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THE ALGEBRA OF LOGIC 
 
 BY 
 
 LOUIS COUTURAT 
 
 AUTHORIZED ENGLISH TRANSLATION 
 
 LYDLA. GILLINGHAM ROBINSON, B. A. 
 ■ With a Preface by PHILIP E. B. JOURDAIN. M. A. (Cantab.; 
 
 -se^- 
 
 CHICAGO AND LONDON 
 THE OPEN COURT PUBLISHING COMPANY 
 
 1914 
 
 - 1 ■ 
 
i&i^ip 
 
 I'E 
 
 Copyright in Grtat Britain under tht Act of 1911. 
 
PREFACE. 
 
 Mathematical Logic is a necessary preliminary to logical 
 Mathematics. "Mathematical Logic" is the name given by 
 Peano to what is also known (after Venn) as "Symbolic 
 Logic''; and Sjraabolic Logic is, in essentials, the Logic of 
 Aristotle, given new life and power by being dressed up in 
 the wonderful — almost magical — armour and accoutrements 
 of Algebra. In less than seventy years, logic, to use an 
 expression of De Morgan's, has so thriven upon symbols and, 
 in consequence, so grown and altered that the ancient logicians 
 would not recognize it, and many old-fashioned logicians will 
 not recognize it The metaphor is not quite correct: Logic 
 has neither grown nor altered, but we now see more of it 
 and more into it 
 
 The primary significance of a symbolic calculus seems to 
 lie in the economy of mental effort which it brings about, and 
 to this is due the characteristic power and rapid development 
 of mathematical knowledge. Attempts to treat the operations 
 of formal logic in an analogous way had been made not in- 
 frequently by some of the more philosophical mathematicians, 
 such as Leibniz and Lambert; but their labors remained little 
 known, and it was Boole and De Morgan, about the middle 
 of the nineteenth century, to whom a mathematical — though 
 of course non- quantitative — way of regarding logic was due. 
 By this, not only was the traditional or Aristotehan doctrine 
 of logic reformed and completed, but out of it has developed, 
 in course of time, an instrument which deals in a sure manner 
 with the task of investigating the fundamental concepts of 
 mathematics — a task which philosophers have repeatedly taken 
 in hand, and in which they have as repeatedly failed. 
 
IV PREFACE. 
 
 First of all, it is necessary to glance at the growth of 
 symbolism in mathematics, where alone it first reached per- 
 fection. There have been three stages in the development 
 of mathematical doctrines: first came propositions with par- 
 ticular numbers, like the one expressed, with signs subsequently 
 invented, by "2 -1- 3 = 5"; then came more general laws hold- 
 ing for all numbers and expressed by letters, such as 
 
 '\a-Yb)c = ac\bc"; 
 
 lastly came the knowledge of more general laws of functions 
 and the formation of the conception and expression "function". 
 The origin of the symbols for particular whole numbers is 
 very ancient, while the symbols now in use for the operations 
 and relations of arithmetic mostly date from the sixteenth and 
 seventeenth centuries; and these "constant" symbols together 
 with the letters first used systematically by VifeTE (1540 — 1603) 
 and Descartes (1596 — 1650), serve, by themselves, to express 
 many propositions. It is not, then, surprising that Descartes, 
 who was both a mathematician and a philosopher, should 
 have had the idea of keeping the method of algebra while 
 going beyond the material of traditional mathematics and 
 embracing the general science of what thought finds, so that 
 philosophy should become a kind of Universal Mathematics. 
 This sort of generalization of the use of symbols for analogous 
 theories is a characteristic of mathematics, and seems to be 
 a reason lying deeper than the erroneous idea, arising from 
 a simple confusion of thought, that algebraical symbols nec- 
 essarily imply something quantitative, for the antagonism 
 there used to be and is on the part of those logicians who 
 were not and are not mathematicians, to symbolic logic. This 
 idea of a universal mathematics was cultivated especially by 
 Gottfried Wilhelm Leibniz (1646 — 17 16). 
 
 Though modern logic is really due to Boole and De 
 Morgan, Leibniz was the first to have a really distinct plan 
 of a system of mathematical logic. That this is so appears 
 from research — much of which is quite recent — into Leibniz's 
 unpublished work, 
 
 The principles of the logic of Leibniz, and consequently 
 
PREFACE. V 
 
 of his whole philosophy, reduce to two': (i) All our ideas 
 are compounded of a very small number of simple ideas which 
 form the "alphabet of human thoughts"; (2) Complex ideas 
 proceed from these simple ideas by a uniform and symmetrical 
 combination which is analogous to arithmetical multiplication, 
 With regard to the first principle, the number of simple ideas is 
 much greater than Leibniz thought; and, with regard to the second 
 principle, logic considers three operations — which we shall meet 
 with in the following book under the names of logical multi- 
 plication, logical addition and negation — instead of only one. 
 
 "Characters" were, with Leibniz, any written signs, and 
 "real" characters were those which— as in the Chinese ideo- 
 graphy — represent ideas directly, and not the words for them. 
 Among real characters, some simply serve to represent ideas, 
 and some serve for reasoning. Egyptian and Chinese hiero- 
 glyphics and the symbols of astronomers and chemists belong 
 to the first category, but Leibniz declared them to be imper- 
 fect, and desired the second category of characters for what 
 he called his "universal characteristic".' It was not in the 
 form of an algebra that Leibniz first conceived his charateristic, 
 probably because he was then a novice in mathematics, but 
 in the form of a universal language or script's It was in 
 1676 that he first dreamed of a kind of algebra of thought,* 
 and it was the algebraic notation which then served as model 
 for the characteristic.^ 
 
 Leibniz attached so much importance to the invention of 
 proper symbols that he attributed to this alone the whole of 
 his discoveries in mathematics.* And, in fact, his infinitesimal 
 calculus affords a most brilliant example of the importance 
 of, and Leibniz's skill in devising, a suitable notation. ? 
 
 Now, it must be remembered that what is usually understood 
 by the name "symbolic logic", and which — though not its 
 name — is chiefly due to Boole, is what Leibniz called a 
 Calculus ratiocinator, and is only a part of the Universal 
 
 ' COUTURAT, La Logique de Leibniz d'apres des documents inedits, 
 Paris, 1901, pp. 431—432, 48. 
 
 2 Ibid., p. 81. 3 Ibid., pp. 51, 78. 4 Ibid., p. 61. 
 
 5 Ibid., p. 83. 6 Ibid., p. 84. 7 Ibid., p. 84 — 87. 
 
VI PREFACE. 
 
