition "All ^ is 5" 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, the case in which the converse is a universal, the subject A comprehended within the predicate B. (2) The case in which the converse is particular. Li 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. VI A is a species of A, and in intension the genus is contained in the species, " lUd., p. 12. «/6id., pp. 93 #. 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 pecuhar 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 A m 5 71 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 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 I A 1, then the possible forms of judg- ment are as follows: (1) ^ = ^ All S is all P. Q p (2) J = -r All S is some P. Now expresses negatively what 1/co 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 00 D p (4) Y ^ I ^°™^ ^ '^ ^'^ P- S P (5) r = Y ►^ome S is some P S P (6) 7- = — Some S is not P. f °° S P (7) — = - All not-S is all P. «. 1 5' See Ibid., Brief iv. The Development of Symholic Logic 31 D p (8) — = - All not-S is some P. °° / e p (9) — = — All not-S is all not-P- 00 00 (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 - = 1 ■ S, -■ = vS, where < ■!) < 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 P TV M Ergo, E = — [All Europeans are mortal] pir Example 2. All plants are organisms P = — ^ A All plants are no animals r = — A Ergo, - = — [Some organisms are not animals] p Xl 32 A Survey of Symbolic Logic p Example 3. All men are rational H = — P All plants are not rational P = R 00 pH Ergo, All plants are no men P = — 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 = piHi'^ is also misinterpreted. It should be, "All plants are not some men". A correct reading would have revealed the invahd operation. Lambert rephed 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 CO 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 Berhn Academy in 1803.^* 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 which S is subsumed, M being the logical "difference" of S in S — M. Consonantly S + 3I, symbohzing 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 iS is ^" is represented by S = A + M The universal negative "No S is ^" is symbohzed 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 " "M^moire sur un nouvel algorithme logique", in Memoires de I'Academie des Sciences de Berlin, 1803, Classe de philosophic speculative, pp. 1-14. See alao his paper, "Reflexions siu la Logique", loc. cit., 1802. The Development of Symbolic Logic 33 illusory particular, one whose cbnverse 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^ M Castillon's explanation of this is that the illusory particular judgment gives us to understand that some S alone is yl, or that S is got from A by ab- straction {S = A ~ 31), 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; one can, then, indicate it by S = ^ ^ M. The fact is, of course, that "Some S is ^ " 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 well be symbolized by S + iV = A + M. We suspect that Castillon's choice of S = ^4 ^ 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 5"— will also give S = A "^ M. Thus "Some S is A" may be derived both from "All S is ^" and from "All A is S", Avhich is a de- sideratum. The illusory negative particular is, correspondingly, S = - A^ 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^ M gives - A = - S^ M. Hence A = S ^ M, which comes back to S = ^ ^ M. A universal gives its subaltern S = A + M gives S = A "^ M, and S = - A + M gives S = - A^ M. And a real particular gives also the converse illusory particular, for ^ = S _ Jl/ gives S = A + 31, 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 sj'mbolized 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 All SisM S = M + P All Sis A :. S = A + {N + P) No M is A M = - A+N All SisM S ^ M + P No SkA :. S = - A + {N + P) All M is A M = A + N Some S is M S = J/ =F P or S = M - - P Some S is ^ :. S = {A+N)=f P or S = - {A + 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 "J/ which is not P". If it mean this, then [+]and [ — ]arenot 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 which nothing is ever added to or subtracted from a determined concept, S or A, except an indeterminate, M or N or P, and this indeter- minate, just because it is indeterminate, conceals the fact that [+ ] and [ — ] are not true inverses. And when the sign [ — ] appears before a determinate, A, it may serve as the sign of negation, because no difficulty arises from supposing the whole of what is negated to be absent, or abstracted. Castillon's calculus is theoretically as unsound as Lambert's, or more so if unsoundness admits of degree. It is quite possible that it was worked out empirically and procedures which give invalid results avoided. The Development of Symbolic Logic 35 Whoever studies Leibniz, Lambert and Castillon cannot fail to be con- vinced that a consistent calculus of concepts in intension is either immensely difficult or, as Couturat has said, impossible. Its main difficulty is not the one which troubled Leibniz and which constitutes the main defect in Lambert's system— the failure of [ + ] and [- ] to behave like true inverses. This can be avoided by treating negative propositions as affirmatives with negative predicates, as Leibniz did. The more serious difficulty is that a calculus of "concepts " is not a calculus of things in adu but only in possibile, and in a rather loose sense of the latter at that. Holland pointed this out admirably in a letter to Lambert.^^ He gives the example according to Lambert's method. All triangles are figures. ?' = tF All quadrangles are figures. Q = qF T Whence, F =-=^, or qT = tO t q 'i ■% and he then proceeds : ^^ "In general, if from A = 7nC and B = nC the conclusion nA = mB be drawn, the calculus cannot determine whether the ideas nA and mB consist of contradictory partial-ideas, as in the foregoing example, or not. The thing must be judged according to the matter." This example also calls attention to the fact that Lambert's calculus, by operations which he continuallj' uses, leads to the fallacy of the undis- tributed middle term. If "some A" is simply some further specification of the concept A, then this mode is not fallacious. And this observation brings down the whole treatment of logic as a calculus of concepts in in- tension like a house of cards. The relations of existent things cannot be determined from the relations of concepts alone. The calculus of Leibniz is more successful than any invented by his continental successors — unless Ploucquet's is an exception. That the long period between him and De Morgan and Boole did not produce a successful system of symbolic logic is probably due to the predilection for this inten- sional point of view. It is no accident that the English were so quickly successful after the initial interest was aroused; they habitually think of logical relations in extension, and when they speak of "intension" it is usually clear that they do not mean those relations of concepts which the "intension" of traditional logic signifies. 55 Deutscher Gelehrter Briefwechsel, i, Brief xxvii. K Ibid., pp. 262-63. 36 A Survey of Symbolic Logic The beginning of thought upon this subject in England is marked by the pubhcation of numerous treatises, all proposing some modification of the traditional logic by quantifying the predicate. As Sir William Hamilton notes/' the period from Locke to 1833 is singularly barren of any real con- tributions to logic. About that time, Hamilton himself proposed the quantification of the predicate. As we now know, this idea was as old at least as Leibniz. Ploucquet, Lambert, Holland, and Castillon also had quantified the predicate. Both Hamilton and his student Thomson men- tion Ploucquet; but this new burst of logical study in England impresses one as greatly concerned about its own innovations and sublimely indifferent to its predecessors. Hamilton quarrelled at length with De Morgan to establish his priority in the matter.'* This is the more surprising, since George Bentham, in his Outline of a Neiv System of Logic, published in 1827, had quantified the predicate and given the following table of propositions: 1. X in toto = Y ex parte; 2. A' in toto I I r ex parte; 3. A' in toto = Y in toto; 4. A' in toto II r in toto; 5. A' ex parte = Y ex parte; 6. A ex parte | | F ex parte; 7. A' ex parte = Y in toto; 8. X ex parte | | 1' in toto. (I I is here the sign of "diversity")- But Hamilton was certainly the center and inspirer of a new movement in logic, the tendency of which was toward more precise analysis of logical significances. Bayne's Essay on the New Analytic and Thomson's Latvs of Thought are the most considerable permanent record of the results, but there was a continual fervid discussion of logical topics in various peri- odicals; logistic was in the air. This movement produced nothing directly which belongs to the history of s,^^nbohc logic. Hamilton's rather cumbersome notation is not made the basis of operations, but is essentially only an abbreviation of language. Solly's scheme of representing syllogisms was superior as a calculus. But " See Discussions on Philosophy, pp. 119 J^. ^' This controversy, begun in 1846, was continued for many years (see various articles in the London Athen(Eum, from 1860 to 1867). It was concluded in the pages of the Con- temporary Review, 1873. The Development of Symbolic Logic 37 this movement accomplished two thmgs for symbolic logic: it emphasized in fact— though not always in name— the point of view of extension, and It aroused interest in the problem of a newer and more precise logic. These may seem small, but whoever studies the history of logic in this period will easily convince himself that without these things, symbolic logic might never have been revived. Without Hamilton, we might not have had Boole. The record of symbolic logic on the continent is a record of failure, in England, a record of success. The continental students habitually emphasized intension; the English, extension. IV. Dti, Mobgan De Morgan'^!' is known to most students of symbolic logic only through the theorem which bears his name. But he made other contributions of permanent value— the idea of the "universe of discourse ",'=° the discovery of certain new types of propositions, and a beginning of the logic of rela- tions. Also, his originality in the invention of new logical forms, his ready wit, his pat illustrations, and the clarity and hvehness of his writing did yeoman service in breaking down the prejudice against the introduction of "mathematical" methods in logic. His important writings on logic are comprised in the Formal Logic, the Syllabus of a Proposed System of Logic, and a series of articles in the Transactiotis of the Cambridge Philo- sophical Society.''^ " Augustus De Morgan (1806-78), A.B. (Cambridge, 1827), Professor of Mathematics in the University of London 1828-31, reappointed 1835; writer of numerous mathematical treatises which are characterized by exceptional accuracy, originality and clearness. Per- haps the most valuable of these is "Foundations of Algebra" (Camb. Phil. Trans., vii, viii) ; the best known, the Budget of Paradoxes. For a list of his papers, see the Royal Society Catalogue. For many years an active member of the Cambridge Philosophical Society and the Royal Astronomical Society. Father of William F. De Morgan, the novelist and poet. For a brief biography, see Monthly Notices of the Royal Astronomical Society, XII, 112. *" The idea is introduced with these words: "Let us take a pair of contrary names, as man and not-man. It is plain that between them they represent everything, imaginable- or real, in the universe. But the contraries of common language embrace, not the whole universe, but some one general idea. Thus, of men, Briton and alien are contraries: every man must be one of the two, no man can be both. . . . The same may be said of integer and fraction among numbers, peer and commoner among subjects of a realm, male and female among animals, and so on. In order to express this, let us say that the whole idea under consideration is the universe (meaning merely the whole of which we are considering parts) and let names which have nothing in common, but which between them contain the whole of the idea under consideration, be called contraries in, or with respect to, that vMwerse." {Formal Logic, p. 37; see also Camb. Phil. Trans., viii, 380.) " Formal Logic: or. The Calculus of Inference, Necessary and Probable, 1847. Here- after to be cited as F. L. 38 A Survey of Symbolic Logic Although the work of De Morgan is strictly contemporary with that of Boole, his methods and sjTnbohsm ally him rather more with his prede- cessors than with Boole and those who follow. Like Hamilton, he is bent upon improving the traditional Aristotelian logic. His first step in this direction is to enlarge the number of typical propositions by considering all the combinations and distributions of two terms. A' and Y, and their negatives. It is a feature of De Morgan's notation that the distribution of each term,^^ and the quality — affirmative or negative — of the proposition are indicated, these being sufficient to determine completely the type of the proposition. That a term A' is distributed is indicated by writing half a parenthesis ibefore or after it, with the horns turned toward the letter, thus: A'), or (A'. -An undistributed term is marked by turning the half-parenthesis the other way, thus: A'(, or )A'. A'))F, for example, indicates the proposition in "which the subject, A', is distributed and the predicate, Y, is undistributed, that is, "All A' is Y". A'()}' indicates a proposition with both terms un- distributed, that is, "Some A^ is Y".'^^ The negative of a term. A', is indi- cated by x; of y by y, etc. A negative proposition is indicated by a dot placed between the parenthetical curves; thus "Some X is not Y" will be A'(-(l'.'^'' Two dots, or none, indicates an affirmative proposition. All the different forms of proposition which De Morgan uses can be generated from two types, the universal, "All . . . is . . .," and the particular, "Some . . . is . . .," by using the four terms, X and its nega- tive, X, Y and y. For the universals we have: Syllabus of a Proposed System of Logic, 1S60. Hereafter to be cited as Syll. Five papers (the first not numbered; various titles) in Camb. Phil. Trans., viii, ix, x. The articles contain the most valuable material, but they are ill-arranged and inter- spersed with inapposite discussion. Accordingly, the best way to study De Morgan is to get these articles and the Formal Logic, note in a general way the contents of each, and then use the Syllabus as a point of departure for each item in which one is interested. "^ He does not speak of "distribution" but of terms which are "universally spoken of" or "particularly spoken of ", or of the "quantity" of a term. «3 This is the notation of Syll. and of the articles, after the first, in Camb. Phil. Trans. For a table comparing the different symbolisms which he used, see Camb. Phil. Trans., IX, 91. " It is sometimes hard to determine by the conventional criteria whether De Morgan's propositions should be classed as affirmative or negative. He gives the following ingenious rule for distinguishing them {Syll, p. 13): "Let a proposition be affirmative which is true of X and X, false of X and not-X or x; negative, which is true of X and x, false of X and X. ^ Thus 'Every X is Y' is affirmative: 'Every X is X' is true; 'Every X is x' is false. But 'Some things are neither X's nor F's' is also affirmative, though in the form of a denial: 'Some things are neither X's nor X's' is true, though superfluous in expression; 'Some things are neither A''s nor x's' is false.'' The Development of Symbolic Logic 39 (1) x))Y MIX is y. (2) x))y All not-X is not- 7. (3) X))y All Z is not-F. (4) x))Y Allnot-Zis Y. and for particulars we have : (5) XQY Some X is F. (6) xQy Some not-X is not-F, (7) XOy Some X is not-F. (8) xQY Some not-X is F. The rule for transforming a proposition into other equivalent forms may be stated as follows : Change the distribution of either term — that is, turn * its parenthetic curve the other way, — change that term into its negative, and change the quality of the proposition. That this rule is valid will appear if we remember that "two negatives make an affirmative", and note that we introduce one negative by changing the term, another by changing the quality of the proposition. That the distribution of the altered term should be changed follows from the fact that whatever proposition distrib- utes a term leaves the negative of that term undistributed, and whatever proposition leaves a term undistributed distributes the negative of that term. Using this rule of transformation, we get the following table of equivalents for our eight propositions : («) (b) (c) id) (1) X))Y = X)-iy =x{iy = xi-)Y (2) x))y =x)-iY =X((F = X(-)2/ (3) X))y =X)-(F = .T((F =x{-)y (4) x))Y =x)-iy =X((2/ = X(-)F (5) X()F = X(-(2/ =x){y = x)-)Y (6) xQy =xi-{Y =X)(F = X)-)2/ (7) XQy =X(-(F = x)(F = x)-)y (8) xQY =x{-{y =X){y = X)-)Y It will be observed that in each line there is one proposition with both terms positive, X and Y. Selecting these, we have the eight different types of propositions: 40 A Survey of Symholic Logic (la) X))Y All J is Y. (2c) X((F Some X is all Y; or, All Y is A". (36) X)-{Y NoXis Y. (4d) A'( • ) 1' Everything is either A^ or Y. (See below.) (5a) X()7 Some X is Y. (6c) X){Y Some things are neither X nor F. (See below.) {Ih) X( • (7 Some A' is not Y. (8d) X)-)Y All A' is not some Y; or, Some Y is not X. Since the quantity of each term is indicated, any one of these propositions may be read or written backwards — that is, with Y subject and X predicate — provided the distribution of terms is preserved. (4d) and (6c) are diffi- cult to understand. We might attempt to read X{-)Y "Some X is not some }'", but we hardly get from that the diiference between X{-)Y and ' A'(-(F, "Some X is not (any) F". Also, A'(-)F is equivalent to uni- versals, and the reading, "Some A' is not some F", would make it par- ticular. A'(-)F is equivalent to a-)) F, "All not-X is F", and to x)-{y, "Xo not-A' is not-F". The only equivalent of these with the terms A' and Y is, "Everything (in the universe of discourse) is either X or F (or both)". (6c), X){Y, we should be Hkely to read "All X is all F", or "Xand Fare equivalent"; but this would bean error,^^ since its equivalents are particular propositions. (6a), xQy, is "Some not-A' is not-F". The equivalent of this in terms of X and F is plainly, "Some things are neither A' nor F". Contradictories^" of propositions in line (1) will be found in line (7); of those in line (2), in line (8); of line (3), in hue (5); of line (4), in line (6). We give those with both terms positive : (la) X))F contradicts (76) X(-(F (2c) X((F " (8d) X)-)F (36) X)-(F " (5a) XQF m X( )F " (6c) X)(F « An error into which it might seem that De Morgan himself has fallen. See e. g., Syll., p. 25, and Camb. Phil. Trans., ix, 98, where he translates X){Y by "All X is all Y ", or "Any one X is any one Y ". But this belongs to another interpretation, the "cumular", which requires X and Y to be singular, and not-Z and not-F will then have common members. However, as we shall note later, there is a real difficulty. ««De Morgan calls contradictory propositions "contraries" (See F. L., p. 60; Syll., p. 11), just as he calls terms which are negative's of one another "contraries". The Development of Symbolic Logic 41 Thus the rule is that two propositions having the same terms contradict one another when one is affirmative, the other negative, and the distribution of terms is exactly opposite in the two cases. The rule for transforming propositions which has been stated and exemplified, together with the observation that any symbolized proposition may be read or written backwards, provided the distribution of the terms be preserved, gives us the principles for the immediate inference of uni- versal from universals, particulars from particulars. For the rest, we have the rule, "Each universal affirms the particulars of the same quahty".^' For syllogistic reasoning, the test of vahdity and rule of inference are as follows: ''" "There is inference : 1. When both the premises are universal; 2. When, one premise only being particular, the middle term has different quantities in the two premises. "The conclusion is found by erasing the middle term and its quantities [parenthetic curves]." This rule of inference is stated for the special arrangement of the syllogism in which the minor premise is put first, and the minor term first in the premise, the major term being the last in the second premise. Since any proposition may be written backward, this arrangement can always be made. According to the rule, X))Y, "All A' is 7", and Y) ■ {Z, "No YisZ", give X) ■ (Z, "No X is Z". X) ■ (!', "No X is r", and 7(- (Z, "Some Y is not Z", give X) • ■ (Z, or X) (Z, which is "Some things are neither X nor Z." The reader may, by inventing other examples, satisfy himself that the rule given is sufficient for all syllogistic reasoning, with any of De Morgan's eight forms of propositions. De Morgan also invents certain compound propositions which give com- pound syllogisms in a fashion somewhat analogous to the preceding: " "1. X)-0)ForbothX;)).FandX)-.)y All X's and some things be- sides are F's.' 2. A'||7orbothX))FandX((F All X's are F's, and all F's are X's. 3. X(0 (F or both X((F and A'C- (-F Among X's are all the F's and some things besides. 4. X) O (For both X)-(F and X)(F Nothing both X and F and some things neither. 6' Syll., p. 16. ■ 88 Syll., p. 19. 69 Syll., p. 22. 42 A Survey of Symbolic Logic 5. A'l • I }' or both .Y)-(rand A'(-)r Nothing both A' and Y and everything one or the other. 6. A'(0)i' or both X(-)Y and A'()}' Everj'thing either A' 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: '" "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 [O]- Thus X)0(Y{0)Z gives X)0)Z: or if nothing be both A' and Y and some things neither, and if everything be either Y or Z and some things both, it follows that all A' 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 AT's and the remainder P's; 5 are Y's and the remainder Q's; 7 are Z's and the remainder R's. XXX P P PP PPPPP YYY YY QQ QQQQQ Z Z Z Z Z Z Z RRRRR We can thus verify the eight complex syllogisms X)0)Y)0)Z P{0)Y)0)Z PiO{Q{0)Z P(0(Q(0(ii P{0)Y)OiR X)0)Y)OiR X)OiQ{o{R X)0{q{0)Z ■"> SylL, p. 23. 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 mference from universals to particulars nor observe the problems there involved.'! 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 propositional 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 7 "." 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." 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: '^ '' 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". '2 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-F, every Y 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. " In one passage (Camb. Phil. T^ans., x, 183) he suggests that the relation of two premises to their conclusion should be symbolized by A B < C. ''* 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 sufBcient 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 loMch is B' means 'A is B'; and '^ 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 — F represent X and Y connected by a transitive copula, Camestres in the second figure is valid, as in Every Z—r, Xo X— F, therefore No A'— 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 F F has existence. Every X is F A' has existence. Every A' — F — is a transitive relation, a of A' — F a is a fraction < or = 1. (Probabihty (3) a of A' — F iS 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.'^ "... I hold the supreme form of the syllogism of one middle term to be as follows: There is the probability a. that A' is in the relation L to F; there is the probability (3 that F is in the relation If to Z; whence there is the probabihty a/S that A' is in the relation L of ilf to Z.'^ "... 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." " ™ Ihid., pp. 177-78. ™ Ihid., p. 339. " lUd., 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/^ 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." " For the logic of relations. A', Y, and Z will represent the class names; L, M, N, relations. X . . LY will signify that A' is some one of the objects of thought which stand to Y in the relation L, or is one of the i's of Y.^° X . LY will signify that A' 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 X'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. A' . . LY and Y . . MZ together give X . . LM Z, but X . . LY and X . . MY 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 Y". (Do not say hastily, " X is uncle of Y". "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 Y.^^ 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 " See ibid., p. 190. " 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." »" 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 ^-1 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- i 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 ilf 's of X: L*M*, an L of every M and of none but M's: LMX*, an L of an if of every X, and so on. Two more symbols are needed. The converse of L is sjTnbolized by L"^. If L is "lover of", i~^ is "beloved of"; if L is "aunt", L~^ is "niece or nephew". The contrary (or as we should say, the negative) of L is symbol- ized by I; 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, i~\ is contained in the converse of M, M~^; and (b) 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)~^ = M^^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*)-^ = M*-^L-K If A' 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* = Urn and L*M = Im*. 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-^ IM* or L*M M*-H-'- ot m-'L-'* LM* or l*m M^-^L'^ or m-^H* IM M'H'' L*M or hn* M-^L-^* or m*-iH Lm m-^L-^ 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 Y. 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 Y is also the ancestor of Y. 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 m))l. Applying this theorem to compound relations, we have: (8') LM))N gives n))lM* and n))L*m. (8") If LM))N, then L-'n))vi and nM-'))l. Proof: li LM))N, then n))lM*. Whence niif-'))Of*M-'. But an Z of 48 A Survey of Symbolic Logic every M of an M'^ of Z must be an I of Z. Hence nM-'-))l. Again; if LM))N, then n))L*m. Whence L-^n))L~^L*vi. But whatever has the relation converse-of-i to an L of none but m's must be itself an m. Hence L~^n})m. De ^lorgan calls this "theorem K" from its use in Baroko and Bokardo. (9) If LM = X, then L))NM-' and M))L-^N. Proof: If LM = N, then LMM-' = NM^^ and L^^LM = L-^N. Now for any A', il/J/-^A' and L~^LX are classes which contain X; hence the theorem. We do not have L = NM~^ and il/ = L-^N, because it is not generallj^ true that MM^^X — X and L-^LX = A'. For example, the child of the parent of A' may not be A' but A"s brother: but the class "children of the parent of A'" will contain X. The relation 31 M~^ or M~^3I will not always cancel out. MM~' and M~^M are always symmetrical relations; if XMM~^ Y then YMM'^^X. If X is child of a parent of Y, then Y is child of a parent of A'. But MM~^ and M~^M are not exclusively reflexive. A'i!/il/~'A' does not always hold. If we know that a child of the parent of A is a celebrated, linguist we may not hastily assumie that A' is the linguist in question. With reference to transitive relations, we may quote : ^^ "A relation is transitive when a relative of a relative is a relative of the same kind; as sjTnbolized in LL))L, whence LLL))LL))L; and so on. "A transitive relation has a transitive converse, but not necessarily a transitive contrary: for L~^L-^ is the converse of LL, so that LL))L gives L~^L~^))L~'-. From these, by contraposition, and also by theorem K and its contrapositions, we obtain the following results: L is contained in LL~^*, hl"^, l~H*, L*~'^L i-i i*i-i, IH*,\-H, L-^L* I IL*,L*1 1-' Lt-H~\ l-^L-^* LL L L-^L-^ L-i L-H, IL~^ I Ll-\ l-'L /-I "I omit demonstration, but to prevent any doubt about correctness of printing, I subjoin instances in words : L signifies ancestor and L-^ descendent. 82 Camh. Phil. Trans., x, 346. For this discussion of transitive relations, De Morgan treats all reciprocal relations, such as XLL~'-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 nfan-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. Anaong 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 : ^^ 1 2 3 4 A'. ..LY X . LY X . ..LY X . LY I y . .iz"- Y . .MZ Y . MZ Y .MZ A', . . 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-'Z X . . . LM-^Z X . . . Lm-'Z X . . Im-'Z Y . ..LX Y . .LX Y . ..LX Y . LX III Y .MZ Y . . MZ Y ..MZ Y . MZ X . . L-hnZ X . . l-^MZ X . . L-^MZ X . . l-hnZ Y . .LX Y . . . LX Y .LX Y ..LX IV z . MY Z. MY Z. .MY Z..MY X . . Hm-'Z X . . L-^m-'Z X . . l-'M-^Z X . . L-'M-'Z 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 A' . IM*Z and X . L*mZ; X . . IM-'Z gives A^ . LM-'*Z and X . Um-^Z, and so on for each of the others. 83 IhU., p. 350. o 50 A Survey of Symbolic Logic When the copula of all three propositions is limited to the same traiisitive relation, L, or its converse, the table of syllogisms will be: ^* X..LY X.LY X..LY I Y..LZ Y..L~'Z Y.L-'Z X . . LZ X . LZ X . L-^Z A' .LY X ..LY X ..LY II Z . .LY Z . .L-^Y Z. LY X .LZ X ..LZ X .L-^Z Y . . : LX Y . .LX Y ..LX III Y . .LZ Y . ..LZ Y . . . L~^Z X .LZ X .L-^Z X . . L-^Z Y ..LX Y .LX Y . .LX IV Z . X L-^Y .LZ Z. X ..L-^Y .L-^Z Z . X . .LY .L-'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, Aristotehan 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 work of Frege and Peano, and more especially of Whitehead and Flussell, 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 : ^^ ^ lUd., p. 354. 85 lUd., p. 358. The Developnient 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 symboHzed. 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 principium et exemplum into imita- tion of Ovmis homo est animal, Sojr^s est homo, etc., yet the algebraist was Hving in the higher atmosphere of syllogism, the unceasing composition of relation, before it was admitted that such an atmosphere existed." ^"^ V. Boole The beginning from which symbolic logic has had a continuous develop- ment is that made by George Boole. ^^ 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 work 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 "> I 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., viii, 384-87, and 393^05.) 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, viii; and Camb. Phil. Trans., x, *355-*358) is also omitted. *' 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 ^lorg&n's Formal Logic.^^ 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 Laivs of Thought, on ivhich are founded the Mathe- matical Theories of Logic and Probability, published in 1854.*^ 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 1."" 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 1-x or x; the operation of electing all the l'"s is similarly represented by 1-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- matics — ;r becomes, in practice, the symbol for the class of all the A''s. Thus .T, 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. l-x-y or xy will represent the operation of electing, first, all the A"s in the "universe", and from this class by a second operation, all the Y's. The result of these two operations will be the class whose members are both A"s and ]"s. Thus xy is the class of the common members of z and y; xyz, the class of those things which belong at once to x, to y, and to z, 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 inx + y must have no common members. The relation may be read, "that which ss See De Morgan's note to the article "On Propositions Numerically Definite", Camh. Phil. Trans., xi (1871), 396. *" London, Walton and Maberly. '» 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 ordmary 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 ",9i 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 — z. li 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, symbohzes 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 - 2/) + (1 - x)y. The first term of this sum is the class which are both x's and y's; the second, those which ard x's but not y's; the third, those which are y's but not a;'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 a;(l - 2/) + (1 - 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 2/ is either y or x. 91 See L. 0} 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 a; or both z and y. (4) z{x — y) = zx — zy That which is both z and {x but not y) is both z and a; but not both z and ?/. (5) li x' = y, then 2a; = zy z + X = z + y x — z = y — z (6) X — 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 .r, upon 1, be repeated, the result is not altered by the repetition : 1-x = 1-x-x = \-x-x-x. . .. Hence we have : (7) ,T^ = X Boole calls this the "index law".^^ All these laws, except (7), hold for numerical algebra. It may be noted that, in logic, "If x = y, then 2a; = zy" is not reversible. At first glance, this may seem to be another difference between numerical algebra and the system in question. But "If 2a; = 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^ = 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. '^ 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''.^^ 92 In Mathematical Analysis of Logic he gives it also in the form x» = x, but in L. of T. he avoids this, probably because the factors of .-k" — x (e. g., x^ — x) are not always logically interpretable. "3 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. 9« L. of T., p. 49. The Development of Symbolic Logic 55 Since x^ = x, x — x'' = 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'^, is equivalent to x; therefore Avhat is both X and not-a; 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^ = 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; ^^ s5 L. of T., pp. 37-38. 56 A Survey of Symbolic Logic hence he appHes 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."^^ 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 symboHsm will now be entirely clear. Classes will be represented by x, y, z, etc.; their negatives, by (1 — .t), (1 — y), etc. That which is both x and y will be xy; that which is x but not y will be x{l — y), etc. That which is x or y but not both, will be x + y, or a;(l — y) + (1 — x)y. That which is x or y or both will be a; + (1 — x)y — i. e., that which is x or not x but y — or xy + x{l - 2/) + (1 - x)y — that which 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 = 0: the only class which 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 anj^ 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 y". 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 lo, to represent an indefinite »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"." The universal afiirmative can also be expressed, without this symbol for the indeterminate, as a;(l — y) = ; " All a; is ?/ " means " That which is x but not y is nothing ' ' . Negative propositions are treated as afiirmative propositions with a negative predi- cate. So the four typical propositions of traditional logic are expressed as follows: ^^ All x is y: x = vy, or, x{l — y) = 0. No X is y: x = v{l —jy), or xy = 0. Some x'lsy: vx = «<(1 — ?/)) M or, v = xy. Some x is not y: lu- = w{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: ^' " 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 iliachinery by " 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 ot 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. " 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.^"" 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 + 5(1 — 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 maj^ be some number other than or 1, or involve such a number. If the function, as given, does not have the form Ax + 5(1 — x), it may be put into that form by observing certain interesting laws which govern coefficients. Let fix) = Ax + Bil - x) Then ,: /(I) = A-1 + B^l - 1) = A And fiO) = A-0 + B{l -0) = B Hence f{x) = /(l)-a. +/(0) •(! - x) 1 + x Thus if fix) = 2- x' /(I) =^1 = 2; m 1 + 2-0 1 "2 fix) =2.x-+ J(l -X) Hence A developed function of two variables, x and y, will have the form: Axy-t-BxH -y) + C'H - x)y + Dil - .x-)(l - y) And for any function, fix, y), the coefficients are determined by the law: fix, y) =/(l, 1)-.T2/+/(1, 0)-a;(l - 2/) +/(0, 1)-(1 - x)y + /(0, 0)-(l -x)il -y) i»» See Math. An. of Logic, pp. 60-69; L. of T., pp. 71-79; "The Calculus of Logic," Camhridge and Dublin Math. Jour., iii, 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) = a-1 + 26-1 = a + 2b /(I, 0) = a-l + 2b-0 = a /(O, 1) = a-0 + 26-1 = 2b /(O, 0) = a-0 + 26-0 = Hence f{x, y) = {a + 2b)xy + axil - 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:"' "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 coefiicients. It is as follows : "1st. To expand any function of the symbols x, y, z — 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 I — 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 x as a factor, change in the original function x into 1; but if it involves 1 — a; 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, 1 — y, 1 — z. Thus the product of the two will have a factor of the form .^(1 — x). And where a; 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 i»i L. of T., pp. 75-76. 60 A Survey of Symbolic Logic (2) by an elimination, to discover what statements (equations), which are inde'pendent of some term .r, 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{1 — a-) = 0, or, Axyi-Bx{l - y) + C{l - x)y+Dil - x){l - 2j) = 0, etc. It is an important property of such equations that, since the product of any two constitiients 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 - F = 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 2 as a function of x and y. X = yz gives 2; = x/y; but this form is not interpretable. We shall, then, The Development of Symbolic Logic 61 develop x/y by the law M y) = fil, 1) -xy +/(1, 0) •a;(l - y) +/(0, 1) • (1 - x)y + f(0,0).(l-x)il-y) By this law : If z = - . then y z = xy+ -a:(l - 2/) + 0(1 - .t)?/ + -(1 - a')(l - y) These fractional coefEcients represent the sole necessary difference of Boole's methods from those at present familiar — a difference due to the presence of division in his system. Because any function can always be de- veloped, and the difFerence between any two developed functions, of the same variables, is always confined to the coefficients. If, then, we can interpret and successfully deal with such fractional coefEcients, one of the difficulties of Boole's system is removed. The fraction 0/0 is indeterminate, and this suggests that a proper inter- pretation of the coefficient 0/0 would be to regard it as indicating an unde- termined portion of the class whose coefficient it is. This interpretation may be corroborated by considering the symbolic interpretation of "All x is y", which is a;(l — y) = 0. If x{l — y) = 0, then x — xy = and x = xy. Whence y = x/x. Developing x/x, we have y = x+ -r (1 — a;). If "All X is y", the class y is made up of the class x plus an undetermined portion of the class not-x. Whence 0/0 is equivalent to an arbitrary parameter v, which should be interpreted as "an undetermined portion of" or as "All, some, or none of". The coefficient 1/0 belongs to the general class of symbols which do not obey the "index law", x^ = x, or its equivalent, the "law of duality", x{l — x) =0. At least Boole says it belongs to this class, though the numerical properties of 1/0 would, in fact, depend upon laws which do not belong to Boole's system. But in any case, 1/0 belongs with the class of such coefficients so far as its logical interpretation goes. Any constituent of a developed function which does not satisfy the index law must he separately equated to 0. Suppose that in any equation w = At + P 62 A Survey of Symbolic Logic w be a "logical sj'mbol", and t be a constituent of a developed function whose coefficient A does not satisfy the index law, A^ = A. And let P be the sum of the remaining constituents whose coefficients do satisfy this law. Then w'- = w, f = t, and P^ = P Since the product of any two constituents of a development is 0, m;2 = {At + Py = AH'' + P2 Hence iv = AH + P Subtracting this from the original equation, {A - A')t = = A(l - A)t Hence since ^(1 — ^) 4= 0, t = Hence any equation of the form w = P + OQ+ ^R+ ^S is equivalent to the two equations ID = P+ vR and S = which together represent its complete solution. It will be noted that a fraction, in Boole's algebra, is always an am- biguous function. Hence the division operation must never be performed: the value of a fraction is to be determined by the law of development only, except for the numerical coefficients, which are elsewhere discussed. We have already remarked that ax = bx does not give a = b, because x may •have the value 0. But we may transform ax = bx into a = bx/x and determine this fraction by the law f{b,x) =/(l, l)-6.r+/(l,0)-6(l -.T)+/(0, 1)-(1 -b)x + ,f(0, 0)-(l -6)(1 -x) V\e shall then have bx a= -=bx+-b{l-x)+~{l - 6)(1 - X) and this is not, in general, equivalent to b. Replacing 0/0 by indeterminate coefficients, v and w, this gives us. If ax = bx, then a = bx + v-b{l — x) + iu • ''1 — 6) (1 — x) The Development of Symbolic Logic 63 And this result is always valid. Suppose, for example, the logical equation rational men = married men and suppose we wish to discover who are the rational beings. Our equation will not give us rational = married but instead rational = married men + v ■ married non-men + to ■ non-married non-men That is, our hypothesis is satisfied if the class "rational beings" consist of the married men together with any portion (which may be null) of the class "married women" and a similarly undetermined portion of the class "unmarried women". If we consider Boole's system as an algebra of and 1, and the fact that for any fraction, x/y, X 1 ~ = xy+ -x{l - 2/) + ^ (1 - x){l - y) we shall find, by investigating the cases (1) a; = 1 and y = I; (2) x = 1, y = 0; (3) x = 0, y = I; and (4) x = 0, y = 0, that it requires these three possible cases : (1) " : 1 (2) 0~ : (3) " 1 ■0 Or, to speak more accurately, it requires that 0/0 be an ambiguous function susceptible of the values and 1. Since there are, in general, only four possible coefficients, 1, 0, 0/0, and such as do not obey the index law, of which 1/0 is a special case, this means that any equation can be interpreted, and the difficulty due to the presence of an uninterpretable division operation in the system has disappeared. And any equation can be solved for any "logical symbol" x, by trans- ferring all other terms to the opposite side of the equation, by subtraction or division or both, and developing that side of the equation. Any equation may be put in the form in which one member is by 64 A Survey of Symbolic Logic bringing all the terms to one side. When this is done, and the equation fully expanded, all the coefBcients which do not vanish may be changed to unity, except such as already have that value. Boole calls this a "rule of interpretation ".'"2 Its validity follows from two considerations: (1) Any constituent of an equation with one member 0, whose coefHcient does not vanish in development, may be separately equated to 0; (2) the sum of the constituents thus separately equated to will be an equation with one member in which each coefficient will be unity. Negative coefficients may be eliminated by squaring both sides of any equation in which they appear. The "logical symbols" in any function are not altered by squaring, and any expression of the form (1 — x), where X is a "logical sj'mbol", is not altered, since it can have only the values and 1 . Hence no constituent is altered, except that its coefficient may be altered. And any negative coefficient will be made positive. No new terms will be introduced by squaring, since the product of any two terms of a developed function is always null. Hence the only change effected by squaring any developed function is the alteration of any negative coef- ficients into positive. Their actual numerical value is of no consequence, because coefficients other than 1 can be dealt with by the method described above. For reducing any two or more equations to a single equation, Boole first proposed the "method of indeterminate multipliers","^ by which each equation, after the first, is multiplied by an arbitrary constant and the equations then added. But these indeterminate multiphers complicate the process of elimination, and the method is, as he afterward recognized, an inferior one. More simply, such equations may be reduced, by the methods just described, to the form in which one member is 0, and each coefficient is 1 . They may then be simply added ; the resulting equation will combine the logical significance of the equations added. Any "logical symbol" which is not wanted in an equation may be ehminated by the method which is famihar to all students of symbohc logic. To eliminate x, the equation is reduced to the form Ax + B{l - x) = The result of elimination will be"* AB =0 i»2 L. of T., p. 90. !»' See Math. An. of Logic, pp. 78-81; L. of T., pp. 11.5-120. "" See L. of T., p. 101. We do not pause upon this or other matters which will be entirely clear to those who understand current theory. The Development of Syviholic Logic 65 By these methods, the difference between Boole's algebra and the classic algebra of logic which grew out of it is reduced to a minimum. Any logical results obtainable by the use of the classic algebra may also be got by Boole's procedures. The difference is solely one of ease and mathematical neatness in the method. Two important laws of the classic algebra which do not appear among Boole's principles are: (1) x + x = X, and (2) x = x + xy These seem to be inconsistent with the Boolean meaning of + ; the first of them does not hold for a: = 1; the second does not hold iov x = I, y = I. But although they do not belong to Boole's system as an abstract algebra, the methods of reduction which have been discussed will always give x in place of X + a; or of a; + xy, in any equation in which these appear. The expansion of x + x gives 2x; the expansion of x + xy gives 2xy + x{l — y). By the method for dealing with coefficients other than unity, 2x may be replaced in the equation by x, and 2xy + x{l — y) by xy + .r(l - y), which is equal to x. The methods of applying the algebra to the relations of logical classes should now be sufficiently clear. The application to ■propositions is made by the famihar device of correlating the "logical symbol", x, with the times when some proposition. A', is true, xy w'll represent the times when A' and Y are both true; a,'(l — y), the times when A' is true and 1' is false, and so on. Congruent with the meaning of + , .t + y will represent the times when either X or Y (but not both) is true. In ord'-r to symbolize the times when A' or Y or both are true, we must write x + il — x)y, or xy + x{\ — y) + {1 — x)y. 1, the "universe", will represent "all times" or "always"; and will be "no time" or "never", x = 1 will represent " X is always true"; x = ov {I — x) = 1, "A' is never true, is always false". Just as there is, with Boole, no symbol for the inclusion relation of classes, so there is no symbol for the implication relation of propositions. For classes, "All A' is 1'" or "A' is contained in Y" becomes x = vy. Cor- respondingly, "All times when X is true are times when Y is true" or "If A' then }'" or "A' implies Y" is x = vy. x = y will mean, "The times when A' is true and the times when Y is true are the same" or "A' implies Y and Y implies A'". The entire procedure for "secondary propositions" is summarized as follows :"= 105 L. of T., p. 178. 6 66 A Survey of Symbolic Logic "Rule. — Express symbolically the given propositions. . . . " Eliminate separately from each equation , in which it is found the indefinite symbol v. "Eliminate the remaining symbols which it is desired to banish from the final solution: always before ehmination, reducing to a single equation those equations in which the symbol or symbols to be eliminated are found. ^ Collect the resulting equations into a single equation [one member of which isO], V = 0. "Then proceed according to the particular form in which it is desired to express the final relation, as 1st. If in the form of a denial, or system of denials, develop the function V, and equate to all those constituents whose coefficients do not vanish. 2ndly. If in the form of a disjunctive proposition, equate to 1 the sum of those constituents M'hose coefficients vanish. 3rdly. If in the form of a conditional proposition having a simple element, as a; or 1 — .r, for its antecedent, determine the algebraic expression of that element, and develop that expression. 4thly. If in the form of a conditional proposition having a com- pound expression, as xy, xy-t- {I — x){l — y), etc., for its antecedent, equate that expression to a new symbol t, and determine t as a developed function of the symbols which are to appear in the consequent. . . . 5thly. ... If it only be desired to ascertain whether a particular elementary proposition .r is true or false, we must eliminate all the symbols but x; then the equation x = 1 will indicate that the proposi- tion is true, x = that it is false, = that the premises are insuf- ficient to determine whether it is true or false." It is a curious fact that the one obvious law of an algebra of and 1 which Boole does not assume is exactly the law which would have limited the logical interpretation of his algebra to propositions. The law li X =^ 1, X = and if a; =1= 0, a: = 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 Developvient of Symbolic Logic 67 preted "a; 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 " propositional 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 "propo- sitional symbols" in the application of the algebra to probability theory gives them the character of "propositional 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 .1"'' 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 ^'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 ^'s are also B's and (2) the C's consist of the .^I's which are not B's together with the B's which are not .4's. Required: the probabihty that if one of the A's be taken at random, it will also be a C. By premise (2) C = A{1 - J3) + (l - A)B Since A, B, and C are "logical symbols", A^ - A and /1(1 — A) = Q. Hence, AC = A'(l - B) + A(l - A)B = ^(1 - B). The A's which are also C's are identical with the A's which are not -B's. Thus the probability that a given A is also a C is exactly the probabihty that it is not a.B\ or by premise (1), 1 — mjn, 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 i»« Chap. l&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, 5^7 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., xi, 396-411. 6S 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 .r represent the class of cases (in a given total) in which the event A' occurs — or those which "are favorable to" the occurrence of A'.^"^ And let p be the probabihty, numerically expressed, of the event A'. 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 A' — then p = 1 — the probability of A' is "certainty". If a; = 0, then j) = 0. Further, the class of cases in which A' does not occur, will be expressed by 1 — x; the probability^ that A' will not occur is the numerical 1 — p. Also, .T + (1 — x) = 1 and p+ (l — p) = 1- This parallelism extends likewise to the combinations of two or more events. If x represent the class of cases in which A' occurs, and y the class of cases in which }' occurs, then xy will be the class of cases in which A' and Y both occur; x{l — y), the cases in which A' occurs without Y; (1 — x)y, the cases in which Y occurs without A'; (1 — x){l — y), the cases in which neither occurs; .r(l — y) + y{l — x), the cases in which A' or Y occurs but not both, and so on. Suppose that A' and Y are " simple " and "independent" events, and let p be the probability of X, q the prob- abihty 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 - .t)(1 -y) (1 - p)(l - q) x{l - ?/) + (1 - x)y p{l - g) + (1 - p)q Etc. etc. In fact, this paralleHsm is complete, and the following rule can be formulated: ^"^ 1°' Boole prefers to consider x as representing the times when a certain proposition, asserting an occurrence, will be true. But this interpretation comes to exactly the same thing. 108 L. of T., p. 258. The Development of Symbolic Logic 69 'li p, q, r, . . . are the respective probabilities of unconditioned simple events, x, y, z, . . . , the probability of any compound event V will be [V], this function [V] being formed by changing, in the function V, the symbols X, y, z, . . . into p, q, r, . . . . "According to the well-known law of Pascal, the probability that if the event T' occur, the event V will occur with it, is expressed by a fraction whose numerator is the probabihty of the joint occurrence of V and V, and whose denominator is the probability of the occurrence of V. We can then extend the rule just given to such cases: "The probability that if the event V occur, any other event V will [V V] also occur, will be , where [V V] denotes the result obtained by multiplying together the logicail functions V and V, and changing in the result X, y, z, . . . into p, q, r, . . . ." The inverse problem of finding the absolute probability of an event when its probability upon a given condition is known can also be solved. Given: The probabilities of simple events x, y, z, . . . are respectively p, q, r, . . . when a certain condition V is satisfied. To determine : the absolute probabilities I, m, n, . . . oi x, y, z, . . . . By the rule just given, m^ \yV]^ [zV]^ . [V] ^' [V] '^' [V] *■' And the number of such equations will be equal to the number of unknowns, I, m, n, . . . to be determined. ^°^ The determination of any logical expres- sion of the form xV is peculiarly simple since the product of x into any developed function V is the sum of those constituents of V which contain x as a factor. For example: if F = xyz+x{l — y)z+ (1 - x)y{l — 2) + (1 - a;)(l - y)z, xV = xyz + x{l — y)z yV = xyz+il - x)y{l - z) zV = xyz + x{l — y)z+ (1 — x){l - y)z Thus any equation of the form [xV] [V = V i»'On certain difficulties in this connection, and their solution, see Cayley, "On a Question in the Theory of Probability" (with discussion by Boole), Phil. Mag., Ser. iv, XXIII (1862), 352-65, and Boole, "On a General Method in the Theory of Probabilities'', ibid., XXV (1863), 313-17. 