 Characteristic. In symbolic logic Leibniz enunciated the principal 
 properties of what we now call logical multiplication, addition, 
 negation, identity, class-inclusion, and the null-class; but the 
 aim of Leibniz's researches was, as he said, to create "a kind 
 of general system of notation in which all the truths of reason 
 should be reduced to a calculus. This could be, at the same 
 time, a kind of universal written language, very diiferent from 
 aU those which have been projected hitherto; for the char- 
 acters and even the words would direct the reason, and the 
 errors — excepting those of fact — would only be errors of 
 calculation. It would be very difficult to invent this language 
 or characteristic, but very easy to learn it without any 
 dictionaries". He fixed the time necessary to form it: "I think 
 that some chosen men could finish the matter within five 
 years"; and finally remarked: "And so I repeat, what I have 
 often said, that a man who is neither a prophet nor a prince 
 can never undertake any thing more conducive to the good 
 of the human race and the glory of God". 
 
 In his last letters he remarked: "If I had been less busy, 
 or if I were younger or helped by well-intentioned young 
 people, I would have hoped to have evolved a characteristic 
 of this kind"; and: "I have spoken of my general characteristic 
 to the Marquis de I'Hopital and others; but they paid no 
 more attention than if I had been telling them a dream. It 
 would be necessary to support it by some obvious use; but, 
 for this purpose, it would be necessary to construct a part 
 at least of my characteristic; — and this is not easy, above all 
 to one situated as I am". 
 
 Leibniz thus formed projects of both what he called a 
 characteristica universalis, aud what he called a calculus ratio- 
 cinator; it is not hard to see that these projects are inter- 
 connected, since a perfect universal characteristic would 
 comprise, it seems, a logical calculus. Leibniz did not publish 
 the incomplete results which he had obtained, and conse- 
 quently his ideas had no continuators, with the exception of 
 Lambert and some others, up to the time when Boole, De 
 Morgan, Schroder, MacCoLL, and others rediscovered his 
 theorems. But when the investigations of the principles of 
 
PREFACE. VII 
 
 mathematics became the chief task of logical symbolism, the 
 aspect of symbolic logic as a calculus ceased to be of such 
 importance, as we see in the work of Frege and Russell. 
 Frege's symbolism, though far better for logical analysis than 
 Boole's or the more modern Peano's, for instance, is far 
 inferior to Peano's — a symbolism in which the merits of 
 internationality and power of expressing mathematical theorems 
 are very satisfactorily attained— in practical convenience. 
 Russell, especially in his later works, has used the ideas of 
 Frege, many of which he discovered subsequently to, but 
 independently of, Frege, and modified the symbolism of Peano 
 as little as possible. Still, the complications thus introduced 
 take away that simple character which seems necessary to 
 a calculus, and which Boole and others reached by passing 
 over certain distinctions which a subtler logic has shown us 
 must ultimately be made. 
 
 Let us dwell a little longer on the distinction pointed out 
 by Leibniz between a calculus ratiocinator and a characteristica 
 universalis or lingua characteristica. The ambiguities of ordi- 
 nary language are too well known for it to be necessary for 
 us to give instances. The objects of a complete logical 
 symbolism are: firstly, to avoid this disadvantage by providing 
 an ideography, in which the signs represent ideas and the 
 relations between them directly (without the intermediary of 
 words), and secondly, so to manage that, from given premises, 
 we can, in this ideography, draw all the logical conclusions 
 which they imply by means of rules of transformation of 
 formulas analogous to those of algebra, — in fact, in which 
 we can replace reasoning by the almost mechanical process 
 of calculation. This second requirement is the requirement 
 of a calculus ratiocinator. It is essential that the ideo- 
 graphy should be complete, that only symbols with a well- 
 defined meaning should be used— to avoid the same sort of 
 ambiguities that words have — and, consequently, that no 
 suppositions should be introduced implicitly, as is commonly 
 the case if the meaning of signs is not well defined. Whatever 
 premises are necessary and sufficient for a conclusion should 
 be stated explicitly. 
 
VIII PREFACE. 
 
 Besides this, it is of practical importance, — though it is 
 theoretically irrelevant, — that the ideography should be concise, 
 so that it is a sort of stenography. 
 
 The merits of such an ideography are obvious: rigor of 
 reasoning is ensured by the calculus character; we are 
 sure of not introducing unintentionally any premise ; and 
 we can see exactly on what propositions any demonstration 
 depends. 
 
 We can shortly, but very fairly accurately, characterize the 
 dual development of the theory of symbolic logic during the 
 last sixty years as follows: The calculus raiiocinator aspect 
 of symbolic logic was developed by Boole, De Morgan, 
 Jevons, Venn, C. S. Peirce, Schroder, Mrs. LaddFranklin 
 and others; the lingua characteristica aspect was developed 
 by Frege, Peano and Russell. Of course there is no hard 
 and fast boundary-line between the domains of these two 
 parties. Thus Peirce and Schroder early began to work at 
 the foundations of arithmetic with the help of the calculus of 
 relations; and thus they did not consider the logical calculus 
 merely as an interesting branch of algebra. Then Peano paid 
 particular attention to the calculative aspect of his symbolism. 
 Frege has remarked that his own symbolism is meant to be 
 a calculus raiiocinator as well as a lingua characteristica, but 
 the using of Frege's symbolism as a calculus would be rather 
 like using a three-legged stand-camera for what is called 
 "snap-shot" photography, and one of the outwardly most 
 noticeable things about Russell's work is his combination of 
 the symbolisms of Frege and Peano in such a way as to 
 preserve nearly all of the merits of each. 
 
 The present work is concerned with the calculus raiiocinator 
 aspect, and shows, in an admirably succinct form, the beauty, 
 symmetry and simplicity of the calculus of logic regarded as 
 an algebra. In fact, it can hardly be doubted that some such 
 form as the one in which Schroder left it is by far the best 
 for exhibiting it from this point of view.' The content of the 
 
 1 Cf. A. N. Whitehead, A Treatise en Universal Algebra wilh Appli- 
 calivis, Cambridge, 1 898. 
 
PREFACE. IX 
 
 present volume corresponds to the two first volumes of 
 Schroder's great but rather prolix treatise.' Principally owing 
 to the influence of C. S. Peirce, Schroder departed from 
 the custom of Boole, Jevons, and himself (1877), which 
 consisted in the making fimdamental of the notion oi equality, 
 and adopted the notion of subordination or inclusion as a 
 primitive notion. A more orthodox Boolian exposition is 
 that of Venn', which also contains many valuable historical 
 notes. 
 