70 A Survey of Symbolic Logic is readily determined as a numerical equation. Boole gives the following example in illustration: '^° "Suppose that in the drawings of balls from an urn, attention had only been paid to those cases in which the balls drawn were either of a particular color, 'white,' or of a particular composition, 'marble,' or were marked by both of these characters, no record having been kept of those cases in which a ball which was neither white nor of marble had been drawn. Let it then have been found, that whenever the supposed condition was satisfied, there was a probability p that a white ball would be drawn, and a probability q that a marble ball would be drawn: and from these data alone let it be required to find the probabihty m that in the next drawing, without refer- ence at all to the condition above mentioned, a white ball will be drawn; .also a probability n that a marble ball will be drawn. "Here if x represent the drawing of a white ball, y that of a marble ball, the condition V will be represented by the logical function xy + x{l - y)+ (1 - x)y Hence we have Whence xV = xy + x{l — y) = x yV = xy +{1 - x)y = y [xV] = m, [yV\ = n and the final equations of the problem are mn + m(l — n) + (1 — m)n n mn + m{\ — n) + (1 — m)n = g from which we find f* q — I ]) + q — 1 m = , n = q p ... To meet a possible objection, I here remark that the above reasoning does not require that the drawings of a white and a marble ball should be independent, in virtue of the physical constitution of the balls. "In general, the probabilities of any system of independent events being given, the probability of any event X may be determined by finding a logical equation of the form X = A+QB+^C+^D "" L. of T., p. 262. I have slightly altered the illustration by a change of letters. The Development of Symbolic Logic 71 where A, B, C, and D are functions of the symbols of the other events. As has already been shown, this is the general type of the logical equation, and its interpretation is given by X = A + vC, where v is arbitrary and D = By the properties of constituents, we have also the equation, A+B+C+D = 1 and, since D = 0, A+B+C = 1 A+ B + C thus gives the 'universe' of the events in question, and the prob- abilities given in the data are to be interpreted as conditioned hy A +B + C = 1, since D = is the condition of the solution x = A + vC. If the given probabihty of some event S is p, of T is q, etc., then the supposed 'absolute' probabilities of S, T, etc., may be determined by the method which has been described. Let V = A+B+C, then [sV] [tV] where [sV], [tV], etc. are the "absolute probabilities" sought. These, being determined, may be substituted in the equation rrob. w = — z=- — which will furnish the required solution. " The term vC will appear only in cases where the data are insufficient to determine the probability sought. Where it does appear, the limits of this probability may be determined by giving v the limiting values, and 1. Thus Lower limit of Prob. iv = -^ ,. . [A + C]" Upper limit = — Ty. — With the detail of this method, and with the theoretical difficulties of its appHcation and interpretation, we need not here concern ourselves. Suffice it to say that, with certain modifications, it is an entirely workable method and seems to possess certain marked advantages over those more generally in use. It is a matter of surprise that this immediately useful appHcation of symbolic logic has been so generally overlooked. 72 A Survey of Symbolic Logic VI. Jevons It has been shown that Boole's "calculus of logic" is not so much a system of logic as an algebra of the numbers and 1, some of whose ex- pressions are capable of simple interpretation as relations of logical classes, or propositions, and some of whose transformations represent processes of reasoning. If the entire algebra can, with sufficient ingenuity, be inter- preted as a system of logic, still Boole himself failed to recognize this fact, and this indicates the difficulty and unnaturalness of some parts of this interpretation. Jevons"' pointed a way to the simplification of Boole's algebra, dis- carding those expressions which have no obvious interpretation in logic, and laying down a procedure which is just as general and is, in important respects, superior. In his first book on this subject, Jevons says: "^ "So long as Professor Boole's system of mathematical logic was capable of giving results beyond the power of any other system, it had in this fact an impregnable stronghold. Those who were not prepared to draw the same inferences in some other manner could not quarrel with the manner of Professor Boole. But if it be true that the system of the foregoing chapters is of equal power with Professor Boole's system, the case is altered. There are now two systems of notation, giving the same formal results, one of which gives them with self-evident force and meaning, the other by dark and symbolic processes. The burden of proof is shifted, and it must be for the author or supporters of the dark system to show that it is in some way superior to the evident sj^stem." He sums up the advantages of his system, compared with Boole's, as follows : "^ " 1 . Every process is of self-evident nature and force, and governed by laws as simple and primary as those of Euclid's axioms. "2. The process is infalhble, and gives us no uninterpretable or anom- alous results. "3. The inferences may be drawn with far less labor than in Professor Boole's system, which generally requires a separate computation and development for each inference." 111 William Stanley Jevons (1835-1882), B.A., M.A. (London), logician and economist; professor of logic and mental and moral philosophy and Cobden professor of political economy in Owens College, Manchester, 1866-76; professor of political economy, Uni- versity College, London, 1876-80. 112 Pure Logic, or the Logic of Quality apart from Quantity, p. 75. 112 lUd., p. 74. The Development of Symbolic Logic 73 The third of these observations is not entirely warranted. Jevons unduly restricts the operations and methods of Boole in such wise that his own procedure is often cumbersome and tedious where Boole's would be expeditious. Yet the honor of first pointing out the simplifications which have since been generally adopted in the algebra of logic belongs to Jevons. He discards Boole's inverse operations, a — h and ajh, and he interprets the sum of a and h as "either a or h, where a and h are not necessarily exclusive classes". (We shall symbolize this relation by a + 6: Jevons has A + B OT A-\ ■B.y^'^ Thus, for Jevons, a+ a = a, whereas for Boole a+ a is not interpretable as any relation of logical classes, and if it be taken as an expression in the algebra of and 1, it obeys the usual arithmetical laws, so that a + a = 2a. As has been indicated, this is a source of much awk- ward procedure in Boole's system. The law a-¥ a = a eliminates numerical coefficients, other than and 1, and this is a most important simplification. Jevons supposes that the fundamental difference between himself and Boole is that Boole's system, being mathematical, is a calculus of things taken in their logical extension, while his own system, being "pure logic", is a calculus of terms in intension. It is true that mathematics requires that classes be taken in extension, but it is not true that the calculus of logic either requires or derives important advantage from the point of view of intension. Since Jevons's system can be interpreted in extension without the slightest difficulty, we shall ignore this supposed difference. The fundamental ideas of the system are as follows: (1) a b denotes that which is both a and b, or (in intension) the sum of the meanings of the two terms combined. (2) a+b denotes that which is either a or b or both, or (in intension) a term with one of two meanings."'^ (3) a = b a is identical with b, or (in intension) a means the same as b. (4) -b Not-&, the negative of b, symbolized in Boole's system by 1 - b.'^" (5) According to Jevons, indicates that which is contradictory or "excluded from thought". He prefers it to appear as a factor rather than 1" A + B in Pure Logic; A' \' B in the other papers. (See Bibl.) 115 Jevons would add "but it is not known which". (See Pure Logic, p. 25.) But this is hardly correct; it makes no difference if it is known which, since the meaning of a + h does not depend on the state of our knowledge. Perhaps a better qualification would be "but it is not specified which". no Jevons uses capital roman letters for positive terms and the corresponding small italics for their negatives. Following De Morgan, he calls A and a "contrary" terms. 74 A Survey of Symbolic Logic by itself.^" The meaning given is a proper interpretation of the symbol in intension. Its meaning in extension is the null-class or "nothing", as with Boole. Jevons does not use any symbol for the "universe", but writes out the "logical alphabet". This "logical alphabet", for any number n of ele- ments, a, b, c, . . . , consists of the 2" terms which, in Boole's system, form the constituents of the expansion of 1. Thus, for two elements, a and b, the "logical alphabet" consists of ab, a-b, -ab, and -a-b. For three terms, x, y, z, it consists of x y z, xy -z, x -y z, x -y -z, x y -z, -x -y z, and -X -y -z. Jevons usually writes these in a column instead of adding them and putting the sum = 1 . Thus the absence of 1 from his system is simply a whim and represents no real difference from Boole's procedure. The fundamental laws of the system of Jevons are as follows: (1) It a = b and b = c, then a = c. (2) ab = ba. (3) a a = a. (4) a -a = 0. (5) a + b = b + a. (6) a+ a = a. (7) a + = a. This law is made use of but is not stated. (8) a{b + c) = ab+ ac and {a + b)(c + d) =ac + ad+bc + bd. (9) a + a b = a. This law, since called the "law of absorption", allows a direct simplification which is not possible in Boole. Its analogue for multiplication a{a+ b) = a follows from (8), (3), and (9). The law of absorption extends to any number of terms, so that we have also a + ab + ac + ab d + = a (10) a = a{b + -b) {c-^-c) This is the rule for the expansion of any term, a, with reference to any other terms, b, c, etc. For three terms it gives us a = a{b + -b){c + -c) = abc + ab-c + a-bc + a-b-c This expansion is identical with that which appears in Boole's system, except for the different meaning of + . But the product of any two terms of such an expansion will always have a factor of the form a -a, and hence, by (4), will be null. Thus the terms of any expansion will always represent classes "' See Pure Logic, pp. 31-33. The Development of Symbolic Logic 75 which are mutually exclusive. This accounts for the fact that, in spite of the different meaning of + , developed functions in Boole's system and in Jevons's always have the same form. (11) The "logical alphabet" is made up of any term plus its negative, a + -a. It follows immediately from this and law (10) that the logical alphabet for any number of terms, a,h, c, . . . , will be (a + -a) (& + -h) (c + -c) . . . and will have the character which we have described. It corresponds to the expansion of 1 in Boole's system because it is a developed function and its terms are mutually exclusive. A procedure by which Jevons sets great store is the "substitution of similars", of a for h orh for a when a = h. Not only is this procedure vahd when the expressions in which a and h occur belong to the system, but it holds good whatever the rational complex in which a and h stand. He considers this the first principle of reasoning, more fundamental than Aristotle's dictum de omni et nullo}^^ In this he is undoubtedly correct, and yet there is another principle, which underlies Aristotle's dictum, which is equally fundamental — the substitution for variables of values of these variables. And this procedure is not reducible to any substitution of equivalents. The only copulative relation in the system is [ = ] ; hence the expression of simple logical propositions is substantially the same as with Boole: All a'lsh: a = ah No a is 6 : a = a-b Some a is 6: c a = c ab or U a = V ab " U" is used to suggest "Unknown". The methods of working with this calculus are in some respects simpler than Boole's, in some respects more cumbersome. But, as Jevons claims, they are obvious while Boole's are not. Eliminations are of two sorts, "intrinsic" and "extrinsic". Intrinsic eliminations may be performed by substituting for any part of one member of an equation the whole of the other. Thus from a = b c d, we get a = acd = abd = ac = ad This rule follows from the principles a a = a, ab = b a, and if a = 6, ac = be. For example If a = b c d a- a = b c d-b c d = bb-c c-d d = be d-d = ad. "* See Substitution of Similars, passim. 76 A Survey of Symbolic Logic Also, in cases where a factor or a term of the form a(b + -h), or of the form a -a, is involved, eliminations may be performed by the rules a{h + -h) = a and a -a = 0. Extrinsic elimination is that simplification or "solution" of equations which may occur when two or more are united. Jevons does not add or multiply such equations but uses them as a basis for striking out terms in the same "logical alphabet". This method is equivalent, in terms of current procedures, to first forming the expansion of 1 (which contains the terms of the logical alphabet) and then putting any equations given in the form in which one member is and "subtracting" them from the expansion of 1. But Jevons did not hit upon the current procedures. His own is described thus: ^'^ " 1 . Any premises being given, form a combination containing every term involved therein. Change successively each simple term of this into its contrary [negative], so as to form all the possible combinations of the simple terms and their contraries. [E. g., if a, b, and c are involved, form the "logical alphabet" of all the terms in the expansion of (a + -a) (6 + -&) (c + -c) . ] "2. Combine successively each such combination [or term, as a be,] with both members of a premise. When the combination forms a con- tradiction [an expression having a factor of the form (a -a) ] with neither side of a premise, call it an included subject of the premise; when it forms a contradiction with both sides, call it an excluded subject of the premise; when it forms a contradiction with one side only, call it a contradictory com- bination or subject, and strike it out. "We may call an included or excluded subject a possible subject as distinguished from a contradictory combination or impossible subject. "•3. Perform the same process with each premise. Then a combination is an included subject of a series of premises, when it is an included subject of any one; it is a contradictory subject when it is a contradictory subject of any one; it is an excluded subject when it is an excluded subject of every premise. "4. The expression of any term [as a or b] involved in the premises consists of all the included and excluded subjects containing the term, treated as alternatives [in the relation + ]. "5. Such expressions may be simpHfied by reducing all dual terms [of "' Pure Logic, pp. 44-46. The Development of Symbolic Logic 77 the form a{b + -b)], and by intrinsic elimination of all terms not required in the expression. " 6. When it is observed that the expression of a term contains a com- bination which would not occur in the expression of any contrary of that term, we may eliminate the part of the combination common to the term and its expression. . . . "7. Unless each term of the premises and the contrary of each appear in one or other of the possible subjects, the premises must be deemed in- consistent or contradictory. Hence there must always remain at least two possible subjects. "Required by the above process the inferences of the premise a = b c. "The possible combinations of the terms a, b, c, and their contraries are as given [in the column at the left, which is, for this case, the 'logical alphabet']. Each of these being combined with both sides of the premise, we have the following results: ab c ab c = ab c abc included subject fab-c ab -c = ab c -c = a b -c contradiction a -be a-b e = ab-b e = a-b c contradiction a -b -c a -h-e = ab-b e-c = a -b -c contradiction ^-a be ■9-= a-ab c = -ab c ^#- -a b c contradiction -a b -c = a-ab -e = -ab e -e = -a b -c excluded subject -a -b e = a-a-b c = -ab -b c = -a -b e excluded subject -a -b -c = a -a -b -c = -ab-b c-e = -a -b -c excluded subject "It appears, then, that the four combinations ab-c to -abc are to be struck out, and only the rest retained as possible subjects. Suppose we now require an expression for the term -b as inferred from the premise a = h c. Select from the included and excluded subjects such as contain -b, namely -a -b c and -a -b -e. " Then -b = -a-b e + -a -b -e, but as -a c occurs only with -b, and not with b, its contrary, we may, by Rule 6, eliminate -b from -a-b c; hence -b = -ae +-a -b -c." This method resembles nothing so much as solution by means of the Venn diagrams (to he explained in Chapter III). The "logical alphabet" is a list of the different compartments in such a diagram; those marked "contradiction" are the ones which would be struck out in the diagram by transforming the equations given into the form in which one member is 0. 78 A Survey of Symbolic Logic The advantage which Jevons claims for his method, apart from its obvious- ness, — namely, that the solutions for different terms do. not require to be separately performed, — is also an advantage of the diagram, which exhibits all the possibilities at once. If any problem be worked out by this method of Jevons and also that of Boole, it will be found that the comparison is as follows: The "logical alphabet" consists of the terms which when added give 1, or the universe. Any term marked "contradiction" will, by Boole's method, have the coef- ficient or 1/0; any term marked "included subject" will have the coef- ficient 1; any marked "excluded subject" will have the coefficient 0/0, or v where v is arbitrary. If, then, we remember that, according to Boole, terms with the coefficient 1/0 are equated to and thus eliminated, we see that the two methods give substantially the same results. The single important difference is in Boole's favor: the method of Jevons does not distinguish decisively between the coefficients 1 and v. If, for example, the procedure of Jevons gives x = .r -y z, Boole's will give either ,r = -y z or X = V -y z. One further, rather obvious, principle may be mentioned : '^° Any subject of a proposition remains an included, excluded, or con- tradictory subject, after combination with any unrelated terms. This means simply that, in any problem, the value of a term remains its value as a factor when the term, is multiplied by any new terms which may be introduced into the problem. In a problem involving a, h, and c, let a -6 c be a "contradictory" term. Then if x be introduced, a -hex and a-h c -x will be "contradictory". On the whole Jevons 's methods are likely to be tedious and have little of mathematical nicety about them. Suppose, for example, we have three equations involving altogether six terms. The "logical alphabet" will consist of sixty-four members, each of which will have to be investigated separately for each equation, making one hundred and ninety-two separate operations. Jevons has emphasized his difference from Boole to the extent of rejecting much that would better have been retained. It remained for others, notably Mrs. Ladd-Frankhn and Schroder, to accept Jevons's amended meaning of addition and its attendant advantages, yet retain Boole's methods of development and similar methods of elimination and solution. But Jevons should have credit for first noting the main clue to this simplification — the laws a-¥ a = a and a + ab = a. ™ Pure Logic, p. 48. The Development of Symbolic Logic 79 YII. Peiece The contributions of C. S. Peirce'"' to symbolic logic are more numerous and varied than those of any other writer — at least in the nineteenth century. He understood how to profit by the work of his predecessors, Boole and De Morgan, and built upon their foundations, and he anticipated the most important procedures of his successors even when he did not work them out himself. Again and again, one finds the clue to the most recent developments in the writings of Peirce. These contributions may be summed up under three heads: (1) He improved the algebra of Boole by distinguishing the relations which are more characteristic of logical classes (such as multiplication in Boole's algebra) from the relations which are more closely related to arithmetical operations (such as subtraction and division in Boole). The resulting algebra has certain advantages over the system of Jevons because it retains the mathematical methods of develop- ment, transformation, elimination, and solution, and certain advantages over the algebra of Boole because it distinguishes those operations and relations which are always interpretable for logical classes. Also Peirce introduced the "illative" relation, "is contained in", or "implies", into symbohc logic. (2) Following the researches of De Morgan, he made marked advance in the treatment of relations and relative terms. The method of deahng with these is made more precise and "mathematical", and the laws which govern them are related to those of Boole's algebra of classes. Also the method of treating "some" and "all" propositions as sums (2) and products (n) respectively of "propositions" containing variables was here first introduced. This is the historic origin of "formal implication" and all that has been built upon it in the more recent develop- ment of the logic of mathematics. (3) Like Leibniz, he conceived symbolic logic to be the science of mathematical form in general, and did much to revive the sense of logistic proper, as we have used that term. He worked out in detail the derivation of various multiple algebras from the calculus of relatives, and he improved Boole's method of applying symbolic logic to problems in probability. "1 Charles Saunders Peirce (1839-1914), son of Benjamin Peirce, the celebrated mathematician, A.B. (Harvard, 1859), B.S. (Harvard, 1863), lecturer in logic at Johns Hopkins, 1890- ?. For a number of years, Peirce was engaged in statistical researches for the U. S. Coast Survey, and was at one time head of the Department of Weights and Measures. His writings cover a wide variety of topics in the history of science, meta- physics, mathematics, astronomy, and chemistry. According to WiUiam James, his articles on "Some lUustrations of the Science of Logic", Pop. Sd. Mo., 1877-78, are the source of pragmatism. 80 A Survey of Symbolic Logic V\e shall take up these contributions in the order named. The improvement of the Boolian algebra is set forth mainly in the brief article, "On an Improvement in Boole's Calculus of Logic "/^- and in two papers, "On the Algebra of Logic "."^ It will be remembered that Boole's calculus has four operations, or rela- tions: a+b indicates the class made up of the two mutually exclusive classes, a and b; [ — ] is the strict inverse of [ + ], so that if x + h = a, then x = a — b; ax b or ab denotes the class of those things which are common to a and b; and division is the strict inverse of multiplication, so that ii x b = a, then X = a/b. These relations are not homogeneous in tj'pe. Boole's [+] and [ — ] have properties which approximate closely those of arithmetical addition and subtraction. If [?i].r indicate the number of members of the class X, [n]a+ [n]b = [n](a+ b) because a and b are mutually exclusive classes, and every member of a is a member of (a + b) and every member of 6 is a member of (a + b). This relation, then, differs from arithmetical addition only by the fact that a and b are not necessarily to be regarded as numbers or quantities. Simi- larly, [n]a — [n]b = [n](a — b) But in contrast to this, for Boole's axb or ab, [7i]a X [7i]b = [n]{ab) will not hold, except for and 1 : this relation is not of the type of its arith- metical counterpart. And the same is true of its inverse, a/b. Thus, in Boole's calculus, addition and subtraction are relations of the same type as arithmetical addition and subtraction; but multiplication and division are different in type from arithmetical multiplication and division. Peirce rounds out the calculus of Boole by completing both sets of these relations, adding multiphcation and division of the arithmetical type, and addition and subtraction of the non-arithmetical type.'^* The general character of these relations is as follows : 122 Proc. Amer. Acad., vii, 250-61. This paper will be referred to hereafter as "Boole's Calculus ". 123 Amer. Jour. Math., in (1880), 15-57, and vii (1885), 180-202. These two papers will be referred to hereafter as Alg. Log. 1880, and Alg. Log. 1885, respectively. 124 "Boole's Calculus," pp. 250-54. The Development of Symbolic Logic 81 A. The "Non-Arithmetical" or Logical Relations (1) a+b denotes the class of those things which are either a's or b's or bothP^ (2) The inverse of the above, a \-b, is such that if x + b = a, then X = a \-b. Since x and b, in x + b, need not be mutually exclusive classes, a \- b is an ambiguous function. Suppose a; + 6 = a and all 6 is x. Then a \- b = X, and a \- b = a Thus a\-b has an upper limit, a. But suppose that x + b = a and no b is X. Then a \- b coincides with a — b (a which is not b) — i. e., a \- b = x, and a \- b = a — b Thus a \- b has a lower limit, a — 6, or (as we elsewhere symboHze it) a -b. And in any case, a \- b is not interpretable unless all b is a, the class b contained in the class a. We may summarize all these facts by a \- b = a-b + v ab+ [0]-ab where v is undetermined, and [0] indicates that the term to which it is prefixed must be null. (3) a b denotes the class of those things which are both a's and b's. This is Boole's a b. (4) The inverse of the preceding, a/b such that iib x = a, then x = a/b. This is Boole's a/b. a/b is an ambiguous function. Its upper limit is a + -b; its lower limit, a.'^^ It is uninterpretable unless b is contained in a — i. e., a/b = a b + v-a-b+ [0] a-b B. The "Arithmetical" Relations (5) a -\- b denotes the class of those things which are either a's or b's, where a and b are mutually exclusive classes. This is Boole's a + b. a + b = a-b + -ab+ [0]ab (6) The inverse of the preceding, a — b signifies the class "a which is not b". As has been mentioned, it coincides with the lower limit of a \-b. (7) a X b and a -^ b are strictly analogous to the corresponding relations 126 peirce indicates the logical relations by putting a comma underneath the sign of the relation: that which is both a and 6 is a, h. 128 Peirce indicates the upper limit by a : b, the lower limit by a -^ 6. These occur only in the paper "Boole's Calculus". 