 We will finally make two remarks. 
 
 When Boole (cf S 2 below) spoke of propositions deter- 
 mining a class of moments at which they are true, he really 
 (as did MacColl") used the word "proposition" for what we 
 now call a "prepositional function" A "proposition" is a 
 thing expressed by such a phrase as "twice two are four" or 
 "twice two are five", and is always true or always false. But 
 we might seem to be stating a proposition when we say: 
 "Mr. Wn,LiAM Jennings Bryan is Candidate for the Presidency 
 of the United States", a statement which is sometimes true 
 and sometimes false. But such a statement is like a mathe- 
 matical function in so far as it depends on a variable — the 
 time. Functions of this kind are conveniently distinguished 
 from such entities as that expressed by the phrase "twice 
 two are four" by calling the latter entities "propositions" and 
 the former entities "propositional functions" : when the variable 
 in a propositional function is fixed, the function becomes a 
 proposition. There is, of course, no sort of necessity why 
 these special names should be used; the use of them is 
 merely a question of convenience and convention. 
 
 In the second place, it must be carefully observed that, in 
 S 13, o and I are not defined by expressions whose principal 
 
 • Vorlesungen iiber die Algebra der Logik, Vol. I., Leipsic, 1 890; 
 Vol. II, 1891 and 1905. We may mention that a much shorter Abriss 
 of the work has been prepared by EuGEN MiJLLER. Vol. Ill (1895) of 
 Schr6der's work is on the logic of relatives founded by De Morgan and 
 C. S. Peirce, — a branch of Logic that is only mentioned in the con- 
 cluding sentences of this volume. 
 
 2 Symbolic Logic, London, 1881; 2nd ed., 1894.. 
 
X PREFACE. 
 
 copulas are relations of inclusion. A definition is simply the 
 convention that, for the sake of brevity or some other con- 
 venience, a certain new sign is to be used instead of a group 
 of signs whose meaning is already known. Thus, it is the 
 sign of equality that forms the principal copula. The theory 
 of definition has been most minutely studied, in modem times 
 by Frege and Peano. 
 
 Philip E. B. Jourdain. 
 
 Girton, Cambridge. England. 
 
CONTENTS. 
 
 Preface „ 
 
 Bibliography 
 
 1. Introdaction 
 
 2. The Two Interpretations of the Logical Calculus 
 
 3. Relation of Inclusion 
 
 4. Definition of Equality 
 
 5. Principle of Identity 
 
 6. Principle of the Syllogism 
 
 7. Multiplication and Addition 
 
 8. Principles of Simplification and Composition 
 
 9. The Laws of Tautology and of Absorption 
 
 10. Theorems on Multiplication and Addition 
 
 11. The First Formula for Transforming Inclusions into Equalities 
 
 12. The Distributive Law 
 
 13. Definition of o and I 
 
 14. The Law of Duality 
 
 15. Definition of Negation 
 
 16. The Principles of Contradiction and of Excluded Middle 
 
 17. Law of Double Negation 
 
 18. Second Formula for Transforming Inclusions into Equalities 
 
 19. Law of Contraposition 
 
 20. Postulate of Existence 
 
 21. The Developments of o and of i 
 
 22. Properties of the Constituents 
 
 23. Logical Functions 
 
 24. The Law of Development 
 
 25. The Formulas of De Morgan 
 
 26. Disjunctive Sums 
 
 27. Properties of Developed Functions 
 
 28. The Limits of a Function 
 
 29. Formula of Poretsky 
 
 30. Schroder's Theorem 
 
 31. The Resultant of Elimination 
 
 32. The Case of Indetermination 
 
 Page- 
 Ill 
 XIII 
 3 
 3 
 4 
 6 
 
 9 
 II 
 12 
 14 
 15 
 16 
 
 17 
 20 
 21 
 23 
 24 
 25 
 26 
 
 27 
 28 
 29 
 29 
 30 
 32 
 34- 
 34 
 37 
 38 
 39 
 41 
 43 
 
XII CONTENTS. 
 
 Page. 
 
 33. Sums and Products of Functions 44 
 
 34. The Expression of an Inclusion by Means of an Indeterminate 46 
 
 35. The Expression of a Double Inclusion by Means of an Inde- 
 
 terminate 48 
 
 36. Solution of an Equation Involving One Unknown Quantity ... 50 
 
 37. Elimination of Several Unknown Quantities 53 
 
 38. Theorem concerning the Values of a Function 55 
 
 39. Conditions of Impossibility and Indetermination 57 
 
 40. Solution of Equations Containing Several Unknown Quantities 57 
 
 41. The Problem of Boole 59 
 
 42. The Method of Poretsky 61 
 
 43. The Law of Forms 62 
 
 44. The Law of Consequences 63 
 
 45. The Law of Causes 67 
 
 46. Forms of Consequences and Causes 69 
 
 47. Example: Venn's Problem 70 
 
 48. The Geometrical Diagrams of Venn 73 
 
 49. The Logical Machine of Jevons 75 
 
 50. Table of Consequences 76 
 
 51. Table of Causes 77 
 
 52. The Number of Possible Assertions 79 
 
 53. Particular Propositions 80 
 
 54. Solution of an Inequation with One Unknown „ „ ... 81 
 
 55. System of an Equation and an Inequation 83 
 
 56. Formulas Peculiar to the Calculus of Propositions 84 
 
 57. Equivalence of an Implication and an Alternative 85 
 
 58. Law of Importation and Exportation 88 
 
 59. Reduction of Inequalities to Equalities 90 
 
 60. Conclusion 92 
 
 Index 95 
 
BIBLIOGRAPHY.' 
 
 George Boole. The Mathematical Analysis of Logic (Cam- 
 bridge and London, 1847). 
 
 — An Investigation of the Laws of Thought (London and 
 Cambridge, 1854). 
 
 W. Stanley Jevons. Fure Logic (London, 1864). 
 
 — "On the Mechanical Performance of Logical Inference" 
 {Philosophical Transactions, 1870). 
 
 Ernst Schroder. Der Oferationskreis des Logikkalkuls 
 (Leipsic, 1877). 
 
 — Vorlesungen iiber die Algebra der Logik, Vol. I (1890), 
 Vol. II (1891), Vol. Ill: Algebra und Logik der Relative 
 (1895) (Leipsic).^ 
 
 Alexander Macfarlane. Principles of the Algebra of Logic, 
 with Examples (Edinburgh, 1879). 
 
 John Venn. Symbolic Logic, 1881; 2nd. ed., 1894 (London).^ 
 Studies in Logic by members of the Johns Hopkins Uni- 
 versity (Boston, 1883): particularly Mrs. Ladd-Franklin, 
 O. Mitchell and C. S. Peirce. 
 
 A. N. Whitehead. A Treatise on Universal Algebra. Vol. I 
 (Cambridge, 1898). 
 
 — "Memoir on the Algebra of Symbolic Logic" {American 
 Journal of Mathematics, Vol. XXIII, 1901). 
 
 I This list contains only the works relating to the system of Boole 
 and Schroder explained in this work. 
 
 » EuGEN MuLLER has prepared a part, and is preparing more, of 
 the publication of supplements to Vols. II and III, from the papers left 
 by Schr6der. 
 
 3 A valuable work from the points of view of history and bibliog- 
 raphy. 
 
XIV BIBLIOGRAPHY. 
 
 EuGEN MtJLLER. Ober die Algebra der Logik: I. Die Grund- 
 lagen des Gebietekalkuls ; 11. Das Eliminationsproblem und 
 die Syllogistik; Programs of the Grand Ducal Gymnasium 
 of Tauberbischofsheim (Baden), 1900, 1901 (Leipsic). 
 
 W. E. Johnson. "Sur la theorie des egalites logiques" {Biblio- 
 thlqtie du Congres international de Philosophie. Vol. Ill, 
 Logique et Histoire des Sciences; Paris, 1901). 
 
 Platon Poretsky. Sept Lois fondamentales de la theorie des 
 ^galitis logiques (Kazan, 1899). 
 
 — Quelques lois ultfrieures de la theorie des egalitis logiques 
 (Kazan, 1902). 
 
 — "Expos6 elementaire de la theorie des egalites logiques \ 
 deux termes" {Revue de Mitaphysique et de Morale. Vol. VIII, 
 1900.) 
 
 — "Theorie des egalites logiques a trois termes" {Bibliotfuque 
 du Congres international de Philosophie. Vol. III. {Logique 
 et Histoire des Sciences. (Paris, 1901, pp. 201 — 233). 
 
 — Thiorie des non-igalitis logiques (Kazan, 1904). 
 
 E. V. Huntington. "Sets of Independent Postulates for the 
 Algebra of Logic" {Transactions of the American Mathe- 
 matical Society, Vol. V, 1904). 
 
THE ALGEBRA OF LOGIC. 
 
1. Introduction. — The algebra of logic was founded by 
 George Boole (1815 — 1864); it was developed and perfected 
 by Ernst Schroder (1841 — 1902). The fundamental laws 
 of this calculus were devised to express the principles of 
 reasoning, the "laws of thought". But this calculus may be 
 considered from the purely formal point of view, which is 
 that of mathematics, as an algebra based upon certain prin- 
 ciples arbitrarily laid down. It belongs to the realm of 
 philosophy to decide whether, and in what measure, this 
 calculus corresponds to the actual operations of the mind, 
 and is adapted to translate or even to replace argument; 
 we cannot discuss this point here. The formal value of this 
 calculus and its interest for the mathematician are absolutely 
 independent of the interpretation given it and of the appli- 
 cation which can be made of it to logical problems. In 
 short, we shall discuss it not as logic but as algebra. 
 
 2. The Two Interpretations of the Logical Cal- 
 culus. — There is one circumstance of particular interest, 
 namely, that the algebra in question, like logic, is susceptible 
 of two distinct interpretations, the parallelism between them 
 being almost perfect, according as the letters represent con- 
 cepts or propositions. Doubtless we can, with Boole and 
 Schroder, reduce the two interpretations to one, by con- 
 sidering the concepts on the one hand and the propositions 
 on the other as corresponding to assemblages or classes; since 
 a concept determines the class of objects to which it is 
 applied (and which in logic is called its extension), and a 
 proposition determines the class of the instances or moments 
 of time in which it is true (and which by analogy can also 
 be called its extension). Accordingly the calculus of con- 
 
 I* 
 
4 LOGICAL CALCULUS AND INCLUSION. 
 
 cepts and the calculus of propositions become reduced to 
 but one, the calculus of classes, or, as Leibniz called it, the 
 theory of the whole and part, of that which contains and 
 that which is contained. But as a matter of fact, the cal- 
 culus of concepts and the calculus of propositions present 
 certain differences, as we shall see, which prevent their com- 
 plete identification from the formal point of view and conse- 
 quently their reduction to a single "calculus of classes". 
 
 Accordingly we have in reality three distinct calculi, or, 
 in the part common to all, three different interpretations of 
 the same calculus. In any case the reader must not forget 
 that the logical value and the deductive sequence of the 
 formulas does not in the least depend upon the inter- 
 pretations which may be given them, and, in order to 
 make this necessary abstraction easier, we shall take care to 
 place the symbols "C. I." {conceptual interpretation) and "P. I." 
 [propositional interpretation) before all interpretative phrases. 
 These interpretations shall serve only to render the formulas 
 intelligible, to give them clearness and to make their mean- 
 ing at once obvious, but never to justify them. They may 
 be omitted without destroying the logical rigidity of the 
 system. 
 
 In order not to favor either interpretation we shall say 
 that the letters represent terms; these terms may be either 
 concepts or propositions according to the case in hand. 
 Hence we use the word term only in the logical sense. 
 When we wish to designate the "terms" of a sum we shall 
 use the word summand in order that the logical and mathe- 
 matical meanings of the word may not be confused. A term 
 may therefore be either a factor or a summand. 
 
 3. Relation of Inclusion. — Like all deductive theories, 
 the algebra of logic may be established on various systems 
 of principles'; we shall choose the one which most nearly 
 
 ' See Huntington, "Sets of Independent Postulates for the Algebra 
 
 of Logic", Transactions of the Am. Math. Soc, Vol. V, 1904, pp. 288 309. 
 
 [Here he says: "Any set of consistent postulates would give rise to a 
 corresponding algebra, viz., the totality of propositions which follow 
 
RELATION OF INCLUSION. 5 
 
 approaches the exposition of Schroder and current logical 
 interpretation. 
 
 The fundamental relation of this calculus is the binary 
 (two -termed) relation which is called inclusion (for classes), 
 subsumption (for concepts), or implication (for propositions). 
 We will adopt the first name as affecting alike the two logical 
 interpretations, and we will represent this relation by the 
 sign < because it has formal properties analogous to those 
 of the mathematical relation << ("less than") or more exactly 
 <^ , especially the relation of not being symmetrical. Because 
 of this analogy Schroder represents this relation by the sign =$ 
 which we shall not employ because it is complex, whereas 
 the relation of inclusion is a simple one. 
 