7 82 A Survey of Symbolic Logic of arithmetic. They have no such connection with the corresponding "logical" relations as do a + 6 and a — b. Peirce does not use them except in applying this system to probability theory. For the "logical" relations, the following familiar laws are stated :i^' a + a = a a a = a a+b = b + a ab = b a {a+b) + c = a+ {b + c) (ab)c = a{b c) {a + b)c = a c + b c ab+ c = (a + c){b + c) The last two are derived from those which precede. Peirce's discussion of transformations and solutions in this system is inadequate. Any sufficient account would carry us quite beyond what he has given or suggested, and require our report to be longer than the original paper. We shall be content to suggest ways in which the methods of Boole's calculus can be extended to functions involving those relations which do not appear in Boole. As has been pointed out, if any function be developed by Boole's laws, fix) =fil)-x*f{0)--x, fix, y) = 2' Peirce's symbol is — < which he explains as meaning the same as ^ but being sim- pler to write. J29 Mermrirs of the Amer. Acad., n. s., ix (1867), 317-78. 130 "Description of a Notation for the Logic of Relatives," he. cit., pp. 334-35, 338-39, 342. 131 In this paper, not-x is symbolized by n^, "different from every x," or by a ^. 84 A Survey of Symbolic Logic (18) x + = X. (19) a;+l = 1. (20) (fi) = 0. (23) If p{l))a " The chief points of difference between this modified calculus of prob- abilities and the original calculus of Boole are as follows: (1) Where Boole puts p, q, etc. for the "probability of a, of b, etc.", in passing from the logical to the arithmetical interpretation of his equa- tions, Peirce simply changes the relations involved from logical relations to the corresponding arithmetical relations, in accordance with the foregoing, and lets the terms a, b, etc. stand for the frequency of the a's, b's, etc. in the system under discussion. (2) Boole has no symbol for the frequency of the a's amongst the b's, which Peirce represents by «&. As a result, Boole is led to treat the probabilities of all unconditioned simple events as independent — a pro- cedure which involved him in many difficulties and some errors. (3) Peirce has a complete set of four logical operations, and four analogous operations of arithmetic. This greatly facilitates the passage from the purely logical expression of relations of classes or events to the arithmetical expression of their relative frequencies or probabilities. Probably there is no one piece of work which would so immediately reward an investigator in symbolic logic as would the development of this calculus of probabilities in such shape as to make it simple and practicable. Except for a monograph by Poretsky and the studies of H. MacColl,"-^" the subject has lain almost untouched since Peirce wrote the above in 1867. Peirce's contribution to our subject is the most considerable of any up to his time, with the doubtful exception of Boole's. His papers, however, are brief to the point of obscurity: results are given summarily with little or no explanation and only infrequent demonstrations. As a consequence, the most valuable of them make tremendously tough reading, and they have never received one-tenth the attention which their importance de- serves."^ If Peirce had been given to the pleasantly discursive style of De ^lorgan, or the detailed and clearly accurate manner of Schroder, his work on symbolic logic would fill several volumes. '^" Since the above was written, a paper by Couturat, posthumously published, gives an unusually clear presentation of the fundamental laws of probability in terms of symbolic logic. See BM. I'l Any who find our report of Peirce's work unduly difficult or obscure are earnestly requested to consult the original papers. The Development of Symbolic Logic 107 VIII. Developments since Peiece Contributions to symbolic logic which have been made since the time of Peirce need be mentioned only briefly. These are all accessible and in a form sufficiently close to current notation to be readily intelligible. Also, they have not been superseded, as have most of the papers so far discussed; consequently they are worth studying quite apart from any relation to later work. And finally, much of the content and method of the most important of them is substantially the same with what will be set forth in later chapters, or is such that its connection with what is there set forth will be pointed out. But for the sake of continuity and perspective, a summary account may be given of these recent developments. We should first mention three important pieces of work contemporary with Peirce's later treatises.^^^ Robert Grassmann had included in his encyclopedic Wissenschaftslehre a book entitled Die Begriffslehre oder Logik,^^^ containing (1) Lehre von den Begriffen, (2) Lehre von den Urtheilen, and (3) Lehre von den Schliissen. The Begriffslehre is the second book of Die Formenlehre oder Mathematik, and as this would indicate, the development of logic is entirely mathematical. An important character of Grassmann's procedure is the derivation of the laws of classes, or Begriffe, as he insists upon calling them, from the laws governing individuals. For example, the laws a + a = a and a- a = a, where a is a class, are derived from the laws e + e = e, e-e = e, ei-e2 = 0, where e, ei, Ci represent individuals. This method has much to commend it, but it has one serious defect — the supposition that a class can be treated as an aggregate of individuals and the laws of such aggregates proved generally by mathematical induction. As Peirce has observed, this method breaks down when the number of individuals may be infinite. Another difference between Grassmann and others is the use throughout of the language of intension. But the method and the laws are those of extension, and in the later treatise, there are diagrammatic illustrations in which "concepts" are represented by areas. Although somewhat incomplete, in 162 Alexander MacFarlane, Principles of the Algebra of Logic, 1879, gives a masterly presentation of the Boolean algebra. There are some notable extensions of Boole's methods and one or two emendations, but in general it is the calculus of Boole unchanged. Mac- Farlane's paper "On a Calculus of Relationship" {Proc. Roy. Soc. Edin., x, 224-32) re- sembles somewhat, in its method, Peirce's treatment of "elementary relatives". But the development of it seems never to have been continued. "3 There are two editions, 1872 and 1890. The later is much expanded, but the plan and general character is the same. 108 A Survey of Symbolic Logic other respects Grassmann's calculus is not notably different from others which follow the Boolean tradition. Hugh MacCoU's first two papers on "The Calculus of Equivalent Statements"/^* and his first paper "On Symbolical Reasoning "/^^ printed in 1878-80, present a calculus of propositions which has essentially the properties of Peirce's, without n and S operators. In others words, it is a calculus of propositions, like the Two-Valued Algebra of Logic as we know it today. And the date of these papers indicates that their content was arrived at independently of Peirce's studies which deal with this tonic. In fact, MacColl writes, in 1878, that he has not seen Boole. "^ The calculus set forth in MacColl's book, Symbolic Logic and its Applications,^'^'' is of an entirely different character. Here the funda- mental symbols represent prepositional functions rather than propositions; and instead of the two traditional truth values, "true" and "false", we have "true", "false", "certain", "impossible" and "variable" (not cer- tain and not impossible). These are indicated by the exponents r, t, e, 1], 6 respectively. The result is a highly complex system, the fundamental ideas and procedures of which suggest somewhat the system of Strict Implication to be set forth in Chapter V. The calculus of Mrs. Ladd-Franklin, set forth in the paper "On the Algebra of Logic" in the Johns Hopkins studies,"^^ differs from the other systems based on Boole by the use of the copula v Where a and b are classes, a v6 represents "a is-partly b", or "Some a is h", and its negative, a V &, represents " a is-wholly-not-& ", or " No a is & ". Thus a v 6 is equiva- lent to a 6 =t= 0, and awb to ab = Q. These two relations can, between them, express any assertable relation in the algebra, acb will be a v -b, and a = b IS represented by the pair, (a v-6)(-a v6). For propositions, a V 6 denotes that a and b are consistent — a does not imply that b is false and b does not imply that a is false. And a v 6 symbolizes " a and b are inconsistent" — if a is true, b is false; if b is true, a is false. The use of the terms "consistent" and "inconsistent" in this connection is possibly mis- leading: any two true propositions or any two false propositions are con- I" (1) Proc. London Math. Soc, ix, 9-20; (2) ibid., ix, 177-86. '«5 Mind, V (1880), 45-60. '«« Proc. London Math. Soc, ix, 178. 1" Longmans, 1906. i«* The same volume contains an interesting and somewhat complicated system by O. H. Mitchell. Peirce acknowledged this paper as having shown us how to express uni- versal and particular propositions as II and 2 functions. B. I. GUman's study of relative number, also in that volume, belongs to the number of those papers which are important in connecting symbolic logic with the theory of probabilities. The Development of Symbolic Logic 109 sistent in this sense, and any two propositions one of which is true and the other false are inconsistent. This is not quite the usual meaning of "consistent" and " inconsistent "—it is related to what is usually meant by these terms exactly as the "material implication a c6 is related to what is usually meant by "b can be inferred from a". That a given class, x, is empty, or a given proposition, x, is false, x = 0, may be expressed by a; v oo, where «> is "everything" — in most systems represented by 1. That a class, y, has members, is sj-mbolized by 2/ v oo. This last is of doubtful interpretation where y is a proposition, since Mrs. Ladd-Franklin's system does not contain the assumption which is true for propositions but not for classes, usually expressed, "If a; =t= 0, then X = 1, and if a: =H 1, then a; = 0". a; v » may be abbreviated to a;v, ab V CO to ab V , and t/vqo to 2/v,ccZvoo to cc?v, etc., since it is always understood that if one term of a relation v or v is missing, the missing term is «> . This convention leads to a very pretty and convenient opera- tion: V or V may be moved past its terms in either direction. Thus, (avb) = (abv) = {vab) and (x'^y) = (xyv) = i^x y) But the forms (vab) and ( v a; 2/) 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, No ais b. avb, Some ais b. avb, Some a is not b. ■ a v-b, 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 o is & and all b is c, then all a is c, " will be represented by {a V -b) (b V -c) V (avc) where v , or v , within the parentheses is interpreted for classes, and v between the parentheses takes the propositional interpretation. This ex- pression may also be read, "'All a is 6 and all 6 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 or a-bv or abv or abv or a-bv 110 A Survey of Symbolic Logic "The three propositions, 'All a is b', 'All 6 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 -6) (6 V -c) V (a V -c) "If all a is 6 and all b is c, then all a is c"; (2) (a V -b) (a v -c) v (& v -c) "If all a is 6 and some a is not c, then some b is not c"; (3) (6v-c)(av-c) v(av-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 '^c)(avc) 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.^^^ 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 =t= 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), Schr5der 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. i6» 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; A^olume 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."" 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 (Aussagenkalkid) 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 . . . symboHze "elements" or individuals."' These are distinct from one another and from 0. 2) V = A+B + C + D+ .. P symboHzes 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 P. 4) P = Si i "o For an excellent summary by Schroder, see Abriss der Algebra der Logik ; ed. Dr. Eugen Muller, 1909-10. Parts i and ii, covering Vols, i and ii of Schroder's Vorlesungen, have so far appeared. 1" The propositions here noted will be found in Vorlesungen iiber die Algebra der Logik, III 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 P in a determined order. 6) {i = j) = (i -.j = j : i), (^ 4= j) = (i : i 4= J : i) for every i and j. 7) i : i + Pairs of elements of 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". P = {A ■.A) + iA ■.B) + {A :€) + ... + {B •.A) + {B ■B) + {B -.0) + ... 9) + {C ■.A) + {C ■.B) + {C -.0) + ... + . P represents the universe of binary relatives. 10) P = S,S.- (i : i) = SiS, [i : j) = S^,- (i : j) 9) and 10) may be summarized in a simpler notation: 1 = Si,- i : j = yl : ^ + ^ : 5 + ^ : C + + B:A + B -.B + B : C + . 11) + C:^ + C:5 + C: C+ . 12) i : j : h will symbolize an "individual ternary relative". 13) P = 2aS,S,- (i : i : /i) = S^-, i : j : /» 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,- an {i : j) where a,,- 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. /= Siy(z =j){i:j) = Si(t:z)i'2 N= ^iiii=¥j){i:j) {ah)ij = aijbii {a + h)ij = an + bij -an = {-a)ij = -(a,,-) (a\b)ij = Sa flift b/,j (a t b)ii = lih (am + b^,) The general laws which govern propositional functions, or Atissngen- schemata, such as {_ab)i,; Zhttihbhi, Uh (aih + bhj), Uattn, l^a an, etc., are as follows : Au symbolizes any statement about u; UuAu will have the value 1 in case, and only in case, Au = 1 for every u; XuAu will have the value 1 if there is at least one u such that Au = 1. That is to say, UuAu means " Au for every u", and S,jy4„ means " Au for some u". a) UuAu cA^c ZuAu, -[liuAu] c -A^ c -[UuAu] (The subscript u, in a and 13, represents any value of the variable u.) y) -[UuAu] = S„ -Au, -[S„4„] = n„ -Au 8) If Au is independent of u, then UuAu = A, and S„^„ = A. e) UuiA cBu) = (AcUuBu), UuiAucB) = (S„^„ cB) f) n„,„ or UuU,{AucB;) = CSuAuCU^B^) r,) l^uiAucB) = (UuAucB), 2uiAcBu) = U c S,B„) e) S„,„ or S„S„(^„c5„) = (UuAuCXyB,) f n„(^„ = 1) = (n,^„ = 1), Uu{Au = 0) = (s„^„ = 0) 1 -EuiAu = 0) = (UuAu = 0), S„(^„ = 1) = i^uAu = 1) "2 We write I -where Schroder has 1'; N where he has 0'; {a\ b) for (a; 6); (a t &) for (a ^ 6); -a for a; ^a for a. + ac). But a[a {b + c)] = {a a){b + c) = a {b + c). And a (b + a c) = a (a c + b). Hence 'a (b + c) ca (a c + b). But by lemma 3, a (ac + b) cac + ab. And a c + ab = ab + a c. Hence a (b + c) cab + 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. ^ 5-51 {a + b){c + d) = (a c + b c) + (a d + b d). [5-5] {a + b){c + d) = (a + b) c + (a + b) d = (ac + b c) + (a d + b d). 5-52 a + b c = {a + b)(a + c). (Correlate of 5-5) [5-51] {a + b)(a + c) = (a a + b a) + {a c + b c) = [(a + ab) + ac] + b c. But [5-4] (a + ab) + 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: ^ See "Sets of Independent Postulates, etc.", he. cit., p. 300, footnote. 10 130 A Survey of Symbolic Logic 5-6 a-l = a = 1-a. [1-5] a-0 = 0. Hence a--l = 0. But [1-61] if a-l = 0, then a-l = a. 5-61 acl. [1-9] Since a-l = a, ad. 5-62 a + = a = + a. -a—0 = -a-l = -a. Hence [3 - 2] a -h = -(-a--0) = -(-a) = a. 5-63 Oca. O-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] Oca holds alwaj's. Hence Q.E.D. 5-7 li a + b = X and a = 0, then b = x. lia = 0, a + b = + b = b. 5-71 1{ ab = X and a = 1, then b = ,r. If a = 1, ab = 1-b = b. 5-72 a + b = is equivalent to the two equations, « = and b — 0. If a = and 6 = 0, then a+& = + = 0. And a a + b = 0, -a-b = -{a + b) = -0 = 1. But if -a -6 = 1, a = a-l = a{-a-b) = {a-a) -b = 0--b = 0. And [5-7] if a-t-6 = and a = 0, then b = 0. 5-73 a & = 1 is equivalent to the two equations, a = 1 and b = 1. If a = 1 and b = 1, then ab = 1-1 = 1. And if a 6 = 1, -a + -b = -{ab) =-1=0. Hence [5-72] -a = and -b = 0. But [3-2] if -a = 0, a = 1, and if -b = 0, & = 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 anj^ 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+ -b) = ab + a -b. And [4-8] b + -b = I. Hence a{b + -b) = a-1 = a. 5-85 a + b = a + -ab. [5-8]b = ab + -ab. Hence a + b = a+ {ab + -ab) = {a + ab) + a-b. But [5-4] a + a 6 = 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 I = {a + -a){b + -b){c + -c)... [4-8] 5-93 a = a + ab + ac + ad+... [5-4] 5-931 a = a {a + b)(a + c){a + d). . . [5-41] 5-94 a{b + c + d+...)=ab + ac + ad+.. [5-5] 5-941 a+bcd... = {a + b){a + c)(a + d) . . . [5-52] 5-95 -(a + b + c +...)= -a -b-c .. . If the theorem hold for n terms, so that -(ai + a2 + . . . + a„) = -ai -Ui . . . -an then it will hold for ?i + 1 terms, for by 3 - 4, -[{ai + ai+ . . + a„) + a„+i] = -(ai ■^ a2 + . . . -KXn) • -cin^ 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-h -c . . . [4-8, 5-951] 5-97 a + b + 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, & = 1, c = 1, . . . [5-73] 5-98 a-b c d = a b-ac-ad . . . [1-2] a a a a . . . = a. 5 • 981 a+ {b + c + d+ . . ) = (a + b) + (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/(a;), ^{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 z or its negative, -x, that $(.r, 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, o = a (x -H -x) = a X + a-x = {a x + a -x) {y + -y) = axy + ax-y + a-xy + a-x -y, etc. a x + d-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 a; = 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 + -ab + -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 + -b) + -ab+ (a + -a) -b = ab + a-b + -ab + -a-b = 1 whatever the value of a or of &. 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 -r. The decision what 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 2;, 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 Ax + 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 .r, (2) those which involve -r, (3) those which involve both, and (4) those which involve neither.'' Since the Distributive Law, 5 • 5, allows us to collect the coefficients of X, of -X and of {x -x) , the most general form of such an expres- sion is ■p X + q-x + r {x -.r) + 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-ir s, gives the required reduction. The normal form of a function of ti + 1 variables, ^\X\, X-z, . . Xn, Xn+l) may be defined as the expansion by the Distributive Law of /(a-l, X2, . . X„)-Xn+1 +/ '{Xl, X2, . . . Xr,)--Xn+l where/ and/ ' are each some function of the n variables, Xi, x^, . a:„, 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.* Thus the normal form of a function of two variables, (a;i, X2, . . Xk, Xk+i) be any function of k + 1 variables. By 6-11, for some/ and some/', $(a;i, X2, .i-k, Xk+i) = f{xi, X2, . . Xk)-Xk+i + /'(a-i, X2, ... Xk)--Xk+i (1) Since this last expression may be regarded as a function of Xk+i in which the coefficients are the functions / and / ', [6-3] fixu X2, ... Xk) x/ '{xi, X2, . . . Xk) c $(.ri, X2, . . Xk, Xk+i) The Classic, or Boole-Schroder, Algebra of Logic 141 Let Ai{$}, A^l^}, A4$}, etc., be here the coefficients in $; Ai{f}, Mf}, Mf\, etc., the coefficients in/; and Ai{f'}, A,{f'], As{f'}, etc., the coefficients in / '. If IlA{f} c/ and n^{/'} c/', then [6-3] n^{/!xn^{/')c/x/' and, by (1), IJ^l/} > \ Xl Xo . . . Xn Ak ] f*Bki I -Xj -l-o . . -X„ J I -Xi -X-i . . . -Xn , , { Xi X2 . . . Xn = (Ak + Bk) -\ L ~X\ -X2 -Xn And since addition is associative and commutative, the sum of the two functions is equivalent to the sum of the sums of such corre- sponding terms, pair by pair. 6-43 The product of two functions of the same variables, ^{xi, x^, ... Xn) and ■^(a-'i, X2, ... x„), is another function of these same variables, F{Xi, Xi, . . Xn), such that the coefficient of any term in F is the product of the coeffi- cients of the corresponding terms in $ and ^. Let Ajci ' ' ' " r and Bkl ' ' " r be any two L -Xi -X2 . . . -Xn J L -Xi -X2 . . -Xn J corresponding terms in $ and ^. Xl X2 . . . Xn\ „ \ Xi X2 . . . Xn f ^Bk] -Xl -Xi . . . -Xn J L -Xl -X2 . . -Xn = {Ak X Bk) Xn I -Xl -; X2 ■ —Xn By 6-15, $ and ^ 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 cro^s-products of terms in $ and ^ 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 b = 0. [2-2] a = b is equivalent to the pair, acb and b ca. [4-9] a cb is equivalent to a-b = 0, and b ca to -ab =0. And [5 • 72] a -6 = and -a 6 = are together equivalent to a -b + -ab = 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, .T2, . . . x„) = 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 $(a-i, a-2, . . . x„) and SE'Ccci, x^, . . . x„) be any two functions of the same variables, then $(a-i, X2, . . . Xn) = ^(.x'l, X2, . . x„) is equivalent to F{xi, X2, . . . x„) =0, where f is a function such that if Ai, Ai, Ai, etc., be the coefficients in $, and Bi, B^, B3, etc., be the coef- ficients of the corresponding terms in S^, then the coefficients of the corre- sponding terms in F will be (^1 -Bi + -Ai Bi), {A2 -B2 + -A2 B2}, (A^ -B3 + -.4.353), etc. The Classic, or Boole-Schroder, Algebra of Logic 145 By 7- 1, $ = ^ is equivalent to ($ x-'i') + (-$ x ^) =0. By 6-41, -$ and -^ are functions of the same variables as $ and SI'. Hence, by 6-43, $ x-^ and -$ x ^ will each be functions of these same variables, and by 6-42, ($ x-^) + (-$ x ^) will also be a function of these same variables. Hence $, ^, -$, -^, $ x-^, -$ x ^, 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 $, and Bk the corresponding coefficient in SI>, then by 6-41, the corresponding coefficient in -$ will be -Ak and the corresponding coefficient in -^ will be -5/,. Hence, by 6-4,3, the corresponding coefficient in $x-^ will be Ai -Bk, and the corresponding coefficient in -$ x N[' will be -AkBk- Hence, by 6-42, the corresponding coefficient in ($ x-^) + (-$ x^) will be Ak -Bk + -AkBk. Thus ($ x--*^) + (-$ X -^if) 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. ^ 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 + -at. If a = 0, a-t + -at = 0--t+l-t = t. And U t = a~t+-at, then [7 • 1] {a-t + -at) -t+ {at + -a-t) t = = 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 = & is equivalent to if = {ab + -a-b) t+ (a -b + -a b) -t. [7-1] a = b is equivalent to a-b + -ab = 0. And [6 • 4] -{a -h + -ab) = ab + -a -b. Hence [7-15] Q.E.D. The number of equations equivalent to a given equation and expressible ' See Se-pt lois fandamentales de la theorie des egaliles logiques, Chap. i. 11 146 A Surrey of Symbolic Logic in terms of n elements will be half the number of distinct fmictions which can be formed from n elements and their negatives, that is, 2"72- The sixteen distinct functions expressible in terms of two elements, a and b, are: a, -a, b, -b, (i. e., a -a, h-b, etc.), 1 (i- e., a + -a, b + -b, etc.), ab, a -b, -a b, -a -b, a + b, a + -b, -a + h, -a + -b, ab + -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 + -ab; I = 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 cl, i. e., t = 1-t + 0--i. 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 ex c-A. [5 -72] A X + B -X = is equivalent to the pair, A x = and B -X = 0. [4 • 9] 5 -a; = is equivalent to S c x. And A X = Q is equivalent to x -(-.4) = 0, hence to x c -A. 7-21 A solution in the form H ex cK 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 Ocxcl, 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 cxcK, will give also H c-xc K. But U Hex and H c-x, then [i-Q] H x = H and H-x = H. Hence [1-62] H = 0. And if a;c/v and -xcK, then [5-33] x + -xcK, and [4-8, 5-63] K = 1. Hence H ex c K will be equivalent to c a; 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 Ax + B -x =0 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 ex c -A. Hence if 5 = -.1, it is equivalent to B ex cB and to -A ex e-A, and hence [2-2] to a; = 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; and 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 = he pos- sible in the algebra, and hence that its solution be possible, is A B = 0. By 6-3, ABeAx + B-x. Hence [5-65] ii A x + B -x = 0, then A B = 0. Hence if .4 5 =1= 0, then A x + B -x = must be false for all values of x. And Ax + B -x = and the solution B ex c-A are equivalent. A B = is called the "equation of condition" ol 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 e -A is equivalent to Ax + B -x = 0, whether ^ a; + 5 -a; = be true, false, or impossible. The sense in which ^45=0 conditions other forms of the solution of ^4 a; + jB -a; = 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 Ax + B -X = 0.) For example, if ab X + {a + b) -X = then (a + b) ex c -{a b). But the equation of condition is ab (a + b) = ab = 0, or, -(a b) = 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 ab = 0. Again suppose a x + b -x + c = Expanding c with reference to x, and collecting coefiicients, we have (a + c) x + {b + c) -X — and the equation of condition is (a + c){b + c) = ab + ac + bc + c = ab + c = The solution is b + c cx 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 cx c-a But here again, the solution b cx 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 a; + & -t + c = 0, then by 5-72, a a; + 6 -.r = 0, and any addition to the solution by retaining c will be indeterminate. All terms which involve neither the unlmown 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 = Q'ls equivalent to x = -A x-*- B -x. [7-15] Ax + B -X = is equivalent to x = (A x + B -x) -X + {-A X + -B -,!■) 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." 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 5 c a; c -A. IMany 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 + 5 -,r = is satisfied by x = B -u + -A u, ov 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 Q-4:, U X = B -11 + -A u, then -x = -B -u + A u. Hence if a; = B -u + -A ii, then Ax + B-x = A iB-u + -Bu) + B i-B-u + Au) = AB-u + ABu = AB Hence if ^-15=0 and x = B -u + -A u, then whatever the value of u, Ax + B-x = 0. (6) Suppose x known and such that Ax + 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 = {Ax + -A -x) u+ {B -X + -B x) -u = The condition that this equation hold for some value of tt is, by 7 • 221, {A X + -A -x){B -x + -Bx) = A-Bx + -A B -x = This condition is satisfied if Ax + B -x = 0, for then A{B + -B) X + (A+-A) B-x = AB + A-BX + -A B -x = and by 5-72, A -B x + -A B -x = 0. (c) If ^ 5 = 0, then B-zi + -Au = B + u -A, for: If .4 5 = 0, then A B u = 0. Hence B-u + -Au = B-u + -A {B + -B) u + .4 B u = B-u+{A+-A) Bu + -A-Bu = B {-u + u) + -A -B u = B + -A-BU. But [5-85] B + -A-BU = B+u -A. 1" See above, p. 77. 150 ,1 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 .V in the original equation. We then have A {B -u + -A u) + B i-B -u + Au) = AB-u + ABu = A B And U A B = 0, the solution is verified for every value of u. That the solution, .t = B-i) + -A u — B-^-u-A, means the same as B « Z Q3* X . . . =0 h h h h h h or n[EJ"/]xn[ZQ.''] = o r h r h And by 7-51, the solution of ^{xx, Xzy ... Xn) = for Xk is n[ZQ/]c.T.cz-[Z^/] The Classic, or Boole-Schroder, Algebra of Logic 161 r h r h And by 5-95, -[£ P/] = IJ --P."