 In the system of principles which we shall adopt, this 
 relation is taken as a primitive idea and is consequently 
 indefinable. The explanations which follow are not given 
 for the purpose of defining it but only to indicate its meaning 
 according to each of the two interpretations. 
 
 C. I.: When a and b denote concepts, the relation a <i b 
 signifies that the concept a is subsumed under the concept b; 
 that is, it is a species with respect to the genus b. From 
 the extensive point of view, it denotes that the class of a's 
 is contained in the class of ^'s or makes a part of it; or, 
 more concisely, that "All a's are ^'s". From the comprehen- 
 sive point of view it means that the concept b is contained 
 in the concept a or makes a part of it, so that consequently 
 the character a implies or involves the character b. Example: 
 "All men are mortal"; "Man implies mortal"; "Who says 
 man says mortal"; or, simply, "Man, therefore mortal". 
 
 P. I. : When a and b denote propositions, the relation a<^b 
 signifies that the proposition a implies or involves the prop- 
 osition b, which is often expressed by the hypothetical 
 judgment, "If a is true, b is true"; or by "a implies b"\ or 
 more simply by "a, therefore b". We see that in both inter- 
 
 from these postalates by logical deductioDS. Every set of postulates should 
 be free from redundances, in other words, the postulates of each set 
 should be independent, no one of them deducible from the rest."] 
 
6 DEFINITION OF EQUALITY. 
 
 pretations the relation <C may be translated approximately 
 by "therefore". 
 
 Remark. — Such a relation as "<z <; 3" is a proposition, 
 whatever may be the interpretation of the terms a and b. 
 Consequently, whenever a <C relation has two like relations 
 (or even only one) for its members, it can receive only the 
 propositional interpretation, that is to say, it can only denote 
 an implication. 
 
 A relation whose members are simple terms (letters) is 
 called a primary proposition; a relation whose members are 
 primary propositions is called a secondary proposition, and 
 so on. 
 
 From this it may be seen at once that the propositional 
 interpretation is more homogeneous than the conceptual, 
 since it alone makes it possible to give the same meaning 
 to the copula <^ in both primary and secondary prop- 
 ositions. 
 
 4. Definition of Equality. — There is a second copula 
 that may be defined by means of the first; this is the 
 copula = ("equal to"). By definition we have 
 
 whenever 
 
 a <^b and b <^a 
 
 are true at the same time, and then only. In other words, 
 the single relation fl = ^ is equivalent to the two simulta- 
 neous relations a <ib and b <^a. 
 
 In both interpretations the meaning of the copula = is 
 determined by its formal definition: 
 
 C. I.: a = b means, "All a's are ^'s and all ^'s are a's"; 
 in other words, that the classes a and b coincide, that they 
 are identical.' 
 
 P. I., a ^ b means that a implies b and b implies a; in 
 
 I This does not mean that the concepts a and b have the same 
 meaning. Examples: "triangle" and "trilateral", "equiangular triangle" 
 and "equilateral triangle". 
 
DEFINITION OF EQUALITY. 7 
 
 Other words, that the propositions a and b are equivalent, 
 that is to say, either true or false at the same time.' 
 
 Remark. — The relation of equality is symmetrical by very 
 reason of its definition: a = ^ is equivalent to b = a. But 
 the relation of inclusion is not symmetrical: a < ^ is not 
 equivalent to b <^a, nor does it imply it. We might agree 
 to consider the expression a"^ b equivalent to b <Z.a, but 
 we prefer for the sake of clearness to preserve always the 
 same sense for the copula <[• However, we might translate 
 verbally the same inclusion a <Z,b sometimes by "a is con- 
 tained in b" and sometimes by "b contains a". 
 
 In order not to favor either interpretation, we will call 
 the first member of this relation the antecedent and the 
 second the consequent. 
 
 C. L: The antecedent is the subject and the consequent is 
 the predicate of a universal affirmative proposition. 
 
 P. L : The antecedent is the premise or the cause, and the 
 consequent is the consequence. When an implication is trans- 
 lated by a hypothetical (or conditional) judgment the ante- 
 cedent is called the hypothesis (or the condition) and the 
 consequent is called the thesis. 
 
 When we shall have to demonstrate an equality we shall 
 usually analyze it into two converse inclusions and demon- 
 strate them separately. This analysis is sometimes made also 
 when the equality is a datum (a premise). 
 
 When both members of the equality are propositions, it 
 can be separated into two implications, of which one is 
 called a theorem and the other its reciprocal. Thus when- 
 ever a theorem and its reciprocal are true we have an 
 equality. A simple theorem gives rise to an implication 
 whose antecedent is the hypothesis and whose consequent is 
 the thesis of the theorem. 
 
 It is often said that the hypothesis is the sufficient condition 
 of the thesis, and the thesis the necessary condition of the hy- 
 
 I This does not mean that they have the same meaning. Example: 
 "'The triangle ABC has two equal sides", and "The triangle ABC has 
 two equal angles". 
 
» IDENTITY AND SYLLOGISM. 
 
 pothesis; that is to say, it is sufficient that the hypothesis be 
 true for the thesis to be true; while it is necessary that the 
 thesis be true for the hypothesis to be true also. When a 
 theorem and its reciprocal are true we say that its hypoth- 
 esis is the necessary and sufficient condition of the thesis; 
 that is to say, that it is at the same time both cause and 
 consequence. 
 
 5. Principle of Identity. — The first principle or axiom 
 of the algebra of logic is the principle of identity, which is 
 formulated thus: 
 
 (Ax. I) a<C,a, 
 
 whatever the term a may be. 
 
 C. I.: "All a's are a's", i. e., any class whatsoever is con- 
 tained in itself. 
 
 P. I.: "a implies a", i. e., any proposition whatsoever im- 
 pUes itself. 
 
 This is the primitive formula of the principle of identity. 
 By means of the definition of equality, we may deduce from 
 it another formula which is often wrongly taken as the ex- 
 pression of this principle: 
 
 a = a, 
 whatever a may be; for when we have 
 
 a <^a, a <C<i, 
 we have as a direct result, 
 
 a ^ a. 
 C. I.: The class f is identical with itself. 
 P. I.: The proposition a is equivalent to itself. 
 