- h h 7 ■ 54 The condition that m equations in n unknowns, each of the form F{Xx, X2, ... X„) = f{Xl, X2, ... Xn) may be regarded as simultaneous, is as follows: Let the coefficients in F\ in the equation F' = f\ be P,\ P,\ P,\ . . . Q,\ Q,\ Q,\ . . ., and let the coefficients of the corresponding terms in / ', in the equation F'- = / S be Mi\ MsS Ms\ . . . Ni\ N2\ iVsS . . . ; let the coefficients of the corresponding terms in F\ in the equation F' = f^, be P,^ Pi", Pi\ . . . Qi", Q/, Q,'', ..., and let the coefficients of the corresponding terms in / ^ be Mi^, M^,^, Mz^, . . . Ni^, Ni^, Ni^, . . . ; let the coefficients of the corresponding terms in F'", in the equation P™ = / "=, be Pi", Pj™, Pa", . . . Qi", Q^"", Qs"", ■ ■ ., and let the coefficients of the corresponding terms in / " be Mi", M^'^, Ms", . . iVi*", iVa", iVs", The condition then is n [ E (-P/ -Mr' + -Pr" M/)] X n [ Z iQr' -N," + -Q/ iV/)] = r h r h or if AJ" represent any coefficient in P*, whether P or Q, and P/ represent the corresponding coefficient in / '', whether M or N, the condition is r h And the solution which m such equations give, on this condition, for any one of the unknowns, xu, is as follows: Let P^ and Mr' be the coefficients of those terms, in any one of the equations F' = / ^ in which Xh is positive, and let Qr'' and iV/ be the coefficients of the terms, in P* = / ', in which x^^ is negative. The solution then is n [E (Q.' -^r' + -qr' ^/)] C a:, C E [IT (-P.' i>^/ + -Fr' "M/)] r h r h or a;, = n [ E (Q^' --^^^^ + "6^' ^^')] + ^ • Z (II (-P/ M/ + -Pr' -M/)] r h r h By 7-13, P^Ca^i, 3:2, ... Xn) = f'ixi, Xi, ... Xn) is equivalent to ^(a;i, Xi, ... a;„) = 0, where >I> is a function such that if Q/ and iV/ be coefficients of corresponding terms in F' and / *, the coefficient of the corresponding term in ^ will be Qr' -Nr' + -Qr' Nr', and if Pr' and Mr' be coefficients of corresponding terms in F' and / ^ the coefficient of the corresponding term in '^ will be Pr' -Mr' + -Pr' Mr'. And -{Pr' -Mr' + -Pr' Mr') = Pr' Mr' + -Pr' -Mr'. Hence the theorem follows from 7 • 53. 12 162 A Survey of Symbolic Logic F{xi, X2, . . . Xn) = /(a-i, .1-2, ... a;„) 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. ^^ Given any equation connecting the symbols x, y, . . u\ 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 $(.r, y, . .) = ^(w, z, . . . ) ; to determine t in terms of iv, 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 = /(.t, 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 b^^ 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,b=0, c = 0, ...,a+b — 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 + b = 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 we have a = and 6 = 0, we have also a = b, or ab = 0, or a + b = 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,'^ 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 ii a = b, and therefore a-b + -ab = and ab + -a-b = 1, then any t is such that a-b + -ab ct 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 ct, 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 m which we choose to introduce. Similarly, if tcy, then tcy + 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 = 6, then t = {ab + -a -b + u) t + v {a -b + -ab) -t, where u and v are arbitrary. [7- 1 • 12J If a = b, then a-b + -ab = and ab + -a-b = 1. Hence {ab + -a-b + u) t+ v {a-b + -ab) -t = {1 +u) t + vO--t ^^ 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. li A X + B -X = 0, then AB = {-Ax + -B -x + u) AB + V {A X + B -x){-A + -B) = uAB + v{A-Bx + -AB -x). Since u and v are both arbitrary and may assume the value 0, there- fore AB = 0. '2 Sept Ims, etc., Chap. xii. 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 i) = 1, we have AB = A-BX + -AB -X Letting u = A and v = B, we have AB = AB + -AB-x And so on. But it will be found that every one of the equivalents oi A B which the law gives will be null. Choosing x for our t, we should get the solution. li A X + B -X = 0, then x = (-A x + -B -x + u) x + v (Ax* B -x) -x = {-A + u) 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 a; But if B -X c X, then B -x = {B -x) x = 0, ov B ex (2) Hence, (1) and (2), Bex c-A. When M = 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, li a = h, and therefore, by the Law of Forms, t = {ab + -a-b) t+ {a-b + -a b) -t the lower limit of t, 0, is expressed as a-b + -a b, and the upper limit of t, 1, is expressed as ab + -a -b. In the Law of Consequences, the lower limit, 0, is expressed as v {a-b + -ab), that is, in all possible ways 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 ab + -a-b. Since an expression of the form t = {ab + -a-b) t+ (a-b + -a h) -t The Classic, or Boole-Schroder, Algebra of Logic 165 or of the form t = {ab + -a-b + u) t+v ia-b + -ab) -t determines t only in tiie sense of thus expressing its hmits, 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", and the number of combinations formed from them will be 2^". Therefore, since pi + pz + pa + ■ • . =0 gives pi = 0, p2 = 0, pa = 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: '^ If all a is &, a-b = 0, and all 6 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 = ab = a (b c + -b -e) = b -c + a (b c + -b -c) = ac = b -c + ac = a {-b + c) +-ab -c = ab c = b -c + ab c = a (b c + -b -c) +-ab -c = ac + -ab -c = ab c + -ab-c The Inverse Problem of Consequences. — Just as the Law of Conse- quences expresses any inference from a = 6 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 ab + -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 {ab + -a -b) = 1, then ab + -a-b = 1, and if a-b + -ab + 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 SufRcient Conditions." 7-7 If for some value of u and some value of v t = V {ab + -a -b) t+ (a-b + -ab + u) -t, then a = b. If t = V (ab + -a-b)t+{a-b + -ab + u) -t, then [7-1, 5 • 72] [v (ab + -a -b) t+ {a-b + -ab + u) -t] -t = = {a -b + -ab + u) -t = {a -b + -ab) -t+u-t = i^ Ibid., pp. 98 ff. " Ibid., Chap, xxiii. 166 A Survey of Symbolic Logic Hence (a -b + -ab)-t = (1) Hence also [5-7] t = v {ab + -a -b) t, and [4 ■ 9] t--[v iab + -a -b)] = = t {-v + a-b + -ab) = t-v+ {a-b + -ab) t Hence [5 ■ 72] {a-b + -ab)t = (2) .By (1) and (2), (a -b + -a b) {t + -t) = = a-b + -ab. Hence [7 • 1] a = 6. 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 7i 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 =t= 6 is equivalent to "a = b is false ". Def. We might use a symbol also for " a cb is false ". But since a c 6 is equiva- lent to ah = a and to o -& = 0, its negative may be represented by a 6 4= o 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 o & = 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 =t=, the more important forms of this principle are: (1) "If a = b, then c = d", gives also, "If c =t= d, then a #= 6 ". (2) "Ifa = 6, then c = dandh = fc", gives also, "If c + d, then a + b", and "If A + A-, then a + b". (.3) "li a = b and c = d, then h = k", gives also, " If a ^ 6 and h ^ k, then c =1= d", and "li c = d and h 4= k, then a ^ b". The Classic, or Boole-Schroder, Algebra of Logic 167 (4) "a = b is equivalent to c = d", gives also, "a ^ b is equivalent toe + d". (5) "a = b is equivalent to the set, c = d, h = k, . . .," gives also, "a =1= 6 is equivalent to 'Either c =|= d or A =|= fc, or . . . ' ". '^ 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. ^Ye 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,^^ we shall discuss another mode of developing mathematical logic — the logistic method — which 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= 6. [2-1] 8-12 If a + c =i= 6 + c, then a =t= &. [3-37] 8-13 a =j= 6 is equivalent to -a 4= -b. [3-2] 8-14 a + b =^ b, a b =^ a, -a + b ^ I, and a -6 #= are all equivalent. [4-9] 8-15 li a + b = X and b ^ x, then a =i= [5-7] 8- 151 If a = and b ^ x, then a + b =^ x. [5-7] 8-16 li ab = X and b =# x, then a =j= 1. [5-71] 15 "Either . . . or . . " is here to be interpreted as not excluding the possibility that both should be true. " Chap. IV, Sect, vi, and Chap. v. 168 A Survey of Symbolic Logic 8- 161 If a = 1 and b ^ x, then ab =^ x. [5-71] 8-17 If a + 6 + and a = 0, then 6 + 0. [5-72] 8-18 If a 6 4= 1 and a = 1, then 6 =t= 1. [5-73] 8 ■ 17 allows us to drop null terms from any sum =t= 0. In this, it gives a rule by which an equation and an inequation may be combined. Suppose, for example, a + 6 =f= and x = 0. a + b = {a + b){x + -x) = ax + b x + a.-x + b -x. Hence ax + b x + a-x + b -x =f= 0. But if a; = 0, then a x = and b x = 0. Hence [8-17] a-x + b-x =)= 0. 8-2 If a + 0, then a + 6 =t= 0. [5-72] 8'21 If a + 1, then a 6 =f= 1. [5-73] 8-22 If a 6 =f= 0, then a 4= and 6+0. [1-5] 8-23 If a + 6 + 1, then a =f 1 and 6 =# 1. [4-5] 8-24 If a 6 4= a; and a = x, then 6 =t= a;. [1-2] 8-25 If a =t= and a c b, then 6 + 0. [1-9] If a c 6, then ab = a. Hence if a + and a cb, then a 6 + 0. Hence [8-22] 6 + 0. 8-26 a + 6 + is equivalent to "Either o + or 6 + ". [5-72] 8-261 ai + a2 + 03 + . . + is equivalent to "Either ai + or 02 + or as + 0, or ... ". 8-27 a 6 + 1 is equivalent to "Either a + 1 or 6 + 1 ". [5-73] 8-271 Oi 02 03 • • • +1 is equivalent to "Either ai + 1 or 02 + 1 or as + 1 or . . . ". The difference between 8-26 and 8-27 and their analogues for equa- tions — 5-72 a + 6 = is equivalent to the pair, a = and 6 = 0, and The Classic, or Boole-Schroder, Algebra of Logic 169 5 • 73 a 6 = lis equivalent to the pair, a = 1 and b = 1— points to a neces- sary difPerence between the treatment of equations and the treatment of mequations. 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 + 6 + is a consequence of the pair, a 4= and 6 =t= 0. Equivalent Inequations of Different Forms.— The laws of the equiva- lence of inequations follow immediately from their analogues for equations. 8-3 a 4= 6 is equivalent to a -6 + -a 6 =(= 0. [7-1] 8-31 a =i= 1 is equivalent to -a =|= 0. [7-12] 8-32 If $(a;i, X2, . . . .t„) and ^{xi, X2, ... a;„) be any two functions of the same' variables, then $(a;i, X2, ... .T„) =j= 4= ", and -^ C + -5 D 4= is equivalent to "Either -A C + ov -B D + 0". Hence the entire condition of the system is expressed by AB = and -AC + -BD ^0 And [8 • 5] the solution of the inequation, i-AC + AD)x + {BC + -BD)-x^Q, is x + {-A-C + A-D)x + iBC + -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 Inveese Operations, "Subtraction" and "Division" It is possible to define "subtraction" { — } and "division" { : j in the algebra. Let a — b he x such that b + x = a. And let a -.b he 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 : 6 is an element in K; nor (2) If a and b are elements in K, then a — 6 is an element in X. If a : 6 is an element, y, then for some y it must be true that b y = a. But ii b y = a, then, by 2 • 2, achy and, by 5-2, acb. Thus if a and b he so chosen that a c 6 is false, then a : h cannot be any element in K. To give a : 6 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 — 6 be ah element, X, in K, then for some x, it must be true that b + x = a. But \i b + x = a, then, by .2 • 2, b + x ca and, by 5-21, 6 c a. Thus if a and b he so chosen that b ca 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 = .t; is satisfied by any x such that -a c x. For 1 — a = a; is equiva- lent to a; + 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 = 2/, or a 2/ = 0, is satisfied by any y such that y c-a. ay = and y c-a are equivalent. Finally, these relations can always be otherwise expressed. The value of a : 6 is the value of y in the equation, b y = a. by = ais equivalent to -ab 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 + -b) = ab + u-a-b, where u is undetermined. The value of a — 6 is the value of x in the equation, b + x = a. b + x = a is equivalent to -ab + -a x + a -b -X = 174 A Survey of Symbolic Logic The equation of condition here is, -ab =0. And the solution, on this condition, is X = a-b + v a = a-b + vab, 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 xb or ab, is that region which is com- mon to a and b. If a and b do not "overlap ", then a 6 is the null-region, 0. The "sum", a + b, denotes the region which is either a or 6 (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 b denote the same region, acb signifies that a lies wholly within b, that a is included or contained in b. It should be noted that whenever a = b, acb and be a. Also, aca 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 famihar with Euler's diagrams, invented a century earher than Boole's algebra. "All a is b" is represented by a circle a wholly within a circle b; " No a is 6 " by two circles, a and b, which nowhere intersect; "Some o is 6" and "Some a is not b" by intersecting circles, sometimes with an asterisk to indicate that division of the diagram which represents the proposition. The defects of this style of diagram are obvious : All a is 6 No a is 6 Some a is 6 Some a is not b Fia. 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. ^ 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? For two elements, a and b, 1 = (a + -a) (b + -b) = a b + a -b + -a b + -a ~b 1 See Vemi, 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 + 6. 2 See above, Chap, ii, propositions 4-8 and 5-92. Applications of the Boole-Schroder Algebra For three elements, a, b, and c, 177 1 = {a + -a){b + -b){c + -c) =abc + ab-c + a-bc + -abc + a-b-c + -ab -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 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 b. 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, -{ab + -a-b), in the above, will be the sum of the other two divisions of the diagram, a-b + -a b. If it be desired to 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 -b, or -a -b -c, etc., according to the number of elements involved. —1 ^^a — - f 1 7 or E is F. (3) Either ^i is 5 or i; is not F. Let X i= A is B; y = C is D; z = E is F. (1) .11 + -^ = 1. (2) y + z = l, or -y -z = 0. (3) a; + -3 = 1, or -a: z = 0. By (1), x + -y {z + -z) = x + -yz + -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-yz = x = l. 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.^^ 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. IV = Motion exists. t = Gravitation is necessarily absent. V = A vacuum is necessary. The premises then are: ,'l) X c{y + t), or X -y -t = 0. ^2) tzc -IV, or tz IV = 0. (3) y cv, or y -V = 0. (4) V c -X, or ?) .'c = 0. (5) X cz, or X -z = 0. And since gravitation cannot be both present and absent, (6) yt = 0. Combining these equations : x-y-t+tziv + y-v + vx + x-z + yt = (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 22 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) x y = x y = 0, or x c-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 ab = and a-b = ; From (7), (-y -t+tz + y-v + v + -z + yt) iv x + {tz + y -v + y t) w -X + ("2/ -t + y -V + V + -Z + y t) -w x + {y -v + y t) -w -X = (8) Here the coefficient oi w x reduces to 1, for [5-85], y -1J + 1J = y + v, and tz + -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 tax =0, or wc-x, xc-w. -t Applications of the Boole-Schroder Algebra 219 None of the other coefficients in (8) reduces to 1. Hence the conckision 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 imphcations involved in given data is to state the data entirely in terms of the relation c , and, remembering that "If ach and hcc, then ace", as well as "ach 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 (.t; y) and the class of couples (y; x), since not all relations are symmetrical and X Ry 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 will be sufficient. The "product ", RxS,ot R S, will represent the class of all those couples (x; y) such that x Ry 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 Ry and x S y, holds. The negative of R, -R, will be the class of couples {x; y) for which X Ry is false. The null-relation, 0, will be the null-class of couples. If the class of couples {t; u) for which t Ru 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 wis 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, t and u, such that t Ruis 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 Ry is true is also such that .i- S y is true; or, to put it otherwise, that the class of couples {x ; y) for which x Ry is true is included in the class of couples (m; v) for which u S vis true. Perhaps the most satisfactory reading of i? c S is " The presence of the relation R implies the presence of the relation S". R = S, being equivalent to the pair, RcS and S cR, 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 vis 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 iJ and S are relations (that is, if there is a class of couples {x; y) such that x Ry is true, and a class of couples (m; v) such that u S r. is true), then RxS is a. relation (that is, there is a class of couples {w; z) such that IV Rz and w S z are both true). If R and S be such that there is no couple (tv; z) for which iv Rz and w S z both hold, then RxS is the null-relation, — i. e., the null-class of couples. 1 • 2 The class of couples {x ; y) for which x R y and x Ry both hold is simply the class of couples for which x Ry holds. 1 • 3 The class of couples denoted hy RxS is the same as that denoted by S X R — namely, the class of couples {x ; y) such that x Ry and x S y are both true. 1-4 The class of couples {x; y) for which xRy, xSy, and xTy all hold is identically the same in whatever order the relations be combined — i. e., Rx{SxT) = {RxS) xT. 1-5 R xO = — i. e., the product of the class of couples for which x Ry holds and the null-class of couples is the null-class of couples. 1-6 For every relation, R, there is a relation -R, the class of couples for which x Ry is false, and -R is such that: 1-61 If the relation Rx-S is null (that is, if there is no couple such that X Ry is true and x S y is false), then RxS = R (that is, the class of couples for which x Ry is true is identically the class of couples for which X Ry and x S y are both true) ; and Applications of the Boole-Schroder Algebra 221 1-Q2 If RxS = Rand R x-S = R, then R = 0— i. e., if the class of couples for which x Ry and x S y are both true is identically the class of couples for which x Ry is true, and if also the class of couples for which X Ry is true and x S y is false is identically the class of couples for which X Ry is true, then the class of couples for which x Ry is true is null. 1-71= -0 — i. e., the universal class of couples is the negative of the null-class of couples, within the universe of discourse of couples. 1-8 R + S = -{-Rx-S) — i. e., the class of couples (x;y) such that at least one of the two, x Ry and x S y, is true is the negative of the class of couples for which x Ry and x S y are both false. 1-9 Ri(S = Ris equivalent to Re S — i. e., if the class of couples {x\ y) for which x Ry and x S y are both true is identical with the class of couples for which x Ry is true, then the presence of R implies the presence of S; and if the presence of R implies the presence of S, then the class of couples {x; y) for which x Ry is true is identical with the class of couples for which X Ry and x S y are both true.^^ ^^ For a further discussion of the logic of relations, see Chap, iv, Sect. v. CHAPTER IV SYSTE?*IS BASED ON MATERIAL IMPLICATION We are concerned, in the present chapter, with the "calcukis of propo- sitions" or calculus of "material implication", and with its extension to prepositional functions. We shall discover here two distinct modes of procedure, and it is part of our purpose to set these two methods side by side. The first procedure takes the Boole-Schroder Algebra as its foundation, interprets the elements of this system as propositions, and adds to it a postulate which holds for propositions but not for logical classes. The result is what has been called the "Two-Valued Algebra", because the additional postulate results in the law: For any x, ii x ^ 1, then x = 0, and if a; =t= 0, then x = 1. This Two-Valued Algebra is one form of the calculus of propositions. The extension of the Two-Valued Algebra to propositions of the form tpXn, where Xn is an individual member of a class composed of xi, x«, x^, etc., gives the calculus of prepositional functions. n and 2 functions have a special significance in this system, and the relation of "formal implication", Ilx{ and g are elements in K, then p = g is an element in A'. [7-1] p — q is equivalent to p-q + -p q = 0, and hence [9-1] to -(p-q + -pq). Hence [1-6, M, 3-35] Q.E.D. (7) p = q is equivalent to " p is equivalent to g". [2-2] p = g is equivalent to "pcq and qcp". By (5) above, "pcq and qcp" is equivalent to "If p, then g, and if g, then p". And this is equivalent to "p is equivalent to q''. (8) If p and g are elements in iv, then p =t= g is an element in K. [9-02] (p + g) =-(p = g). Hence, by (6) above and 1-6, Q.E.D. (9) p + g is equivalent to "p is not equivalent to q". By (4) and (2) above, Q.E.D. (10) p + g is equivalent to "At least one of the two, p and g, is true. [1-8] p + g = -i-p-q). By (4) and (2) above, -(-p-q) is equivalent to "It is false that (p is false and g is false) ". And this is equivalent to "At least one of the two, p and g, is true". In consideration of the above theorems, we can henceforth write "... c . . ." for "If . . . , then . . .", "... = ..." for ". . . is equivalent to ...","... + .. ." for "Either . . . or . . .", etc., for we have proved that not only all expressions formed from elements in K and the relations x and + are elements in K, but also that expressions which in- volve c, and =, and =^= are elements in the system of the Two-Valued Algebra. The equivalence of "If . . . , then ..." with ". . . c . . .", of "Both . . . and ..." with ". . . x . . .", etc., is no longer a matter of interpretation but a consequence of 9-01, p = (p = 1). Also, we can go back over the theorems of Chapter II and, considering them as propositions of the Two-Valued Algebra, we can replace "If . . . , then . . .", etc., Systems Based on Material Implication 227 by the symbolic equivalents. Each theorem not wholly in symbols gives a corresponding theorem which is wholly in symbols. But when we consider the Boole-Schroder Algebra, without the additional postulate, 9-01, this procedure is not vahd. It is vahd only where 9 -01 is one of the postulates — i. e., only in the system of the Two-Valued Algebra. Henceforth we shall write all our theorems with p cqior "If p, then q", p = q for "p is equivalent to q", etc. But in the proofs we shall frequently use "If ... , then ..." instead of "... c ... ", etc., because the symbolism sometimes renders the proof obscure and makes hard reading. (That this is the case is due to the fact that the Two-Valued Algebra does not have what we shall hereafter explain as the true "logistic" form.) 9-15 + 1. 0=0. Hence [9-13] + 1. 9-16 (p 4= g) = i-p = q) = ip = -q). (1) li p = q and p = I, then g =(= 1 and [9-13] q = 0. And if p = 1, [3-2] -p = 0. Hence -p = q. (2) If p + g and p 4= 1, then [9-13] p = 0, and [3-2] -p = 1. Hence ii p =\= q, then g + 0, and [9 • 14] g = 1 = -p. (3) If -p = g and g = 1, then -p = 1, and [3-2] p = 0. Hence [9-15] p =i= g- (4) li-p = q and g #= 1, then -p #= 1, and [9 ■ 13] -p = 0. Hence [3-2] p = 1, and p 4= g- By (1) and (2), if p =t= g, then -p = q. And by (3) and (4), if -p = g, then p =t= g. Hence p =¥ q and -p = q are equivalent. And [3-2] (-P = q) = (p = -g). This theorem illustrates the meaning of the relation, =, in the calculus of material implication. If p ^ q, then either p = I and g = or p = and g = 1. But if y> = 1, then -p = 0, and if p = 0, then -p = 1. Hence the theorem. Let p represent "Caesar died", and g represent "There is no place like home". If "Caesar died" is not equivalent to "There is no place hke home", then "Caesar did not die" is equivalent to "There is no place like home". The equivalence is one of truth values — { = 0) or { = Ij — not of content or logical significance. 9.17 p = (p = l) = (p^O) = {-p = 0) = i-p + 1). [9-0M3-14-16] 9-18 -p = (2? = 0) = (p + 1) = i-p = 1) = i-p + 0). [9-M3-14-16] 228 ^-1 Survey of Symbolic Logic 9-2 (p = 1)(2> = 0) = 0. [2-4] p-p = 0. And [9-01] p = (p = 1); [9-1] -p = {p = 0). No proposition is both true and false. 9-21 (p + l)(p + 0) = 0. [2-4] -pp = 0. And [9-18] -p = (p + 1); [9-17] p = (p 4= 0). 9-22 (p = 1) + (p = 0) = 1. [4:-8]p + -p = 1. Hence [9-01 -1] Q.E. D. Every proposition is either true or false. 9-23 (p4= l)+(p + 0) = 1. [4-8, 9-OM] Theorems of the same sort as the above, the proofs of which are obvious, are the following: 9-24 ipq) = ipq=l) = (pq + O) = {p = l)(g = 1) = {p + 0) (g + 0) = (p + 0)(g = 1) = (p = l)(g + 0) = -i-p + -q) = (-p + -g = 0) = [(p = 0) + (g = 0) =0] = lip =h 1) + (q + 1) = 0], etc., etc. 9-25 (p + g) = (p + g = 1) = 0) + g + 0) = (p = 1) + (g = 1) = (2^ + 0) + (g + 0) = -i-p -q) = [(p = 0)(g = 0) = 0] = [iv + l)(g + 1) + 1], etc., etc. These theorems illustrate the variety of ways in which the same logical relation can be expressed in the Two-Valued Algebra. This is one of the defects of the system — its redundancy of forms. In this respect, the alternative method, to be discussed later, gives a much neater calculus of propositions. We turn now to the properties of the relation c . We shall include here some theorems which do not require the additional postulate, 9-01, for the sake of bringing together the propositions which illustrate the meaning of "material implication". 9-3 (peg) = (-p+g) = {p -q = 0). [4-9] (peg) = (p-g = 0) = (-p + g = 1). [9-01] (-p + g= 1) = (-p + g). "p materially implies g" is equivalent to "Either p is false or g is true", and to "It is false that p is true and g false". Since peg has been proved to be an element in the system, " It is false that p materially implies g" may be symbolized by -(p e g). Systems Based on Material Implication 229 9-31 -ipcq) = {-p+q = 0) = (p-q). [3-4] -(-p+g) = p-q. And [9-3] -{pcq) = -(-p + g). [9-02] -i-p+q) = {-p+q = 0). "p does not materially imply g" is equivalent to "It is false that either p is false or q is true", and to "p is true and g false". 9-32 (p = 0)c(pcg). [5-63] Ocg. Hence Q.E.D. If p is false, then for any proposition g, p materially imphes q. This is the famous — or notorious — theorem: "A false proposition implies any proposition". 9-33 (g= l)c(2jcg). [5-61] pel. Hence Q.E.D. This is the companion theorem: "A true proposition is implied by any proposition". 9-34 -(peg) c(p = 1). The theorem follows from 9-32 by the rednctio ad absurdurn, since if -(pcq), then [9-32] p =f= 0, and [9-14] p = 1. If there is anj^ proposition, g, which p does not materially imply, then p is true. This is simply the inverse of 9-32. A similar consequence of 9 • 33 is : 9-35 -(pcg)c(g = 0). If -{pcq), then [9-33] g + 1, and [9-1.3] g = 0. If J) does not materially imply q, then g is false. 9-36 -(pcq) c{pc-q); -(p cq) c {-p c q); -{p c q) c {-p c-q). [9 •34- 35] If -ipcq), then p = 1 and g = 0. [3-2] If p = 1, -p = 0, and if g = 0, then -q = 1. [9 • 32] If -p = 0, then -peg and -p c -g. [9 ■ 331 If -g = 1, then p e -g. If p does not materially imply g, then p materially implies the negative, or denial, of g, and the negative of p implies g, and the negative of p implies the negative of g. If "Today is Monday" does not materially imply "The moon is made of green cheese", then "Today is Monday" implies "The moon is not made of green cheese", and "Today is not Monday" implies "The moon is made of green cheese ", and "Today is not Monday" implies "The moon is not made of green cheese". Some of the peculiar properties of material implication are due to the 230 .1 Survey of Symholic Logic fact that the relations of the algebra were originally devised to represent the system of logical classes. But 9-36 exhibits properties of material implication which have no analogy amongst the relations of classes. 