 6. Principle of the Syllogism. — Another principle of 
 the algebra of logic is the principle of the syllogism, which 
 may be formulated as follows: 
 
 (Ax. II) {a <^b) (b<^c)<^{a<^ c). 
 
 C. I.: "If all a's are ^'s, and if all ^'s are f's, then all a's 
 are r's". This is the principle of the categorical syllogism. 
 
 P. I.: "If a implies b, and if b implies c, a implies c." 
 This is the principle of the hypothetical syllogism. 
 
MULTIPLICATION AND ADDITION. g 
 
 We see that in this formula the principal copula has al- 
 ways the sense of implication because the proposition is a 
 secondary one. 
 
 By the definition of equality the consequences of the 
 principle of the syllogism may be stated in the following 
 formulas ': 
 
 {a <b) {b = cX{a<i c), 
 
 {a = b) (b<c)<^{a<:c), 
 
 [a^b) {b=^cX{a = c). 
 
 The conclusion is an equality only when both premises 
 are equalities. 
 
 The preceding formulas can be generalized as follows: 
 {a<:b) (b<c) (^<^)<(a<^), 
 (a = ^) (6^ c) {c = d)<i{a = d). 
 
 Here we have the two chief formulas of the sorites. Many 
 other combinations may be easily imagined, but we can have 
 an equality for a conclusion only when all the premises are 
 equalities. This statement is of great practical value. In a 
 succession of deductions we must pay close attention to see 
 if the transition firom one proposition to the other takes place 
 by means of an equivalence or only of an implication. There 
 is no equivalence between two extreme propositions unless 
 all intermediate deductions are equivalences; in other words, 
 if there is one single implication in the chain, the relation 
 of the two extreme propositions is only that of implication. 
 
 7. Multiplication and Addition. — The algebra of logic 
 admits of three operations, logical multiplication, logical addition, 
 and negation. The two former are binary operations, that is 
 to say, combinations of two terms having as a consequent a 
 third term which may or may not be different from each of 
 them. The existence of the logical product and logical sum 
 of two terms must necessarily answer the purpose of a 
 
 I Strictly speaking, these formulas presuppose the laws of multi- 
 plication which will be established further on; but it is fitting to cite 
 them here in order to compare them with the principle of the syllogism 
 from which they are derived. 
 
10 MULTIPLICATION AND ADDITION. 
 
 double postulate, for simply to define an entity is not enough 
 for it to exist. The two postulates may be formulated thus: 
 (Ax. III}. Given any two terms, a and b, then there is a 
 term / such that 
 
 and that for every value of x for which 
 
 X <^a, X <^b, 
 we have also 
 
 x<:p. 
 
 (Ax. VJ). Given any two terms, a and b, then there exists 
 
 a term s such that 
 
 a < J, ^ < f, 
 
 and that, for any value of x for which 
 
 a <Cx, i <Cx, 
 we have also 
 
 s <^x. 
 
 It is easily proved that the terms / and j determined by 
 the given conditions are unique, and accordingly we can 
 define i/ie product ad and t/ie sum a + * as being respec- 
 tively the terms / and s. 
 
 C. I.: I. The product of two classes is a class / which 
 is contained in each of them and which contains every 
 (other) class contained in each of them; 
 
 2. The sum of two classes a and ^ is a class s which 
 contains each of them and which is contained in every (other) 
 class which contains each of them. 
 
 Taking the words "less than" and "greater than" in a meta- 
 phorical sense which the analogy of the relation <C with the 
 mathematical relation of inequality suggests, it may be said 
 that the product of two classes is the greatest class contained 
 in both, and the sum of two classes is the smallest class 
 which contains both.^ Consequently the product of two 
 
 I According lo another analogy Dedekind designated the logical sum 
 and product by the same signs as the least common multiple and greatest 
 common divisor i}Vas sind und was solUn dieZahkn? Nos. 8 and 17, 1887. 
 [Cf. English translation entitled Essays on Number (Chicago, Open Court 
 Publishing Co. 1901, pp. 46 and 48) ] Georg Cantor originally gave 
 them the same designation (Maikemaiiscke Annalen, Vol. XVII, 1880). 
 
SIMPLIFICATION AND COMPOSITION. I I 
 
 classes is the part that is common to each (the class of 
 their common elements) and the sum of two classes is the 
 class of all the elements which belong to at least one 
 of them. 
 
 P. L: I. The product of two propositions is a proposition 
 which implies each of them and which is imphed by every 
 proposition which implies both: 
 
 2. The sum of two propositions is the proposition which 
 is implied by each of them and which implies every prop- 
 osition implied by both. 
 
 Therefore we can say that the product of two propositions 
 is their weakest common cause, and that their sum is their 
 strongest common consequence, strong and weak being used 
 in a sense that every proposition which implies another is 
 stronger than the latter and the latter is weaker than the 
 one which impUes it. Thus it is easily seen that the product 
 of two propositions consists in their simultaneous affirmation: 
 "a and b are true", or simply "a and b" ; and that their 
 sum consists in their alternative affirmation, "either a or b 
 is true", or simply ''a or b". 
 
 Remark. — Logical addition thus defined is not disjunctive;' 
 tiiat is to say, it does not presuppose that the two summands 
 have no element in common. 
 
 8. Principles of Simplification and Composition. — 
 The two preceding definitions, or rather the postulates which 
 precede and justify them, yield direcdy the following formulas: 
 
 (i) ab<ia, ab<ib, 
 
 (2) {x<a){x<bX{x<:ab), 
 
 (3) a<.a + b, ^ < a + b, 
 
 (4) {a<x)(,b<x)<Z{a + b<x). 
 
 Formulas (i) and (3) bear the name of the principle of 
 simplification because by means of them the premises of an 
 
 ' [Boole, closely following analogy with ordinary mathematics, premised, 
 as a necessary condition to the definition of "x-\-y", that « and > were 
 viutually exclusive. Jevons, and practically all mathematical logicians after 
 him, advocated, on various grounds, the definition of "logical addition" 
 in a form which does not necessitate mutual exclusiveness,] 
 
12 THE LAWS OF TAUTOLOGY AND OF ABSORPTION. 
 
 argument may be simplified by deducing therefrom weaker 
 propositions, either by deducing one of the factors from a 
 product, or by deducing from a proposition a sum (alter- 
 native) of which it is a summand. 
 
 Formulas (2) and (4) are called the principle of composition, 
 because by means of them two inclusions of the same ante- 
 cedent or the same consequent may be combined {composed). 
 In the first case we have the product of the consequents, 
 in the second, the sum of the antecedents. 
 