9-36 is a consequence of the additional postulate, p = (p = !)• For classes, c represents "is contained in": but if a is not contained in h, it does not follow that a is contained in not-6 — a may be partly in and partly outside of 6. 9-37 -{p c q) c (g cjj). [9-36] If -{f cq), then -p c-q, and hence [3-1] qcp- Of any two propositions, p and q, if p does not materially imply q, then q materially implies p. 9-38 {pq)c[{pcq)(qcp)]. [9-24] pq = {p = l)(g = 1). Hence [9-33] Q.E.D. If p and q are both true, then each materially implies the other. 9-39 (-p-g)c[(pcg)(gc2J)]. [9-24] -p-q = {-p = l)(-g = 1) = (p = 0)(g = 0). Hence [9-32] Q.E.D. If p and g are both false, then each materially imphes the other. For any pair of propositions, p and q, there are four possibilities: 1) p = 1, q = 1: J) true, g true. 2) p = 0, q = 0: p false, q false. 3) p = 0, g = 1: p false, g true. 4) p = 1, q = 0: p true, g false. Now in the algebra, cO, 1 c 1, and c 1; but 1 cO is false. Hence in the four cases, above, the material implications and equivalences are as follows : 1) pcq, qcp, p = q. 2) pcq, qcp, p = q. 3) pcq, -{qcp), p ^ q. 4) -{pcq), qcp, p ^ q. This summarizes theorems 9 -31-9 -39. These relations hold regardless of the content or meaning of p and g. Thus pcq and p = q are not the "implication" and "equivalence" of ordinary logic, because, strictly speak- ing, p and g in the algebra are not "propositions" but simply the "truth values" of the propositions represented. In other words, material impH- Systems Based on Material Implication 231 cation and material equivalence are relations of the extension of proposi- tions, whereas the "implication" and "equivalence" of ordinary logic are relations of intension or meaning. But, as has been mentioned, the material implication, p cq, has one most important property in common with " q can be inferred from p" in ordinary logic; if p is true and q false, pcq does not hold. And the relation of material equivalence, p = q, never connects a true proposition with a false one. These theorems should make as clear as it can be made the exact meaning and character of material implication. This is important, since many theorems whose significance would otherwise be very puzzling follow from the unusual character of this relation. Two more propositions, of some importance, may be given : 9-4 {p qcr) = (qp cr) = [p c{qcr)] = [qc.{pc.r)]. [1-3] pq = qp- Hence [3-2] -{pq)=-{qp), and [-{p q) -^ r\ = [-(g ?) + ?•]• But [9 • 3] [-(p q) + r\ = {pqc r), and [-{q p) + r] = (q per). And [3 ■ 41] [-(p q) + r] = [{-p + -q) + r] = [-p + {-q + r)] = [p c (g c r)] Similarly, [-{q p) + r] = [q c {p c r)]. This theorem contains Peano's Principle of Exportation, lip ?) c r] c [p c (g c r)] "If pq implies r, then p implies that g implies r"; and his Principle of Importation, [pc(gcr)] c[(pg) cr] "If p implies that g implies r, then if p and q are both true, r is true." 9-5 [(p g) c r] = [(p -r) c -g] = [(g -r) c -p]. [9 • 3] [(p g) c r] = [-(p q) + r] = [(-p + -g) + r] = [(-p + r) + -q] = [i-q + r)+ -p] = [-(p -r) + -g] = [-(g -r) + -p]. [9-3] [-(p-r)+-g] = [{p-r)c-q], and [-{q-r)+-p] = [(g-r) c-p]. If p and g together imply r, then if p is true but r is false, g must be false, and if g is true but r is false, p must be false. This is a principle first stated by Aristotle, but especially important in Mrs. Ladd-Franklin's theory of the syllogism. We have now given a sufficient number of theorems to characterize the Two-Valued Algebra — to illustrate the consequences of the additional 232 A Survey of Symbolic Logic postulate p = (p = 1), and the properties of p cq. Any further theorems of the system will be found to follow readily from the foregoing. A convention of notation which we shall make use of hereafter is the following: A sign =, unless enclosed in parentheses, takes precedence over any other sign; a sign c, unless enclosed in parentheses, takes precedence over any + or x ; and the sign + , unless enclosed in parentheses, takes precedence over a relation x . This saves many parentheses and brackets. II. The Calculus of Propositional Functions. Functions of One Variable The calculus of propositional functions is an extension of the Two- Valued Algebra to propositions which involve the values of variables. Fol- lowing Mr. Russell,' we may distinguish propositions from propositional functions as follows: A proposition is any expression which is either true or false; a propositional function is an expression, containing one or more variables, which becomes a proposition when each of the variables is re- placed by some one of its values. There is one meaning of "Today is Monday" for which 'today' denotes ambiguously Jan. 1, or Jan. 2, or ... , etc. For example, when we say '"Today is Monday' implies 'Tomorrow is Tuesday'", we mean that if Jan. 1 is Monday, then Jan. 2 is Tuesday; if Jan. 2 is Monday, then Jan. .3 is Tuesday; if July 4 is Monday, then July 5 is Tuesday, etc. 'Today' and 'tomorrow' are here variables, whose values are Jan. 1, Jan. 2, Jan. 3, etc., that is, all the different actual days. When 'today' is used in this variable sense, "Today is Monday" is sometimes true and sometimes false, or more accurately, it is true for some values of the variable 'today', and false for other values. "Today is Monday" is here a propositional function. There is a quite different meaning of "Today is Monday" for which 'today' is not a variable but denotes just one thing — Jan. 22, 1916. In this sense, if "Today is Monday" is true it is always true. It is either simply true or simph' false: its meaning and its truth or falsity cannot change. For this meaning of 'today', "Today is Monday" is a proposition. 'Today, ' meaning Jan. 16, 1916, is one value of the variable 'today'. When this value is substituted for the variable, then the propositional function is turned into a proposition. ^ See Principles of Mathematics, Chap, vii, and Principia Mathemaiica, i, p. 15. Mr. Russell carries out this distinction in ways which we do not follow. But so far as is here in question, his view is the one we adopt. Principia Mathemaiica is cited hereafter as Principia. Systevis Based on Material Implication 233 We may use px — which is the only conception of "individual" which we require. " It may be urged that Xx + fX-!. + ipxz + . . and (pXi c ipX], ■»■ ipXn + ipx^ + . . and ipxz c tpxx + (f/Xi + 'px^ + . . . , etc., etc. By 10 • 2, if ipx is true for all values of x, then it is true for any given value of X, or "What is true of all is true of any given one". By 10-21, If ipx is true for one given value of x, then it is true for some value of x, or " What is true of a certain one is true of some". It might be thought that the implication stated by 10 • 21 is reversible. But we do not have S ipx c ipx^, because ipXn may be pXt, and "Ztpxc tpx^ would not hold generally. For example, let (. (px^y. . .) + P = ix xipx) cZ^px, and ^^(>px x\px) cl^xj/x Hence [5-34] Z^{(px xxpx) cSx is false or \px is true: and this relation conveys so little information that it is hardly worth while to study its properties. ILi{(px c\px) is the relation of "formal implication" — "For every x,. at least one of the two, ' ipx is false' and 'tpx is true', is a true statement". The negative of Jl^{p{x, ys) x . That is, I[y'Sx(p{x, y) means "For some x and 2/1, v{x-, y) is true, and for some X and 2/2, v{x, y) is true, and for some x and 2/3, p{x, y) is equivalent to "Whatever value of (x, y), in (p{x, y), {x, y)n may be, ^{x, y)n". [10-23] 11-24 Ii.Jlyip{x, y) is equivalent to "Whatever values of x and y, in ,p{x, y), X, and y, may be, ,p{xry,)". [10-23] HJlytpix, y) is equivalent to "Whatever value of x, in Ilyip{x,y), Xr may be, 'ilyip{Xry)" . And Ilyipi^Xry) is equivalent to "Whatever value of y, in {x,y)n" is equivalent to "AYhatever values of x and y, in < { fi^iVs) + 'pixiys) + ipixsys) + . . . } X . . . Etc., etc. Since x is distributive with reference to + , this expression is equal to the sum of the products of each column separately, plus the sum of all the cross-products, that is, to A + { ^^yp{x, y, z) = SxS(j,, z)v{x, y, z) Since UJliy, z) < ni/-?/) x . . [5-98] = Uy{ etc. 12-6 '2(px xUxl/y = U.xpyx1,(px^ = i:^Uy{(px x.xpy) = 'SJIyixpy x ,px) = Uy-2^i(pxxxPy) = HyZ^ixPy x (px). (!) [1-3] i:(pxxUpx) '2,yli.:,{\py X (px) UyU.:, ( (px X ^y) Sj,!!;, ( ipx x ^j/y) Uxf/y xUx + \l/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 x ^t/) and ( ipx + :py) give four different kinds of implication relation: (1) UJJy{( pXnX4/Xn) = ti. "The couple {xm 2/n) belongs to the field, or extension, of the relation deter- mined by (z Rw)" means that Xm R yn is true. 15-02 RcS = U„y[ixRy)cixSy)]. Def. This definition is strictly parallel to 13-02, (a c /3) = n^ (a; € a c a; e j3) because, by 15-01, (x R y) is {x, y) e R and (x S y) is {x, y) e S. A similar remark applies to the remaining definitions. 15-03 iR = S) =U„y[{xRy) = (xSy)]. 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. " See above, pp. 253 ff. Systems Based on Material Implication 271 15-04 RxS = £y[{xRy)x{xSy)]. Dei. The logical product of two relations, R and S, is the class of couples (x, y. such that X has the relation Rto y and x has the relation S to y. If R is "friend of", and S is "colleague of", i? xS 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 ov x has the relation S to y. R+S will be "Either E of or S of". 15-06 -R = xy-{xRy). Def. -R is the relation oi x to y when x does not have the relation Rto y. It is important to note that RxS, R + S, and -R are relations: xiR x S)y, x{R + S)y, and x -R y are significant assertions. The "universal-relation" and the "null-relation" are also definable after the analogy to classes. 15-07 1 = xy [f (a;, y) c f (a;, y)]. Def. x has the universal-relation to y in case there is a function, f, such that f(a;, 2/) cf(a;, ?/),- i. e., in case x and 2/ have ant/ 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 (iySz)]} = Uy -[(x Ry)x{yS z)] = Uy[-{x Ry) + -iy S z)] = Uy[{x -R y) + {y -S 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, -(ii 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 ; ^{R\S) = v.S|"i? for X ^{R \S)z = z{R\ S)x = 2j,[(2 Ry)x(yS x)] = Zy[{y Sx)x{zR y)] = Sj(a- -S y) X {y ^R z)] = .t(-S I ^R)z If X is employer of a benefactor of z, then the relation of 2 to x is "bene- fitted by an employee of". Similarly, -(fl t S) = "S t -i? If X is hater of all non-helpers of z, the relation of 2 to a; 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 I {S + T)]z = 2,{ (.T Ry)x[y{S^ T)z] ] = i:y{{xRy)x[{ySz)^{yTz)]\ = ^y{[{x Ry)xiyS z)] + [(.r Ry)x{yT z)]] = [x{R\S)z\ + [x{R\T)z\ Similarly, {R-i- 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: R\{SxT)'R + a'R However, for the logistic development of mathematics, these properties are of the highest importance. We quote from Princiyia Mathematica: ^^ " 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 Prirwipia 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 op Principia Mathematica We have now presented the extensions of the Boole-Schroder Algebra — the Two-Valued Algebra, prepositional 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 S^a;, Il ~p .V . pv q, ~p V ~g, etc., etc., may be substituted for p or g or r in any assumed proposition or any theorem. Such substitution, for which no postulates would ordinarily be stated, is one of the fundamental operations by which the system is developed. Another kind of substitution which is fundamental is the substitution for any complex of symbols of its defined equivalent, where such exists. This operation is covered by the meaning assigned to " = . . Df ". Only one other operation is used in the development of this calculus of elementary propositions — the operation for which *1-1 and *1-11 are assumed. If by such substitutions as have just been explained there results a complex of symbols in which the main, or asserted, relation is s , and if that part of the expression which precedes this sign is identical with a postulate or previous theorem, then that part of the expression which follows this sign may be asserted as a lemma or new theorem. In other words, a main, or asserted, sign s has, by *1 ■ 1 and *1 • 11, the significant property of "If ... , then . . .". This property is explicitly assumed in the postulates. The main thing to be noted about this operation of inference is that it is not so much a piece of reasoning as a mechanical, or strictly mathematical, operation for which a rule has been given. No Systems Based on Material Implication 285 "mental" operation is involved except that required to recognize a previous proposition followed by the main implication sign, and to set off what follows that sign as a new assertion. The use of this operation does not, then, mean that the processes and principles of ordinary logic are tacitly presupposed as warrant for the operations which give proof. What is the significance of this assumption of the obvious in *1 • 1, *1 • 11, *l-7, *1-71, and *l-72? Precisely this: these postulates explicitly assume so much of the logical operations as is necessary to develop the system, and beyond this the logic of propositions simply is not assumed. To illustrate this fact, it will be well to consider carefully an exemplary proof or two. *2-01 1-: p3 ~2J • => ■ ~J' Dem. Taut — \-: ~p v ~p .d . ~p (1) [(1).(*1-01)] h:p3~p.D.~2J "Taut" is the abbreviation for the Principle of Tautology, *l-2 above. ~p/p 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, 2? 3 ~p is substituted for its defined equivalent, ~p v ~p, and the proof is complete. *2-05 \-: . q:3r .^i p^ q .^ . p^r Dem. Sum — \\~:.q^r.^i~ps/q.o.~p'^r (1) [(1) . (*1-01)] \-: . qDr .3: p^q .-2 .p-Dr Here "Sum" refers to *l-6, above. And (1) is what *l-6 becomes when ~p is substituted for p. Then, by *1-01, poq and p^r are substituted for their defined equivalents, ~pvq 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 \-i.p^q.^: q^r .o.por g 3 r, p 3 I 3g, par "! 7, r ] Dem. Comm L p, q, \-::q^r.D: poq.o.p^rz.^s.p^q.^iq^r.^.p^r (1) [*2 ■ 05] h: ■ 9 3 r . 3 : p 3 g . 3 . p 3 r (2) [(1) . (2) . *1-11] h: ■ 233g.3: g3r .3. par 286 A Survey of Symbolic Logic "Comm" is *2-04, previously proved, which is p.o,q:3r::3:qo.pDr. When, in this theorem, g s r is substituted for p, p^q for q, and por for r, it becomes the long expression (1). Such substitutions are valid by *l-7, *1 -71, and the definition *1 -01 : if 2? is a proposition, ~p is a proposition; if ~p and q are propositions, ~p v g is a proposition; and p d g is the defined equivalent of ~p v g. Thus j^^Q can be substituted for p. If we replace the dots by parentheses, etc., (1) becomes h Kg 3 r) 3 [{p Dq)^(p^ r)] } 3 { (p 3 g) 3 [(g 3 r) = (2? = 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, naturallj', 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. We have now seen that the calculus of propositions in Principia Mathe- 7natica 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 = I, p =# 0, etc., etc., and the absence of the entities and 1, the two systems are identical. Anj- 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 { = 0} and { = Ij. 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 = lis 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. ^^ The comparison of our treatment of propositional functions with the same topic in Principia is not quite so simple.'^ 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 'n.px„c^(px, which is an immediate consequence of 10-21, by 5-33. *9-13 is "If whatever value of x, in ipx, x„ may be, (pXn, then Ilcpa;, " and this implication is contained in the equivalence stated by 10-23. These principles which are assumed in Principia Mathematica are suf- ficient to give all further propositions concerning functions of one variable, without assuming (x) . ipx to be the product of (pxi, q-iqp 3-11 p oq-i q op If p and q are both true, then q If p and q are consistent, then q and p are both true. and p are consistent. 1-2 q p -tp 3-12 q op -i- ~p If g and p are both true, then p If g and p are consistent, then is true. it is possible that p be true. 1-3 p -ip p 3-13 -~p-tpop If p is true, then p and p are If it is possible that p be true, both true. then p is consistent with itself. 1-4 p{qr)-iq{pr) 3-14 p o (q r) -i q o (p r) The correspondence exhibited in the last line seems incomplete. But we should note with care that while P(q r) = qip r) = {p q)r and any one of these may be read " p, q, and r are all true", p o{qor) is not " p, q, and r are all consistent", p o(qor) means " p is consistent with the proposition 'q is consistent with r'" . Let p = "Today is Tues- day"; q = "Today is Thursday"; r = "Tomorrow is Friday". Then gor is true. And it happens to be Tuesday, so p is true. Since p and qor are both true in this case, they must be consistent: p o {q or) is true. But " p, q, and r are all consistent" is false. "Today is Tuesday" is incon- sistent with "Today is Thursday" and with "Tomorrow is Friday". Suppose we represent " p, q, and r are all consistent" by p og or. Then as a fact, p oqor will not be equivalent to p o (g o r) . Instead, we shall have p o q or = p o {q r) = q o {p r) = (p q) o r "p, q, and r are all consistent" is equivalent to "p is consistent with the proposition 'q and r are both true'", etc. We may, then, add two new definitions : 301 pqr = p{qr). Dei. 3 02 p oqor = p o{qr). Def . 3-02 is typical of triadic, or polyadic. strict relations: when parentheses are introduced into them, the relation iriside the parentheses degenerates into the corresponding material relation. In terms of the new notation of 3-01 and 3-02, the last line of the above table would be p qr -i q p r poqor-iqopor which exhibits the analogy more clearly. The System of Strict Implication 301 We must now prove the theorems in the right-hand column of the table 3 11 p o q-t q op 2-74 {qp/p; qp/q}: 1 . 1 -i [- ~(p g) -1 - ~(g ^j)] (1) 1-01: (1) = Q.E.D. 3 12 q Op-i — p 2-74 [qp/p; p/q}: 1-2 -i[- ~{q p) -i - ~p] (1) 1-01: (1) = Q.E.D. 3-13 — p -ip op 2-74 {pp/p}: 1.3-i[-~pH-~ipp)] (1) 1-01: (1) = Q.E.D. 3 14 p o (qr) -i qo (pr) 2-74 {p{qr)/p; q{pr)/q}: 1 ■ i -i - ~[p{q r)] -i - ~[q{p r)] (1) 1.01: (1) = Q.E.D. (In the above proof, the whole of what 1-4 is stated to imply should be enclosed in a brace. But in such cases, since no confusion will be oc- casioned thereby, we shall hereafter omit the brace.) 3 15 p o{qr) = {p q) or = qo {p r) 2-76: 2-9-! -~[p(gr)] =-~[g(pr)] (1) 2 • 76 : 2 ■ 9H - ~[p(g 0] = - ~[(P q)r] (2) 1-01: (2) x(l) = Q.E.D. An exactly similar analogy holds between the material logical sum, p + q, and the strict logical sum, p a g. 3-21 p + q-iq + p 3-31 pAq-iqAp "At least one of the two, p and "Necessarily either p or q" im- q, is true" implies "At least one plies "Necessarily either g or j)"- of the two, q and p, is true". 3-22 p-ip + q 3-32 ~-p-ipAq If p is true, then at least one of If i^ is necessarily true, then the two, p and q, is true. necessarily either p or g is true. 3-23 p + p-ip 3-33 pAp-i~-p If at least one of the two, p and If necessarily either p is true or p, is true, then p is true. pi is true, then p is necessarily true. 3-24 p+ (q + r) -iq+ (p + r) 3-34 p A(q + r) -i q A(p + r) As before, the analogy in the last line seems incomplete, and as before, it really is complete. And the explanation is similar. p+ (q + r) and 302 A Survey of Symbolic Logic q+ ip + r) both mean "At least one of the three, p, q, and r, is true". But p A{q Ar) would not mean " One of the three, p, q, and r, is of necessity true". Instead, it would mean "One of the two propositions, p and 'necessarily either q or r', is necessarily true". To distinguish pA{q + r) from p A{q at) is rather difficult, and an illustration just now, before we have discussed the case of implication, would probably confuse the reader. We shall be content to appeal to his 'intuition' to confirm the fact that "Necessarily one of the three, p, q, and r, is true" is equivalent to "Neces- sarily either p is true or one of the two, q and r, is true" — and this last is p A(q + r). If we chose to make definitions here, similar to 3-01 and 3 • 02, they would be p + q + r = p+ (q + r) and p AqAr = p A{q + r) Proof of the theorems in the above table is as follows: 3-21 p + q -i q + p 1-1 {-q/p; -p/q}: -q-p-i-p-q (1) 2.Q2: {l)-i-{-p-q)-i-(-q-p) (2) 1-05: (2) = Q.E.D. 3-22 p-ip + q Similar proof, using 1 -2 in place of 1-1. 3-23 p + p -ip Similar proof, using 1-3. 3-24 p+ iq-t-r) -i q+ (p + r) 1-4 {-q/p; -piq; -rjr]: -q(-p -r) -i -pi-q -r) (1) 2-62: {l)H-[-p{-q-r)]-t-[-q{-p-r)] (2) 2-51: (2) = -{-p-[-{-q-r)]} ^ -{-g-[-(-p _,)]} (3) 1-05: (3) = p + -(,-q-r)-iq + -{-p-r) (4) 1-05: (4) = Q.E.D. 3-25 p+ (q + r) = (p + q) + r = q+ (p + r) Similar proof, using 2-9 and 2-91, and 1-06. 3-31 p Aq-i q Ap- 1-1 {-q/p; -p/q}: -q-p-i-p-q (1) 2-2: (l)H~(-p-q)^~(-q.p) (2) 1-04: (2) = Q.E.D. 3 32 — p -ip Aq Similar proof, using 1-2 in place of 1-1. The System of Strict IrrtTplication 303 3-33 p Ap -i — p Similar proof, using 1-3. 3-34 p A{q + r) -iqA^p + r) 1-4 {-q/p; -pjq; -)•/?•}: -q{-p -r) -i -p{-q -r) (1) 2-2: (1) -f ~[-p{-q -,•)] -i ~[-q{-p -r)] (2) 2-51: (2) = ~{-p-[-{-q-r)]]^~{-q-[.{-p.r)]] (3) 1 • 04 : (3) = pA -{-q -r)-iqA -{-p -r) (4) 1-05: (4) = Q.E.D. 3-35 p A{q + r) = {p + q) Ar = qA{p+r) Similar proof, using 2-9 and 2-91, and 1-06. Again, an exactly similar analogy holds between material implication, p cq, and strict implication, p-i q. 3-41 (p c g) ^ (-g c -2J) If p materially implies q, then ' q is false' materially implies 'p is false'. 3-42 -p-i(pcq) If p is false, then p> materially implies any proposition, q. 3-43 ipc-p)-i-p If p materially implies its own negation, then p is false. 3-44 [p c {q c r)] -i [q c (p c r)] The comparison of the last line presents peculiarities similar to those noted in previous tables. The significance of 3-54 is a matter which can be better discussed when we have derived other equivalents of p-i (qcr). The matter will be taken up in detail further on. The theorems of this last table, like those in previous tables, are got by transforming the postulates 1-1, 1-2, 1-3, and 1-4. In consideration of the importance of this comparison of the two kinds of implication, we may add certain further theorems which are consequences of the above. 3.45 p-t(.qcp) 3-55 ~-p-i{q-ip) If p is true, then every proposi- If p is necessarily true, then p is tion, q, materially implies p. strictly implied by any proposi- tion, q. 2-62 (p-iq) -i {-q -i -p) If p strictly implies q, then ' q is false' strictly implies 'p is false'. 3-52 -p-i (p-iq) If p is impossible (not self-con- sistent, absurd), then p strictly im- plies any proposition, q. 3-53 (p-i -p) -i ~p If p strictly implies its own nega- tion, then p is impossible (not self- consistent, absurd). 3-54 [p-i{qcr)]-i[q-i{pcr)] 304 A Survey of Symbolic Logic 3-46 {-pcp)-ip 3-56 (-pHp)-i~-p If p is materially implied by its If p is strictly implied by its own own denial, then p is true. denial, then p is necessarily true. 3-47 -{pcq)-ip 3-57 -{p-iq)-t-~p If p does not materially imply If p does not strictly imply any any proposition, q, then p is true. proposition, q, then p is possible (self -consistent) . 3-48 -{pcq)-i-q 3-58 -{p -t q) -i q If p does not materially imply q, If 2^ does not strictly imply q, then q is false. then p is possibly false (not neces- sarily true) . Note that the main or asserted implication, which we have translated " If . . . , then . . . ", is always a strict implication, in both columns. 3-42 and 3 -45-3 -48 are among the best known of the "peculiar" the- orems in the system of Material Implication. For this reason, their ana- logues in which the implication is strict deserve special attention. Let us first note that — p -i {-p -i p) is a special case of 3 ■ 55. This and 3-56 give us at once ~ -P = (.-P ^ P) This defines the idea of "necessity". A necessarily true proposition — e. g., "I am", as conceived by Descartes — is one whose denial strictly implies it. Similarly, p-ii-pcj)) is a special case of 3-45. And this, with 3 • 46, gives p = i-pc p) A true proposition is one which is materially implied by its own denial. This point of comparison throws some light upon the two relations. The negative of a necessary proposition is impossible or absurd. -p -i (p -i -p) is a special case of 3-52. This, with 3-53, gives "P = {P -^ -P) And p -I —p is equivalent to -{p o p). Thus an impossible or absurd propo- sition is one which strictly implies its own denial and is not consistent with itself. Correspondingly, we get from 3 ■ 42 and 3 ■ 43 -P = i.P^-p) A false proposition is one which materially implies its own negation. It is obvious that material implication, as exhibited in these theorems, The Systevi of Strict Implication 305 is not the relation usually intended by "implies", but it may be debated whether the corresponding properties of strict implication are altogether acceptable. We shall revert to this question later. At least, these propo- sitions serve to define more sharply the nature of the two relations. Proof of the above theorems is as follows : 3-46 3-47 3-48 3-41 ipcq)-t {-q c -p) M {p/q; -q/p}: -qp-ip-q 2-62: (l)-i-(23-g)^-(-gp) 2-51: (2) =-(p-9)-,-[-g-(-p)] l-OS: (3) = Q.E.D. 3-42 -p -i {p c q) 1-2 [plq; -q/p]: p -q -i p 2-62: (l)-,-ps-(p-g) 1-03: (2) = Q.E.D. 3-43 {pc -p) -{ -p Similar proof, using 1 -3. 3-44 p c{qcr) -tqclp cr) 3 3 2 1 2 (-pc 3 2 -ipc 2 2 -ipc 3 2 4 62: 51: 03: 03: -r) {q/p; p/q; -r/r}: q{p -r) -i piq ■ !: (l)-i-[p(q-r)]-l-[qip-r)] .: (2) =-{p-[-(q-r)]\-i-{q-[-ip-r)]] ) : (3) = p c-(q-r) -iqc-{p -r) !: (4) = Q.E.D. 3-45 p-i iqcp) ■42 {-p/p; -q/q] : -(-p) -I {-pc-q) •41: i-po-q)-i[-{-q)c-i-p)] •51: (2) = (-pc-g)-l(gcp) ■6: (1) X (3) -!-(-?) -i(gc2J^ ■51: (4) = Q.ED. ■■p) -tp •43 {-p/p}: [-pc-{-p)]-t-i-p) •51: (1) = Q.E.D. :q) -ip •62 {-p/p; pcq/q}: 3^42 -i -(p eg) ^ -(-p) •51: (1) = Q.E.D. ■.q)-i-q ■45 {q/p; p/q}: q-i{pcq) •62: (1)H Q.E.D. (1) (2) (3) (1) (2) (1) (2) (3) (4) (1) (2) (3) (4) (1^ (1) (1) 21 306 A Survey of Symbolic Logic 3-52 "p-iip-iq) 2-1 {-q/q}: p-q-ip (1) 1-8: (1) = -p-t-ip-q) (2) 1-02: (2) = Q.E.D. 3 53 (p -i -p) -j ~p Similar proof, using 1-3. 3-54 [p-i{qcr)]-i[q-i{p oq) ■i{r os)]-i[{'p q) -iir s)] 2-75 {pg/p; rs/g|: [-~(pg)^-~(rs)]-i[(pg)-)(rs)] (1) 1-01: (1) = Q.E.D. 4-33 [(p-}g) -{(/-S 5)] ^ [(peg) c (re 5)] 4-15: [(pcg)^(rcs)]H[(pcg)c(f C5)] (1) 1-6: 4-3 x(l)^ Q.E.D. 4-34 [(p Ag) -j (r A^)] -i [(p+ g) e (r + 5)] Similar proof, using 4-31 in place of 4-3. 4-35 [(p og) -i (r os)] -i [(p g) e (r s)] Similar proof, using 4-32. Note that as a subordinate relation, p o g reduces directly. In theorems 4 -3-4 -32, postulate 1-8 only has been used, and the reduc- tion of strict relations to material relations is incomplete. In theorems 4 -33-4 -35, postulates 1-8 and 1-7 have both been used, and the reduction is complete. In these theorems, dyads of dyads are dealt with. The reduction extends to dyads of dyads of dyads, and so on. We may illustrate this by a single example which is typical. Hypothesis: [{p -iq) -i (-p a g)] H [(p o -g) -i-{p -i q)] To prove: [(p e g) e (-p + g)] c [(p -g) c -(p e g)] (The hypothesis is true, though it has not been proved.) 