 The formulas of the principle of compositioa can be trans- 
 formed into equalities by means of the principles of the 
 syllogism and of simplification. Thus we have 
 
 1 (Syll.) {x<^aV) (a6<a)<{x<a), 
 (Syll.) {x<ad) {ab<b)<{x<b). 
 
 Therefore 
 
 (Comp.) {x < ab) < (a: < a) (« < b). 
 
 2 (Syll.) {a<,a-\-b) (a + b<xX(a<x), 
 (Syll.) {b<a + b) {a + b<:x)<{b<x). 
 
 Therefore 
 
 (Comp.) (a + b<x)<:(a<.x) (*<*). 
 
 If we compare the new formulas with those preceding, 
 which are their converse propositions, we may write 
 ix<ab) = {x<a) {x<b), 
 ia + b<:x)=={a<ix) {b<x). 
 
 Thus, to say that x is contained in ab is equivalent to 
 saying that it is contained at the same time in both a and b; 
 and to say that x contains a -\- b is equivalent to saying 
 that it contains at the same time botli a and b. 
 
 9. The Laws of Tautology and of Absorption. — 
 
 Since the definitions of the logical sum and product do not 
 imply any order among the terms added or multiplied, 
 logical addition and multiplication evidently possess commu- 
 tative and associative properties which may be expressed in 
 the formulas 
 
 ab = ba, I a -i- 15 = ^ -1- a, 
 
 {ab) c = a {be), \ {a + b) + c = a + {6 + c). 
 
THE LAWS OF TAUTOLOGY AND OF ABSORPTION. 1 3 
 
 Moreover they possess a special property which is expressed 
 in the law of tautology: 
 
 a = aa, | a = a -V a. 
 
 Demonstration : 
 I (Simpl.) aa < a, 
 
 (Comp.) (a ■< a) (a <^ a) = (a <; aa) 
 whence, by the definition of equality, 
 
 (aa <^a) (a <C aa) = (a = aa). 
 In the same way: 
 2 (Simpl.) a<Cia + a, 
 
 (Comp.) (a <; a) (a <^a) = (a + a <i a), 
 whence 
 
 (a<Ca + a) (a + a <^ a) = (a •==' a + a). 
 
 From this law it follows that the sum or product of any 
 number whatever of equal (identical) terms is equal to one 
 single term. Therefore in the algebra of logic there are 
 neither multiples nor powers, in which respect it is very 
 much simpler than numerical algebra. 
 
 Finally, logical addition and multiplication possess a 
 remarkable property which also serves greatly to simplify 
 calculations, and which is expressed by the law of absorption: 
 a -\- ab = a, | a (a -Y b) = a. 
 
 Demonstration : 
 
 1 (Comp.) (« <; a) {ab <^ a) <^ (a + ab <Z o), 
 (Simpl.) a <^ a + ab, 
 
 whence, by the definition of equality, 
 
 {a-'r ab<^a) (a <^ a + ab) ^ (a + ab = a). 
 
 In the same way: 
 
 2 (Comp.) (a < a) (a<^a + b)<C[a<Ca (a + b)], 
 (Simpl.) a(a + b)<Za, 
 
 whence 
 
 [a<:^a (a + b)] [a (a + b)<^a] = [a {a + b) = a]. 
 
 Thus a term (a) absorbs a summand (ab) of which it is a 
 factor, or a factor (a + b) of which it is a summand. 
 
14 THEOREMS ON MULTIPLICATION AND ADDITION. 
 
 lo. Theorems on Multiplication and Addition. — We 
 
 can now establish two theorems with regard to the com- 
 bination of inclusions and equalities by addition and multi- 
 plication: 
 (Th. I) {a<6X{a<:<i<:), | {a<i)<{a + c <6 + c). 
 
 Demonstration : 
 
 1 (Simpl.) ac<^c, 
 
 (Syll.) {ac <a) (a <d)< {ac < b), 
 
 (Comp.) {ac<b) (ac < cX (ac < be). 
 
 2 (Simpl.) c<ib + c, 
 (Syll.) {a<b) {l><b + c)<ia<b + c), 
 (Comp.) ia.<:b + £) (c <b + c)<,{a + e <:^b + c). 
 
 This theorem may be easily extended to the case of 
 equalities: 
 
 (a = i>)<Z {ac -=bc), I (a = *)< (a + t = * -h c). 
 (Th. II) {a <b) {c<d)< {ac < bd), 
 
 {a<:b) {c<d)<{a + c<b + d). 
 Demonstration : 
 
 1 (Syll.) {ac<a) {a<b) <:{ac<ib), 
 (Syll.) • {ac<c) {c<dX{ac<d), 
 (Comp.) {ac <^b) {ac<id)<Z {ac < bd). 
 
 2 (Syll.) {a<.b) {b<b-VdX{a<b^d), 
 (Syll.) {c<d) id<:b + d)<{c<b+d), 
 (Comp.) {a<ib + d) (c<id + d) <i {a + c <Zb +d). 
 This theorem may easily be extended to the case in which 
 
 one of the two inclusions is replaced by an equality: 
 {a = b) {c<id)<^{ac<ibd), 
 {a^b) {c<:d)<:{a + c<.b + d). 
 
 When both are replaced by equalities the result is an 
 equality: 
 
 (a = b) {c=dX {ac = bd), 
 {a = b) {c=d)<Z{a + c= b + d). 
 To sum up, two or more inclusions or equalities can be 
 added or multiplied together member by member; the result 
 will not be an equality unless all the propositions combined 
 are equalities. 
 
TRANSFORMING INCLUSIONS. I 5 
 
 ir. The First Formula for Transforming Inclusions 
 into Equalities. — We can now demonstrate an important 
 formula by which an inclusion may be transformed into an 
 equality, or vice versa: 
 
 {a<ib) = {a = ab) \ {a<ib) = {a + b = b) 
 
 Demonstration : 
 
 1. (a < <J)< (a = ab), {a<b)<:{a + b = b). 
 For 
 
 (Comp.) (a<a) (a < ^)< (a < a/J), 
 
 (a<,b) {b<,bX{a + b<b). 
 On the other hand, we have 
 (Simpl.) ab<ia, b<ia-\-b, 
 
 (Def. =) ' (a < ab) {ab <^a) = {a = ab), 
 
 ia + b<b) {b<:,a + b) = {a + b=^b); 
 
 2. (a = a^)<(a<^), (a + b = b)<(a<b). 
 For 
 
 (a^ab) iab<b)<ia<b), 
 ia<:a + b) (a + ^ = <5)<(a<^). 
 Remark. — If we take the relation of equality as a primitive 
 idea (one not defined) we shall be able to define the relation 
 of inclusion by means of one of the two preceding formulas.' 
 We shall then be able to demonstrate the principle of the 
 syllogism.^ 
 
 From the preceding formulas may be derived an inter- 
 esting result: 
 
 {a = b) = (ab == a + b). 
 For 
 r. {a^b)^{a<^b) {b<a), 
 
 (a<^b) = (a = ab), (b<.a) = (a + b = a), 
 (Syll.) (a == ab) (a + b = a) <i(ab = a + b). 
 