2-71 {p-g/p;p-g/g!:[-(p-g)^-(p-g)]-s[~(p-g)H~(p-g)] (1) •1-02, 1-03, 1-04, 1-05, and 2-51: (1) = [(p e g) H (-P + g)] -i [(p ^ g) ^ (-p a g)] (2) The System of Strict Implication 311 1-6: (2) xHyp. -i [{p cq)-i {-p + q)] s [(p o -q) -i -{p -i q)] (3) 2-72 {p -qlp) p -q/q] : [- ~{p -q) s - ~(p -q)] -^ [{p -q) h {p -q)] (4) 1-01, 1-02, 1-03, and 2-51: (4) = [{p O -q) -i -{p -J q)] -J [{p -q) -i -{p c q)] (5) 1-6: (3) X (5) -i [{p c g) -i {-p + g)] -i [{p -q) h -(p c q)] (6) 4-33: (6)-iQ.E.D. In any theorem in which ~p is related to ~q, or - ~p to - ~g, or ~ -p to ~ -g, ~ may be replaced by -. This follows immediately from 2-77. We illustrate briefly the reduction in those cases in which ~r, or - ~r, or ~ -r, is related to p o q, ot p a q, qtc p-iq. 4-36 (p oq-i- ~r) -i (p q-ir) 2-75 {p qlp; r/q} : [- ~{p q) -i~r\-i{pq-i f) (1) l-Ql: {I) = {p oq-i- ~r)-i{p q-ir) (2) 4-15: ipq-ir) -i{pqcr) (3) 1-6: (2) x(3)-tQ.E.D. 4-37 (p Aq-i ~r) -t {p+ q c-r) 2-7 {-p -qlp; rlq] : [~(-p -q) -J ~r] -J [-{-p -q) -i -r] (1) 4 • 15 : [-{-p -q) -! -r] -i [-(-p -g) c -r] (2) 1 • 6: (1) X (2) -1 [~(-p -g) -i ~r] -i [-(-^^ -g) c -r] (3) 1-04 and 1-05: (3) = Q.E.D. A dyad of triadic strict relations, e. g., p o{qr) -iqo {p r), reduces just like a dyad of dyads, because a triadic strict relation is a dyadic strict relation — with a dyadic material relation for one member. But a triad of dyadic strict relations behaves quite difl^erently. Such is postulate 1 • 6, (p-{g)(g-i?-) -^ (p-ir) This does not look like a strict triad, but it is, being equivalent to (p -{ g) -I [(g -lr)c{p-i r)] which obviously has the character of strict triads generally. The sub- ordinate relations in such a triad cannot be reduced by any direct use of 1-8 and its consequences. However, all such relations can be reduced. The method will be illustrated shortly by deriving (pcg)(gcr) c(pcr) from the above. What strict relations, then, cannot be reduced to the corresponding material relations? The case of asserted relations has already been dis- 312 A Survey of Syvibolic Logic cussed. For subordinate relations, the question admits of a surprisingly simple answer. All the relations of the system can be expressed in terms of some product and the various truth values — the truth values of — p, p, — P> ~P' and ~x)- Let us remind ourselves: poq = -~{pq) pq = --(p q) p-iq = "{p -q) v^q = -(p -q) pAq = ~{-p -q) p + q = -i-p -q) The difference between the truth-value of p and that of -p, between -p and — p, between — p and p, does not affect reduction, because — p can be regarded as — (p) or as ~{-p); p as (p) or as - ~{-p); and p is also -(-2^). Hence we may group the various types of expression which can appear in the system under three heads, according to truth-value: f~l or f~ -1 [ ] or [-] [- ~] or [- ~ -] p-tq pcq p = q p = q pAq p + q pq poq -{poq) -ip q) -ip + q) -ipAq) -{p = q) -ip = q) -ipcq) -{p -! g) -p -P p ~-p P -~p In this table, the letters are quite indifferent: replacing either letter by any other letter, or by a negative, or by any relation, throughout the table, gives a valid result. The blank spaces in the table could also be filled; for example, the first line in the third column would be - ~ -{p -g). But, as the example indicates, the missing expressions are more complex than any which are given, and possess little interest. The significance of the table is this: //, in any theorem, two expressions which belong in the same cohimn of this table are connected, then these expressions may be reduced by postulate 1-8 and its consequences. For, by 2-77, a relation of any two in the same column gives the corresponding relation of the corresponding two in either of the other columns. But any theorem which relates expressions which belong in different columns of this table is not thus reducible, since anv such difference of truth-value is ineradicable. This table also sum- The System of Strict Implication 313 marizes the consequences of postulate 1-7: any expression in the table gives the expression on the same line with it and in the next column to the right. It follows that expressions in the column to the left also give the expressions on the same line in the column to the right, since -t is transitive. Just as postulate 1 ■ 7 is the only source of asserted strict relations which are not replaceable by the corresponding material relations, so also the only theorems containing irreducible subordinate relations are con- sequences of 1-7. For this postulate is the only one in which different truth-values are related, and is the only assumed principle by which an asserted (or denied) truth-value can be altered. But if we simply substi- tute - for ~ in 1 • 7, it becomes the truism, -p -j -p. As a consequence, for every proposition in the system which contains strict relations or the truth- values, [~], [ — ], [ — ], or [ ], in any form, in such wise that these truth- values cannot be reduced to the simple negative, [-], or the simple positive (the truth-value of p), by the use 1-8, the theorem which results if we siinply substitute - for ~ in that proposition is a vahd theorem. Or, to put it more clearly, if less accurately; if any theorem involve [~], explicitly or implicitly, in such wise that it cannot be reduced to [-] by the use of 1-8, still the result of substituting ~ for - is valid. For example, 4 • 13, ~ -f» -i -~ p, cannot be reduced by 1 ■ 8 ; — p and - ~2J are irreducibly different truth- values. But substituting - for ~, we have -{-p) -i-(-p), and hence -(-p) c-(-p), or pep. Propositions such as the pair ~ -p -i — p and -(-p) c-{-p) may be called "pseudo-analogues". If we reduce completely, so far as possible, all the propositions which involve [~] or strict relations, by the use of 1 • 7 and 1 • 8 and their consequences, and then take the pseudo- analogues of the remaining propositions, we shall find such pseudo-analogues redundant. They will all of them already be present as true analogues of propositions which are completely reducible. This transformation by means of postulates 1-7 and 1-8, by which strict relations give the cor- responding material relations, may be represented by the substitution scheme pcq, p ^ q, -ipq), p + q,-p, p, -{p) -M = p-iq, p = q, -ipoq), pAq, -p, - -p, -{- -p) We put -{p o q) and -(p q), -(- ~p) and -(p), because p o g as a main rela- tion in theorems is reducible only when it is denied, and — p is reducible only through its negative. As we have now shown (except for triads of dyads, the reduction of which is still to be illustrated), propositions involving 314 A Survey of Symbolic Logic expressions below the line are still valid when the corresponding expressions above the line are substituted. The transformation by {-/~) of all the assumptions and theorems of the system of Strict Implication ivhich can be thv^ completely reduced, and the reiectio7i of remaining propositions which involve, expressions below the line {or the substitution for them of their pseudo-analogues), gives precisely the system of Material Implication. All the postulates and theorems of Material Implication can be derived from the postulates and definitions of Strict Implication: the system of Strict Implication contains the system of Material Implication. We may further illustrate this fact by deriving from previous propositions the postulates and definitions of the calculus of elementary propositions as it appears in Principia Mathematica.^ pcq = -p+ q is theorem 4 • 23. 4 41 p q = -{-p + -q) 1-05 {-pip; -q/q]: -p + -q = -[-(-p)-(-g)] 2-51: (1) =.-p + -q^-ipq) 2-63: (2) =-(-p + -g) =-[-(p9)] 2-51: (3) = Q.E.D. (p = q) = (p<^q)(.q<^p) is the definition, 1-07. 4-42 p + p cp 3-23: p + p-ip 4-15: (l)-i Q.E.D. 4-43 q cp-i- q 1-2 {-q/p; -p/q]: -p-q-i-q 2-61: (l)-iq-i-{-p-q) 1-05: (2) = q-ip + q 4-15: (3)-) Q.E.D. 4-44 p + q cq + p 3-21: p+ q-i q + p 4-15: (1) -J Q.E.D. 4-45 p+ {q + r) cq+ (p + r) 3 • 24 : p+ (q + r) -iq+ ip + r) 4-15: (l)-i Q.E.D. » Pp. 98-101, 114, 120. (Principia, *1-01) (Principia, *3-01) (1) (2) (3) (Principia, *4-01) (Principia, *l-2) (1) (Principia, ^1-3) (1) (2) (3) (Principia, *l-4) (1) (Principia, *l-5) (1) The System of Strict Implication 315 For the proof of the last postulate in the set in Principia Mathematica certain lemmas are needed which are of interest on their own account. 4-51 pqcr = pc{qcr)=qc{pcr) 1-03 {pq/p; r/q]: pqcr = -[{pq)-r] (1) 2 • 91 and 2 • 9 : (1) = pqcr = -[p{q -r)] = -[q{p -r)] (2) 2-51: (2) =pqcr=-{p-[-{q-r)]} =-{g-[-(p-r)]} (3) 1-03: (3) = Q.E.D. 4-52 p q-ir = p-i (qcr) = q-i {p cr) 1-02 {pq/p; r/q]: p q-ir = ~[{pq)-r] Remainder of proof, similar to the above. 4-53 [(pcq)p]-iq 2-4 {pcq/p}: {pcq)-i{pcq) ' (1) 4-52 {pcq/p; p/q; q/r]: (1) = Q.E.D. It is an immediate consequence of 4-53 that " If p is asserted and p c.q is asserted, then q may be asserted", for, by our assumptions, if p is asserted and p cq'is, asserted, then [{p c q)p\ may be asserted. And if this is asserted, then by 4-53 and our operation of "inference", q can be asserted. But note that the relation which validates the assertion of q is the relation -i in the theorem. This principle, deduced from 4-53, is required in the system of Material Implication (see Principia, *1-1 and *1-11). 454 .(~P<=~g) -^ (?cp) 4-3: 2-2HQ.E.D. 4-55 ip -iq) -i (p r cqr) 1-6 {-r/r}: ip-iq)(q-i -r) -i (p -i -r) (1) 4-52: (1) = (p -I g)-! [(?-!-?•) c (2? -i-r)] (2) l-02and2-51: (2) = (p -J g) H [~(g r) c ~(p r)] (3) A-54:: ~{qr) c~(pr) -i {pr cqr) (4) 1-6: (3) X (4) -I Q.E.D. 4-56 {pcq) c{pr cqr) 4-55 {{pcq)p/p]: 4,-bZ-i[{pcq)p\rcqr ■ (1) 2-91: (1) = [{pcq){pr)]cqr (2) 4-51: (2) = Q.E.D. 4-57 pcq = -qc-p 4-3: 2-62 H (peg) s(-gc-p) (1) 4-3: 2-Q -i {-qc-p) -i {pcq) (2) 1-06: (l)x(2) = Q.E.D. 316 A Survey of Symbolic Logic 4-58 (p cq){qc r) c (p c r) 4-56 {-r/r}: {p cq) c{p-r cq~r) (1) 4-57: (1) = (pcg)c[-(g-,-)c-(p-r)] (2) 1-03: (2) = (pcg)c[(gcr)c(pcr)] (3) 4-51: (3) = Q.E.D. 4 ■ 58 is the analogue, in terms of material relations, of 1 • 6. The method by which we pass from 1-6 to 4-58 illustrates the reduction of triads of strict dyads in general. This reduction begins in the first line of the proof of 4-55. Here 1 • 6 is put in the form (P -i q) -i [(g -tr)c{p-t r)] 12 3 4 5 The relation numbered 4 is already a material relation. This is character- istic of strict triads. Relations 3 and 5 are reduced together by some consequence of 1-8, in a form in which the asserted relation is material. Then, as in 4-56, relations 1 and 2 are reduced together by the use of 4-53 as a premise. This use of 4-53 is quite puzzling at first, but will become clearer if we remember its consequence, " If p is asserted and p cqis asserted, then q may be asserted". This method, or some obvious modification of it, applies to the reduction of any triad of strict dyads which the system gives. We can now prove the last postulate for Material Implication. 4-59 {q cr) c [{p + q) c {p + r)] (Principia, *1 • 6) 4-58 {-p/p}: {-pcq)(qcr) c{-pcr) (1) 4-51: (1) = (rycr)c[(-2jcg)c(-pcr)] (2) 4-24: (2) = Q.E.D. These are a sufficient set of symbolic postulates for Material Impli- cation, as the development of that system from them, in Principia Mathe- viatica, demonstrates. However, in the system of Strict Implication, those theorems which belong also to Material Implication are not necessarily derived from the above set of postulates. They can he so derived, but the transformation {-/~j produces them, more simply and directly, from their analogues in terms of strict relations. IV. Extensions of Strict Implication. The Calculus op Consist- encies AND THE Calculus of Ordinary Inference From the symmetrical character of postulate 1 • 8, and from the fact that postulate 1-7 is converted by negating both members, i. e., p-i — p, The System of Strict Implication 317 it follows that, since the transformation {-/~} is possible, an opposite transformation, {~/-}, is possible. And since implications are reversed by negating both members, those expressions which are transformed directly by {-/~) will be transformed through their negatives by {~/-}, while those expressions which are transformed through their negatives by {-/~} will be transformed directly by { ~/- ) . Hence we have ^ -jp-iq), -ip = q), yog, -{p^q)\ -{-y), -{--y), - ~v -{y<^q), -{y = q), yq, -(p + q)\ -i-p), -ii)), v This substitution scheme may be verified by reference to the table on page 312. The transformation {-/«} represents the fact that expressions in the column to the left, in this table, give expressions in the middle column: {~/-} represents the fact that expressions in the middle column give ex- pressions in the column to the right. {-/~j eliminated strict relations: {~/-} eliminates material relations. As in the previous case, so here, a dyad of dyadic relations, or a relation connected with p, -p, ~p, etc., can be transformed by 1 • 8 and its consequences when and only when the connected expressions appear in the same column of that table. Thus the transformation {~/-} is subject to the same sort of limitation as is {-/~}. The transformation {~/-}, eliminating material relations, has already been illustrated in those tables in Section II, in which theorems in terms of strict relations were compared with analogous propositions in terms of material relations. Theorems in the right-hand column of those tables result from those in the left-hand column by the transformation {~/-}. The proofs of 3-11, 3-12, 3-13, 3-14, 3-31, 3-32, 3-33, 3-34, 3 '52, 3-53, and 3-55 indicate the method of this transformation. Theorem 3-54 indi- cates a limitation of it. As we have noted, triadic strict relations are not expressible in terms of strict dyads alone. Consequently, in the case of triadic relations, the transformation {~/-} cannot be completely carried out. This is an important limitation, since postulate 1 • 6, which is necessary for any generality of proof, is a triadic strict relation. It means that any system of logic in which there are no material relations cannot symbolize its oivn oyerations. Since strict relations are the relations of intension, this is an important observation about calculuses of intension in general. The vertical line in the substitution scheme is to indicate that the transformation {~/-} is arbitrarily considered to be complete when no material relations remain in the expression, p and -p will be transformed when connected with a material relation which is transformed; when not 318 A Survey of Symbolic Logic so connected, p and -p remain. They could be transformed in all cases, but the result is needlessly complex and not instructive. The system, or partial-system, which results from the transformation {~/-} may be called the Calculus of Consistencies. It can be generated independently by the following assumptions : Let the primitive ideas be: (1) propositions, p, q, r, etc., (2) -p, (3) ~p, (4) poq, (5) p = q. Let the other strict relations be defined : I. p A g = -{-p o -g) II. p-iq = -{po-q) For postulates assume : III. p oq -iq op IV. q op -i — p V. - "p-ip op VI. p = -i-p) Assume the operations of "Substitution" and "Inference" as before, but in place of "Production" put the following: li p -i qis asserted and q -ir is asserted, then p -ir may be asserted. By this principle, proof is possible without the introduction into the postulates of triadic relations. The system generated by these assumptions is purely a calculus of intensions. It is the same which would result from performing the trans- formation {~/-) upon all the propositions of Strict Implication which admit of it, and rejecting any which still contain expressions, other than p and -p, below the line. It contains, amongst others, all those theorems concerning strict relations (except the triadic ones) which were exhibited in Section II in comparison with analogous propositions concerning material relations. More interest attaches to another partial-system contained in Strict Implication. If our aim be to create a workable calculus of deductive inference, we shall need to retain the relation of the logical product, p q, but material implication, p cq, and probably also the material sum, p+ q, may be rejected as not sufficiently useful to be worth complicating the system with. The ideas of possibility and impossibility also are unnecessary complications. Such a system may be called the Calculus of Ordinary Inference. The following assumptions are sufficient for it. The System of Strict Implication 319 Primitive Ideas: (1) Propositions, p, q, r, etc., (2) -p, (3) p-iq, (4) p q, (5) p = q. Definitions: A. p Aq = -p-iq B. poq = -(p-i -q) C. {p = q) = {p-iq){q-ip) I'D. p+q = -(-p-q)] Optional. Postulates: E. (-P -iq)-t {-q -i p) F. p q-ip G. p-ipp H. p{q r) -i qip r) I. p-i -i-p) J. {p-iq){q-ir)-i{p-ir) K. p q-ip oq^ L. (p q-ir s) = {p oq-ir os) All of these assumptions are propositions of the system of Strict Impli- cation. A. is 4-22, B. is 4-26, C. is 1-06, and D. is 1-05; E. is 2-3, F. is 2-1, G. is 1-3, H. is 1-4, I. is 1-5, J. is 1-06, K. is 4-26, and L. is an im- mediate consequence of 4 • 32 and 4 ■ 35. The Calculus of Ordinary Inference is, then, contained in the system of Strict Implication. It consists of all those propositions of Strict Implication which do not involve the relation of material implication, pcq [or the material logical sum, p + q]- But where, in Strict Implication, we have "p, we shall have, in the Calculus of Ordinary Inference, -{pop) or p-i-p- Similarly " -p will be replaced by -{-p o -p) or -p -J p, and — p by pop or -{p-i -p). In other words, for 'p is impossible' we shall have 'p is not self-consistent' or 'p implies its own negation'; for 'p is necessary' we shall have' 'the negation of p is not self-consistent' or 'the negation of p implies p'; and for 'p is possible' we shall have 'p is self-consistent' or 'p does not imply its own negation'. The Calculus of Ordinary Inference contains the analogues, in terms of p q, p Aq, and p -i q, of all those theorems of Material Implication which are applicable to deductive inference. It does not contain the useless and doubtful theorems such as ''A false proposition implies any proposition", and "A true proposition is implied by any proposition". As a working 320 A Survey of Symbolic Logic system of symbolic logic, it is superior to Material Implication in this respect, and also in that it contains the useful relation of consistency, p o g. On the other hand, it avoids that complexity which may be considered an objectionable feature of Strict Implication. The system of Strict Implication admits of extension to propositional functions by methods such as those exhibited in the last chapter. For the working out of this extension, several modifications of this method are desirable, but, for the sake of brevity, we shall adhere to the procedure which is already familiar so far as possible. In view of this, the outline to be given here should be taken as indicative of the general method and results and not as a theoretically adequate account. Since, as we have demonstrated, the system of Material Implication is contained in Strict Implication, It follows that, with suitable definitions of Ucpx and Sip.r, the whole theory of propositional functions, as previously developed, may be derived from Strict Implication. Il 1". *33-02 a = pR[p = y{{3.x) .xRy}]. Df "Q" is "the relation of (a class) /3 to (a relation) R, when /3 and R are such that /? 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 "d'R", the converse-domain of R, is the class of those things to which something or other has the relation R. li " R" be "precedes", "G.'R" will be the class of those things which are preceded by something or other. *71-03 1->1 = rCR"Q.'Rc1.R"B'Rc1). Df This involves the meaning of " R", of ", and of "1 ". *32-02 R = p£{l3 = -^(xRy)}. Df " R" signifies "the relation of (3 to x, when /3 and x are such that (3 is the class of y's to which x has the relation R. *37-01 R"fi ^ x{('3.y).y€li.xRy\. Df " R"fi" is "the class of x's such that, for some y, y is a member of 0, and x has the relation R to y. In other words, " R"l3" (the R's of the j3'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 (3. If "R" be "precedes", "R"p" will be the class of predecessors of all (any) members of /3. 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-R, is contained in 1; and whatever is such that any member of the class of those-things-which-have-the-relation-iJ-to-anything has the relation i? to it, is contained in 1." Or more freely and intelligibly: "1 ->1" is the class of relations, R, such that if a R p 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 "asm/3" means "There is a one-to- one correspondence of the members of a with the members of |S. "sm" is the relation of classes which are (cardinally) similar. The analj'sis 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 B. 3 We write A ^ B where the text has A non oo 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 conlainbd; and L will be called the container. However, it can happen that the container ai;)d the contained coincide, as for example, if {A and B) = L, apd 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 (constitucntia) , and L is called thai which is constituted {constituium) . Charad. 3. A + B = L signifies that A is in or is contained in L. Scholium. Although A and B maj' 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 Y 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.^ P J£ N /§l ^\'Y ^\''S ''\ ^7- ^'^f- 4- If some term M" is in A and also in B, it » \ ? J is said to be common to them, and they are said to be Y^., ■'^ ^^ ^y.' cominunicating {communicantia) .^ But if they have \""~-.^ ^'■''^'~~~-s^'''' / nothing in common, as A and 'N (the lines RS and ^^^^ \ ^ ^^'' ^S, for example), they are said to be non-communi- "^-^ y'" eating {incommimicantia) . L 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 {detrahi) or taken away {removeri), and A^ 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 A''. Def. 6. If some one term is supposed to coincide with a plurality of terms which are added (positis) or subtracted (remotis), then the plurality of terms are called the con- stituents, and the one term is called the thing constituted.' ^ Lacuna in the text, followed by "significet A, significabit Nihil". * Italics ours. ^ The text here has "communicatia", clearly a misprint. ' Leibniz's idea seems to be that ii A + N = L then L is "constituted" by A and A'', and also liL — A = A then L and A " constitute" N. But it may mean that HL — A = N, then A and N " constitute " L. i\ Two Fragments from Leibniz 375 Scholium. Thus all terms which are in anything are constituents, but the reverse does not hold; for exaniple, L — A = N,m 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 —Am which A'' is. The expressed constitution of A?" is obvious.' 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 -i. 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 loilh a third, are the same with each other. 1{ 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. li A = B and B ^ C, then A =^ C. For if in the proposition B =^ 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 i? and B = C, then also A is in C. For in the proposition A is in fi (true by hyp.) let C be substituted for B. Theorem 3 // terms which coincide be added to the same term, the results will coincide. li A = B, then A + C = B + C. For if in the proposition A +C = A +C (true * This translation is literal : the meaning is obscure, but see the diagram above. 376 ,-1 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. CoHOLLARY. If terms which coincide be added to terms which coincide, the results will coincide. HA = B and L = Af , then A + L = B + M. For (by the present theorem) since L = M, A + L = A + M, a,nd 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 thu-d thing, then that which is in it will be in that same third thing— that is, if yl is in B and B is in C, then also A is in C. For A is in S (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 ior B (since we show that these coincide) we have A + L + M = C. But putting N ior 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 (hy def. 3). Similarly, since B ism 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." 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 + A^. For A is in M (by hyp.) and M isin M + N (by def. 3), hence A is in M + N (by th. 4). Similarly, S is in Af (by hyp.) and N iain M + N (by def. 3), hence B is in ilf + A^ (by th. 4). But ii A is in M + N and B is in M + N, then also (by th. 5) A + B isin M + N. Q.E.D. Theorem 7 If 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 If 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), ' 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. Ttvo 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, suhiradion {de- tmctio) 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" {Homines nisi qua bestiis differt homo, ut in Janibo 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." 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, + A^ — A'' appearing in an expression may be dropped, unless A^ 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 -\- N ~ N] is impossible. Case 1. liA+N-M-N=A-M, and A, N, and M be non-communicating. For here there is nothing in the compensation to be destroyed, -|- A'' — A'', which is also outside it in A or M — that is, whatever is added in -f- A'", however many times it is added, is in -f A'', and whatever is subtracted in — A'^, however many times it is subtracted, is in - N. Therefore (by ax. 2) for 4- A^ — A'' we can put Nothing. Case 2. liA+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. \i A + 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 Z)], then we shall not have C = A - D. Yor let B = E + F + G, laiA 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. " Apparently an allusion to some description of an ape by Grotius. " 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 Synibolic Logic Theorem 10 A sitbiracled term and the remainder are iion-communicating . liL - A = N,l affirm that A and A^ have nothing in common. For by the definition of "subtraction" and of "remainder", everything in L remains in A'' 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 A'' are non-communicating. For P, as well as A'', 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 A''. Then P and A'" 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. li A = A, 1 affirm that it is possible to find two terms, B and A'", such that B Jp N and yet A + B = A + A^. Solution. Choose some term M which shall be contained in A and such that, N being chosen arbitrarily, M is not contained in A^ nor A'' in M, and let i? = Af + A''. And this will satisfy the requirements. Because B = M -\- N (by hyp.) and M and A^ 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 + A^. Theorem 12 Where non-communicating terms only are involved, whatever terms added to coincident terms give coincident terms will he themselves coincide?it. That is, a A -\- B = C -\- D a,nd 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 + fi - 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-fA'", I affirm that B and A'' 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 iV" 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. l)A-^B = P-\-Q -\-M = P-hM-\-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 A'' is in B; hence N and B are communicating. But if, on the same assumption (namely, that P -\- Q -{- M = P -\- M -{- N, OT A is communicating with B) N also be communicating with P -{- M or A, then either A'" 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. A'' will be com- municating with P, and in that case we shall in similar fashion let P = G -|- i? and N = F -j- 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 A'' likewise, then B = N. li A and B are communicating but A and A'" are non-com- municating, then N is in B. And finally, if B communicates with A, and likewise N com- municates with A, then B and N 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 verilate), 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 = iJ'2 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 x4^ ^Y shortest path of a [point] moving from X to '-., ,-'' Y coincides mth that from Y to X. "~- — M--''" Def. 2. Terms which are not the same, that is, terms which cannot always be sub- stituted for one another, are different (diversa). 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 + B^^ 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 vrith another, then that other will he 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 ^ B, then also fi =|= A. // any term he different from another, then that other vMl he 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 vxith 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. UA = S and B = C and C = i3, 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 ^ = S for A /- / • 384 ^ 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 Af and B is in A^, (by the preceding), A ® B is in M ® N. But C is in P, hence (again by the preceding) A 9 B ® C ism M ® N ® 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 ffi 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 © Z) = C, and the requirements are satisfied. For A©C=AffiAffiD(by construc- tion) = A © D (by ax. 2). Similarly, B ® C ,j^ = B © A © D (by construction). But A ffi D is in B ffi A © D (by def. 3). Hence -^'' "" a"^' ""§' A © C is in B © C. Which was to be done." 8Y and YX are disparate. If RS ® SY = YR, then SY © YR will be in XY ffi YR. 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 Q 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",^" which may be called C. But C © A is nothing but "trilateral and circle" again. Consequently, whatever is in C ffi 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].^^ 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 — " Leibniz has carelessly substituted L in the proof where he has D in the proposition and in the figure. We read D throughout. 2" 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 = AorC = B. Hence these three are differ- ent, as the problem requires. Thus A@C=A®A®B (by construction), which (by ax. 2) is = A e 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 e]. 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 e 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 A^, and A^ 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).'^'^ 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, &s L ® N,h\it L ® N = N (by prop. 14).23 And if three are united, asL ®N ® P, then a couple, L ® N, will be joined with one, P. But the couple, L © iV, by themselves will not compose anything new, but one of themselves, namely the latter, A^, 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 S and BinC and C in D and D in E; second, iiA®B = C and C is in D and D in E; third, if A ffi 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, ii 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 "subalternates" 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 aqua fortis" 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,''* 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, spedosa generalis] is nothing but the representation of combinations by means of symbols, and their manipulation, and the discoverable laws of combination are various,^^ 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 S) B] is for us the same as BA [or B © A]. And also we here take no account of repetition — that is ^A [more consistently, A ® A] is for us the same as A. Thus whgieaBi_these laws just mentioned can be used, the present calculus can be applied. Itjs obvious thatjt 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 Q) YX = RYX or P ® B = L, going from R to X. If we suppose this same [point] then to return from X to F 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 \s the same as L— that is, P ® B ® B or RY ® YX © X7 is the same as RY ffi YX. This caution is of much importance in making judgments, by means of the magnitude and motion of those things which generate^^ or describe, concerning the 2* The text here has " . . . f ulminabile persistens in capella " : the correction is obvious. 2*". . . variaeque sint combinandi leges excogitabUes, . . ." "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. ^"Reading "generant" for "generantur" — a correction which is not absolutely neces- Tivo 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.^' 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 ffi YX — RX. Accord- ingly I here denote this real addition by ffi , 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 ffi BX will be the same as YX ffi RX For the straight line RX is always composed by a process of conception (in notionibus). li A is "parallelogram" and B "equiangular" — which are disparate terms — let C be A (B B, ,^-— -, that is, "rectangle". Then "rectangular ^..-'' B^^^^^ parallelogram" is the same as "equiangular /' _,,- ->\^ rectangle ", for either of these is nothing differ- ^/ M / "^ ent from "rectangle "".~ In general, if Maevius v' -^,/ \ is A and Titius B, the pair composed of the Be 1? 7; -^X two men C, then Maevius together with this \\ / ''^-^U-'^/l pair is the same as Titius together with this \V.,__ ^,' /y pair, for in either case we have nothing more ^.^x" ' ^Z than the pair itself. Another solution of ^^^•"""---.C' .--'''-'' this problem, one more elegant hut less general, "^, ,'' can be given if A and B have something in A&O common, and this common term is given and that which is peculiar to each of the terms A and B is also given. For let that which is exclusively A be M, and that which is exclusively B be A^, and let M ffi A^ = I> and let what is common to A and B be P. Then I affirm that A ffi O = B ffi 2), for since A = P ® M and B = P ® N, we have A®D = P@M®Na.D.d also B®D=P®M®N. sary, since a motion which generates a line is also itself generated; but, as the context shows, "generare" and "desoribere" are here synonymous. " Italics ours. BIBLIOGRAPHY The following bibliography contains titles of all the positive contributions to symbolic logic and logistic in the strict sense with which I am acquainted, as well as some, taken from the bibliographies of Venn and Peano, of works which I have not been able to get. But it is not complete. A few of the numerous general discussions of symbolic logic and logistic have also been listed, and certain mathematical books and papers which, though not strictly logistic, are of special interest to the student of that subject. What are considered the most important contributions to symbolic logic, are indicated by an asterisk (*) preceding the title, while certain studies which should be especially helpful to students of this book are indicated by a "dagger" (f). Volume numbers of periodicals are given in bold-faced type. Adamson, E. The logical copula and quantification of the predicate. London, Nutt, 1897. Alsted, J. H. Logicae systema harmonicum. Herbornae Nassoviorum, 1614. Amato, V. La teoria dei numeri reali fondata sui concetti del Russell. Boll. Matem. (Rome), 9 (1910), pp. 198-205. Andreoli, C. Su un nuovo simbolo nell' algebra della logica. Alii Soc. progr. Sci. (Rome), 4. Riunione Napoli, dicemhre 1910, pp. 741 ff. Bain, A. Logic: Part I, Deduction. London, Longmans, 1870. Bardili, C G. Grundriss der ersten Logik. Stuttgart, Loffland, 1800. Baynes, T. S. An essay on the new analytic of logical forms. (Hamilton Prize Essay, 1846.) Edinburgh, Sutherland, 1850. Who discovered the principle of the quantification of the predicate? Contemp. Rev., 22 (1873), pp. 318-23. Benbke, E. C. On the logical meaning of proper names. Proc. Aristot. Soc, 3 (1905), pp. 12-29. Bentham, G. Outline of a new system of logic. London, Hunt, 1827. Bernouilli, J. Parallelismus ratiocinii logici et algebraici (1685): Opera (Geneve, Cramer, 1744), 1, p. 214. Bernstein, B. A complete set of postulates for the logic of classes expressed in terms of the operation "exception", and a proof of the independence of a set of postulates due to del R^. Univ. Calif. Publ. Math., 1 (1914), pp. 87-96. * A set of four independent postulates for Boolian algebras. Trans., Amer. Math. Soc, 17 (1916), pp. 50-52. A simplification of the Whitehead-Huntington set of postulates for Boolian algebras. Amer. Math. Soc Bull, 22 (1916), pp. 458-59. B6cHER, M. The fundamental conceptions and methods of mathematics. Amer. Math. Soc. Bull, 11 (1904), pp. 115-35. Bolzano, B. Wissenschaftslehre : Versuch einer ausftihrlichen Darstellung der Logik. 4 vols. Sulzbach, Seidelschen Buchh., 1837. Neu herausgegeben von Alois Hafier, Leipzig, Felix Meiner, 1914. *BooLE, G. The mathematical analysis of logic. Cambridge, Macmillan, 1847. * The calculus of logic. Camb. and Dublin Math. Jour., 3 (1848), pp. 183-98. On the theory of probabilities, and in particular on Mitchell's problem of the distribution of the fixed stars. Phil. Mag., ser. 4, 1 (1851), pp. 521-30. Further observations on the theory of probabilities. Ibid., ser. 4, 2 (1851), pp. 96-101. 389 390 A Survey of Symbolic Logic An investigation of the laws of thought. London, Walton, 1854. Also re- printed as vol. 2 of Boole's Collected logical works; ed. by Jourdain, Chicago, Open Court Publ. Co., 1916. Solution of a question in the theory of probabilities. Phil. Mag., ser. 4, 7 (1854), pp. 29-32. Reply to some observations published by Mr. Wilbraham . . . Ibid., ser. 4, 8 (1854), pp. 87-91. On the conditions by which the solutions of questions in the theory of prob- abilities are limited. Ibid., pp. 91-98. Further observations relating to the theories of probabilities . . . Ibid., pp. 175-76. On a general method in the theory of probabilities. Ibid., pp. 431-44. On certain propositions in algebra connected with the theory of probabilities. Ibid., ser. 4, 9 (1855), pp. 165-79. On the application of the theory of probabilities to the question of the combina- tion of testimonies or judgments. (Keith Prize Essay.) Trans. Boy. Soc. Edin- burgh, 21 (1857), pp. 597-653. On the theory of probabilities. Phil. Trans. Boy. Soc. (London), 152 (1862), pp. 225-52. On the theory of probabilities. Phil. Mag., ser. 4, 25 (1863), pp. 313-17. Of propositions numerically definite. Trans. Camb. Phil. Soc, 11 (1868), pp. 396-411. Boole, Maky. Symbolical methods of study. London, Kegan, 1884. BoTjRDiLLAT, F. La r^forme logique de Hamilton. Paris, Hachette, 1891. Brown, H. C. The logic of Mr. Russell. Jour, of Phil., Psych, and Sci. Methods, 8 (1911), pp. 85-91. Brunschwieg, L. Les stapes de la philosophic math^matique. Paris, Alcan, 1912. Bryant, Mrs. S. On the nature and functions of a complete symbolic language. Mind, 13 (1888), pp. 188-207. • The relation of mathematics to general formal logic. Proc. Aristot. Soc, 2 (1901), pp. 105-34. BuFFA, P. Alcune formule di logica. Bev. de Mathem. (Turin), 7 (1900), pp. 56-58. Principl di logica. Period. Matem. (Livorno), 16 (1909), pp. 295-303, and 17 (1901), pp. 292-300. BuRALi-FoRTi, C. Teoria delle grandezze: Parte IV del Formulario. Biv. di Matem. (Turin), 3 (1893). SuUe classi derivate a destra e a sinistra. Atti Accad. Torino, 29 (1894). I Humeri negativi. Biv. di Matem., 3 (1893), pp. 138 ff. * Logica matematica. Milano, Hoepli, 1894. Exercise de traduction en symboles de logique mathfimatique. Bull, de Mathem. elimentaire (Turin), 1897. Sur les diff&entes m^thodes logique pour la definition du nombre rdel. Bibl. du Cong. Int. de Phil. (Paris), 1900, 3, pp. 289-307. Sui simboli di logica matematica. II Pitagora, 1900, pp. 65 ff ., 129 ff. Sur l'6galit6 et sur I'introduction des Elements d&iv6s dans la science. Enseign. mathem., 1901, pp. 246-61. Gli enti astratti definiti come enti relativi ad un campo di nozioni. Bend. Accad. Lincei (Rome), ser. 5, 21: 2 (1912), pp. 677-82. Sulla teoria g^nerale delle grandezze e dei numeri. Aiti Accad. Torino, 39 (1903), pp. 256-82. BuscH, M. Anfangsgrtinde der logikalischen Algebra. Tubingen, Cotta, 1768. Cantor, G. Beitrage zur Begrtindung der transfiniten Mengenlehre. Math. Annalen, 46 (1895), pp. 481-512, and 49 (1897), pp. 207-46. Bibliography 391 Theory of transfinite numbers: translation by P. E. B. Jourdain, Chicago, Open Court Publ. Co., 1915. Carroll, Lewis [pseud.] see S. H. Hodgson. Castellano, F. Alcune identity. Rev. de Mathem. (Turin), 7 (1900), pp. 58 ff. Castillon, G. F. Reflexions sur la logique. Berlin Acad. Mem., 1802. * Sur un nouvel algorithme logique. Ibid., 1803. Catley, a. On a question in the theory of probabilities (with discussion by Boole). Phil. Mag., ser. 4, 23 (1862), pp. 352-65. — ■ Note on the calculus of logic. Quar. J. of Pure and Appl. Math., 11 (1871), pp. 282-83. On compound combinations. Mem. Manchester Lit. and Phil. Soc, 26 (1877), pp. 99-100. CiPOLLA, M. Theoria de congruentias intra numeros integros. Rev. de Mathe?n. (Turin), 8 (1905). Specimen de calculo arithmetico integrale. Ibid. *CiAT¥OB.T>, W. K. Types of compound statement involving four classes. Mem. Man- chester Lit. and Phil. Soc, 26 (1879), pp. 81-96. Cohen, M. R. The present situation in the philosophy of mathematics. J. of Phil., Psych. &c., 8 (1911), pp. 533-46. CouTURAT, L. La logique math^matique de M. Peano. Rev. de Metaphysique et de Morale, 7 (1899), pp. 616-46. • Sur une definition logique du nombre. Ibid., 8 (1900), pp. 23-36. Sur la definition du continuum. Ibid., pp. 157-68. L'algebre universelle de M. Whitehead. Ibid., pp. 323-62. t La logique de Leibniz, d'apres des documents inedits, Paris, Alcan, 1901. Histoiie de la langue universelle (en collaboration avec M. Leau.) Paris, Hachette, 1903. ■ Sur I'utilite de la logique algorithmique. C.-R., II Cong. int. de Phil, Geneve 1904. Les principes des mathematiques. Rev. de Met. et de Mar., 12 (1904), pp. 19-50, 211-40, 664-98, 810-44, 13 (1905), pp. 224-56. Definitions et demonstrations mathematiques. Enseign. Mathem., 1905. Les definitions mathematiques. Ibid., pp. 27 ff. Les principes des mathematiques, avec un appendice sur la philosophie des mathematiques de Kant. Paris, Alcan, 1905. t L'algebre de la logique. Paris, Gauthier-Villars, 1905. f . The algebra of logic; translated by L. G. Robinson. Chicago, Open Court Publ. Co., 1914. Pour la logistique: reponse a M. Poincare. Rev. de Met. et de Mor., 14 (1906), pp. 208-50. La logique et la philosophie contemporaine. Ibid., pp. 318-41. f Die Prinzipien der Logik. Encyc. d. phil. Wiss., hrsg. v. A. Ruge, : pp. 137-201. I The principles of logic; translated by B. E. Meyer. Encyc. of Phil. Sci., 1, Logic, pp. 136-239. For logistics. Mind, 22 (1912), pp. 481-523. Des propositions paiticulieres et de la portee existentielle. Rev. de Met. et de Mor., 21 (1913), pp. 256-59. Sur les rapports logique des concepts et des propositions. Ibid., 24 (1917), pp. 15-58. La logique algorithmique et le calcul de probabilites. Ibid., pp. 291-313. Dalgarno, G. Ars signorum, vulgo character universalis et lingua philosophica. Ed. 4. Glasgow, Maitland Club, 1834. 392 A Survey of Symbolic Logic Darjes, J. G. Introductio in artem inveniendi seu logicam theoretico-practicam. Ed. 2. Jenae, 1747. Weg zur Wahrheit. 1776. Davis, E. W. Some groups in logic. Amer. Math. Soc. Bull, 9 (1903), pp. 346-48. Dedekind, R. Was sind und was sollen die Zahlen. Braunschweig, Vierwig, 1888. Stetigkeit und irrationalen Zahlen. Ed. 3. Braunschweig, Vierwig, 1905. — Essays on the theory of numbers. (The two preceding translated by W. W. Beman.) Chicago, Open Court Publ. Co., 1909. De Laguna, T. On certain logical paradoxes. Phil. Rev., 25 (1916), pp. 16-27. Opposition and the syllogism. J. of Phil, Psych., dc, 9 (1912), pp. 393-400. Delbobuf, J. Essai de logique scientifique. Li^ge, Desoer, 1865. Logique algorithmique. Rev. Phil, 2 (1876), pp. 225-52, 335-55, 545-95. Also published separately, Liege et Bruxelles, 1877. t*DEL Re, a. Lezioni di algebra della logioa. Naples, 1907. Sulla indipendenza dei postulati della logica. Rend. Accad. Sci. (Naples), ser. 3, 17 (1911), pp. 450-58. *Db Morgan, A. Formal logic; or. The calculus of inference, necessary and probable. London, Taylor, 1847. * Series of five papers, "On the syllogism, etc." Trans. Camb. Phil. Soc. 1) 8 (1846), pp. 379-408. 2) 9 (1850), pp. 79-127. 3) 10 (1858), pp. 173-230. 4) Ibid. (1860), 331-*358. 5) Ibid. 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M^taphysique et logique math^matique. Rev. de Met. et de Mar., 13 (1905), pp. 589-619. Application de I'algebre de logique k une controverse juridique. Ibid., 14 (1906), pp. 617-25. Sur I'introduction logique k la theorie des foncotins. Ibid., 15 (1907), pp. 186-216. Les principes du calcul fonctionnel. Ibid., 21 (1913), pp. 462-510. Wolf, A. Existential import of categorical predication. Camb. Univ. Press, 1905. WuNDT, W. Logik. Ed. 3, 3 vols. Stuttgart, Enke, 1906-08. Refer especially to vol. 1. Young, G. P. Remarks on Professor Boole's mathematical theory of the Laws of Thought. Canad. Jour., 10 (1865), pp. 161-82. Young, J. W. Lectures on the fundamental concepts of algebra and geometry. New York, Macmillan, 1911. Yule, G. U. On the theory of the consistence of logical class-frequencies and its geo- metrical representation. Phil. Trans. Roy. Soc. (London), ser. A, 197 (1901), pp. 91-133. An introduction to the theory of statistics. Ed. 2. London, Griffin, 1912. INDEX (References here given are in addition to, not inclusive of, those readily suggested bv the Table of Contents). - o- j Absorption, law of, 74. Absurd, see Impossible. Addition, arithmetical, 343; definition of arithmetical bv Peirce, 81 ff., bv Peano, 34.5; relative addition, 91, 95', 275 ff. See also Sum. Alphabet, logical, 74. Aristotle, 231. Bayne, A., 36. Bentham, G., 36. Bernoulli, J., IS. Bernstein, B., 119 Boole, G., 3, 4, 31, 78-79, 118, 137, 207, 211, 212, 217; Jevons on system of, 72; system of, compared with Peirce's, 80 ff.; his general problem, 162. Calculus, 4, 6, 8. See also Classes, Proposi- tions, Propositional functions. Rela- tions. Cantor, G., 4, .5. Carroll, Lewis, see Hodgson, S. G. Castillon, G. F., 4, 18, 32-35. Classes, conception of, 261; application of Boole-Schroder Algebra to, 121-22; calculus of, derived from Strict Impli- cation, 321 ff. CoefKcients, in Boole's system. 59-63; in the Boole-Schroder Algebra, 137 ff. Consistency, relation of, in Mrs. Ladd- Franklin's system, 109; in the system of Strict Implication, 293; meaning of, 333. Contained in (the relation C), 13, 16, 47, 65, 118-19, 262, 270; Peirce on the meaning of, 83-84, 96. Contradictory propositions, 189; as treated by De Morgan, 40, Jevons, 76-77. Contrary propositions, 189; as treated by Jevons, 76-77. Converse, propositions, 191; Converse relations, 46, 74, 91, 273-74. Dalgarno, G., 5. Dedekimd, R., 4, 5, 114, 342. del R6, A., 119. DeMorgan, A., 4, 5, 79, 209; DeMorgan's Theorem, 125, 237, 283. Descriptions, 290, 343. Distribution of terms, 43. Division, in Lambert's .system, 21 ff. ; in Holland's, 30 ff.; in Boole's, 61 ff.; in Peirce's, 81 ff. Duality, law of, in terminology of Boole, 58, in Boole-Schroder Algebra, 126. Either or, meaning of, 213. Elective symbols, 52. Elimination, in Boole's system, 59-60, 64- m Jevons's, 75-76; in the Boole- Schroder Algebra, 1.53 ff., 194 e-relation, 16, 261-62, 270, 345. Equivalence, in the Boole-Schroder Al- gebra, 120; of classes, 262; of relations, 270; of propositions, strict, and mater- ial, 292-93. Euclid, 342. Euler's diagrams, 176. Existence, 12, 14, 186-88, 328-29. Exponents, in the logic of relatives, 87 ff. Extension, Chap. I, Sects. II and III, passim; in Leibniz's system, 13-14; Boole's logic based on, 52; classes in, 184-86; relations in, 219; propositions in, 2.30-31, 328; definition in, 349. See also Intension. Formal implication, see Implication. Formulaire de Mathe'matiques, v, 5 7 115-16,278,280-81,368. Fractions, see Division. Frege, G., 4, 5, 50, 114-15, 273, 351. F-relation, 366. Functions, as developed by Boole, 58, 82; in the Boole-Schroder Algebra, 125, 133 ff. See also Propositional Functions. Geometry, 10, 340-42; 363-67. Grassmann, H., 4. Grassmann, R., 107-08. Hamilton, Sir W., 4, 36-37. Hamilton, W. R., 4. Hilbert, D., 5. Hodgson, S. G., 326; his diagrams, 180 ff. Holland, G. J. von, 18, 29-32, 323. Huntington, E. V., 119, 342. Ideographic language, 2, 6-8, 340. Implication, no symbol for, in Boole's sys- tem, 65; material, 84-85, 214-15, 231, Chap. V, passim, see esp. 303-04 and Sect. V; formal, 243, 328 ff. ; four types of, 259. iSee also Contained in. Impossible, 32, 292, 336-39. ,See also Con- sistency. Inclusion, see Contained in. Inconsistent triad, 195-97. Indeterminate, see Undetermined. Index law, 54. Individual, 324. See also e-relation. Induction, see Mathematical induction. Indv L^ Inference, 57-58, Chap. Ill, Sects. II and III, passim; in the Ingistic method, 352-53; immediate, 14, 33, 39, 41, 77, 190 ff. Intension, 8, 13 ff.. Chap. I, Sects. II and III, passim, 73; postulates for calculus of propositions in, 318; calculus of classes in, 323. Jevons, \y. S., 4, 118, 149, 205. Kant, I., 341. Kempe, A. B., 198. Kircher, A., 5. Ladd-FrankHn, Mrs. C, 78, 108-10; 118, 195, 205, 211, 231. Lambert, J. H., 4, lS-29, 32, 204, 323. Leibniz, G. W., 3, 4, 79, 372. Linear Sets, 365, 367. Logical product, .see Product. Logical sum, see Sum. Logistic, vi, 5-7, 11. Lully, I?.,5. MacCoU, H., 4, 108, 119, 327. Mathematical induction, 29, 131, 236, 351. Multiplication, arithmetical, defined by Peirce, 101-02, by Peano, 345; relative multiplication 86" ff., 275 ff. See also Pioduct, Necessary, 17, Chap. V, passim. Xegative, in the Boole-Schroder Algebra, 119, 124; terms, 3S, 53, 73; classes, 121, 185, 263; relations, 46, 220, 271; propo- sitions, 14-15, 25-26, 30-31, 32-33, 40, 57, 108. 188-89, 213. 292. Null-class, 185-90, 327. See also 0. Null-proposition, see 0. Null-relation, see 0. Number, 80, 101; in Peano's Formulaire, 344; defined in Principia Mathemnticn, 346 ff. 1, see Universe of discourse; see also under Operation, meaning of, 342 ; nature of, in logistic, 358 ff. Order, logical, 3, 342, 364. Peano, G., 50, 115. See also Formulaire de Malhhnatiques. Peirce, C. S., 4, 261, 279. Plato, 4. Ploucquet, C, 4, 18. n and S operators, 79, 97 ff., 140, 234. Pieri, M., 5, 342. Poretsky, P., 114, 145-46, 163-66, 200. Possible, 15, 329. »See also Impossible. Premise, in logistic, 352-53. Primitive concepts, in Leibniz, 7. Primitive ideas and i^ropositions, of Prin- cipia MdiheiiiaUcii, 2.S2, 2.S7-S8; of arithmetic in Peano's Fiiniii.luire, 344- 45. Principia Matliematica: v, 5, 7, 8, 102, 116, 222, 261, 277, 279, 281, 314-16, 324, 361. Probability, Boole's treatment of, 67 ff. ; Peirces' treatment of, 105-06. Product, in Leibniz's system, 12 ff.; in Lambert's, 19; in Boole's, 52; in Peirce's, 81; in the Boole-Schroder Algebra, 119; of classes, 120, 185, 262; of regions in space, 175; of relations, 86 ff., 219, 271, 275; of two functions, 143; strict logical, 293. See also Multi- plication and n and 2 operators. Prepositional functions, 94, 113; meaning of, 232-33 ; range of significance of, 23.'3, 242, 254; in Principia Mathemalica, 287; calculus of, derived from Strict Implication, 320 ff. Quantification of the predicate, 19, 24, 36, 38 ff., 56. See also Undetermined co- efficient. Quaternions, 4; Peirce's logical, 103 ff. Reflii.ctio ad ahsurcj.um, 166-67. Regions in a plane, application of the Boole-Schroder Algebra to, 120-21, 175. Relations, as treated by Lambert, 28-29, by DeMorgan, 37, 45 ff., by Peirce, 85 ff., 102-05, by Schroder, 111 ff.; Peirce's paper on logic of, 100; mean- ing of, in extension, 219, 269; calculus of, compared with calculus of classes, 271; converse, 91, 273-74, 276; powers of, 29, 277; domain, converse domain, and field of, 277-78; calculus of, de- rived from Strict Implication, 323. Relative terms, 277-78; in DeMorgan's system, 45 ff. ; Peirce's treatment of, 85 ff.; Schroder's treatment of. 111 ff. .S'ee also Relations. Royce, .!., vi, 195. Schroder, E., v, 4, 5, 78, 110 ff., 211, 240, 261,279. Science, exact, 7, 370-71. Segner, .1. A., IS. Self-contradictory, see Impos.sible. Sheffer, H. M., 119. S operator, see n and S. operators. Solly, 7, 29, 36. Solutions, in Boole's system, 60-63; in .Jevons's, 76-77; in Peirce's, 98-100; by means of diagrams, 77, 181 ff. ; of some logical problems, 201-12, 21.5-19. See also Equations and Ineqiiaiions in Table of Contents. Square of opposition, 190. Subcontrary propositions, 190. Substitution of similars, in .levons, 75. Subtraction, arithmetical, treated by Peirce, 80-81; in Leibniz's system, 17- 18; in Lambert's, 19; in Castillon's, 32; in Boole's, 53; in Peirce's, 81 ff. Sum, in Leibniz's ^ysfcni, 16 fi. : in Lam- bert's, 19; in rastilton's, 32 ff. ; in Boole's, .52-53; in .levons'.s, 73; in Peirce's, 81-82; in Si-hn.der'.s, 111; in the Boole-Schroder Algebra, 119; of classes, 121. 185, 263; of propositions, 213; of relations, 271; of two functions, 143; of propositional functions, 94; strict logical, 291, 301-02. See also Addition. IihIpx Syllogisms, in Lambert's system, 26 ff.; in Holland's, 31-32; in Castillon's, 34; in DeMorgan's, 41, 49 iT.; in Boole's, 57-58; in Jevons's, 75; in Mrs. Ladd- Franklin's, 109-10, 195-97; Peirce's principle of, 85; application of the Boole-Schroder Algebra to, 181-82, 193-95; in Barbara, 245, interpreted by Kempe, 365; conditional, 197; limitation of , 1, 198-201. System-S, 366. Thomson, W., 36. Tonnies, I. H., 18. Truth value of propositions, 227, 230, 294. Two-valued Algebra, defects of, as a calcu- lus of propositions, 281. Undetermined class, in Holland's system, 30; in Castillon's, 32. See also Undeter- mined coefficient. Undetermined coefficient, in Leibniz's system, 15; in Lambert's, 24 ff.; in 118. See also Prin- Boole's, 50-51; in Jevons's, 75; in Peirce's, 82; in the Boole-Schroder Algebra, 186. Universe of discourse, 37; diagrams of, 177 ff. See also refs. under 0. Variables, 3, 232 ff.; Peirce on, 93; in Princi-pia Mathematica, 289. Venn, J., v, 18, 201, 203, 211; diagrams, 77, 176 ff. Whitehead, A. M., v^ dpia Malhemalica. Wilkins, J., 5. 0: in Boole's system, 52 ff. ; in Jevons's, 73-74; in Peirce's, 82; in Schroder's, 111; in the Boole-Schroder Algebra, 119; in the Calculus of classes, 121, 185 ff., 263; in the system of regions in a plane, 181; in the calculus of proposi- tions, 213-14, 223 ff. ; in the calculus of relations, 218-19, 271; Boole's algebra is an algebra of and 1, 52.