 1 See Huntington, op. cit., § i. 
 
 2 This can be demonstrated as follows: By definition we have 
 (a <^) = (fl = a^), and (i < f) = (* = *<r). If in the first equality we 
 substitute for b its value derived from the second equality, then a = air. 
 Substitute for a its equivalent ab, then ab = abc. This equality is 
 equivalent to the inclusion, ab <C.c. Conversely substitute a for ab; 
 whence we have a < r. Q. E. D, 
 
1 6 DISTRIBUTIVE LAW. 
 
 2. (ab = a + b)<i{a-\-b<^ab), 
 
 (Comp.) {a + b<Cab) = {a<i ab) (b < ab), 
 
 (a < ab) [ab < a) = (a = ab) = (a < 6), 
 (b < ab) (ab <^b) = (b = ab) = (^ < a). 
 
 Hence 
 
 (a^ = a + ^j < (fl < -J) (b<a) = {a = b). 
 
 12. The Distributive Law. — The principles previously 
 stated make it possible to demonstrate the converse distributive 
 law, both of multiplication with respect to addition, and of 
 addition with respect to multiplication, 
 
 ac-\-bc<i{a-\-b)c, ab^c<i{a^c) {b -^ c). 
 
 Demonstration : 
 
 (a < a + 3)< [ac < (a + b)c], 
 {b<ia + b)<i [be < (a + b)c\; 
 whence, by composition, 
 
 [ac < (a + b)c] [be < (a + b)c\ < [ac + (5<: < (a + b)c\. 
 2. {ab<^a)<i{ab -V c<^a + c), 
 
 iab<b)<:{ab + e<Z^ + <:), 
 whence, by composition, 
 {ab+e<^a + c) {ab + c<.b + c)<i[ab + c<^{a->rc) {b + c)\ 
 
 But these principles are not sufficient to demonstrate the 
 direct distributive law 
 
 (a + b) c <i ac + be, (a + c) {b + c)<Cab + c, 
 
 and we are obliged to postulate one of these formulas or 
 some simpler one from which they can be derived. For 
 greater convenience we shall postulate the formula 
 
 (Ax. V). {a + b)c<_ac + bc. 
 
 This, combined with the converse formula, produces the 
 equality 
 
 (a -\- b) c = ac + be, 
 which we shall call briefly the distributive law. 
 From this may be directly deduced the formula 
 (a^b) {c + d) =^ ac -\- bc + ad+ bd, 
 
DEFINITION OF O AND I. 1 7 
 
 and consequently the second formula of the distributive law, 
 
 {a \c) (6 + c) = ad + e. 
 For 
 
 (a + c) (d + c) = ad + ac + dc + c, 
 and, by the law of absorption, 
 
 ac + 6c + c = c. 
 
 This second formula implies the inclusion cited above, 
 
 (a + c) (i+cXal> + c, 
 
 which thus is shown to be proved. 
 
 Corollary. — We have the equality 
 
 ab->r ac ■\- be =(a-\- b) {a + ^ ) (^ + c), 
 for 
 
 (a + 3) (a + c) {b-\-c)^{aArbc) (b + c) = ab + ac+ be. 
 
 It will be noted that the two members of this equality 
 dififer only in having the signs of multiplication and addition 
 transposed (compare S 14)- 
 
 13. Definition of o and i. — We shall now define and 
 introduce into the logical calculus two special terms which 
 we shall designate by o and by i, because of some formal 
 analogies that they present with the zero and unity of arith- 
 metic. These two terms are formally defined by the two 
 following principles which affirm or postulate their existence. 
 
 (Ax. VI). There is a term o such that whatever value 
 may be given to the term x, we have 
 
 o <^ ^. 
 
 (Ax. VII). There is a term 1 such that whatever value 
 may be given to the term x, we have 
 
 X <C 1. 
 
 It may be shown that each of the terms thus defined is 
 - unique; that is to say, if a second term possesses the same 
 property it is equal to (identical with) the first. 
 
1 8 DEFINITION OF O AND I. 
 
 The two interpretations of these terms give rise to para- 
 doxes which we shall not stop to elucidate here, but which 
 will be justified by the conclusions of the theory.' 
 
 C. I.: o denotes the class contained in every class; hence 
 it is the "null" or "void" class which contains no element 
 (Nothing or Naught), i denotes the class which contains all 
 classes; hence it is the totality of the elements which are 
 contained within it. It is called, after Boole, the "universe 
 of discourse" or simply the "whole". 
 
 P. 1.: o denotes the proposition which implies every prop- 
 osition; it is the "false" or the "absurd", for it implies 
 notably all pairs of contradictory propositions, i denotes 
 the proposition which is implied in every proposition; it is 
 the "true", for the false may imply the true whereas the true 
 can imply only the true. 
 
 By definition we have the following inclusions 
 
 o<o, o<i, I < I, 
 
 the first and last of which, moreover, result from the prin- 
 ciple of identity. It is important to bear the second in mind. 
 
 C. I.: The null class is contained in the wholes 
 
 P. I.: The false implies the true. 
 
 By the definitions of o and i we have the equivalences 
 
 (a < o) = (a = o), (i < a) = (a = i), 
 
 since we have 
 
 o <<[ a, a<^ I 
 
 whatever the value of a. 
 
 Consequently the principle of composition gives rise to 
 the two following corollaries: 
 
 {a = o) (6 = o) ^ (a + d ^ o), 
 (a = i) (^ = i) = (ad = i). 
 
 Thus we can combine two equalities having o for a second 
 
 ' Compare the author's Manuel de Logistique, Chap. I., % 8, Paris, 
 1905 [This work, however, did not appear]. 
 
 2 The rendering "Nothing is everything'' must be avoided. 
 
DEFINITION OF O AND I. 1 9 
 
 member by adding their first members, and two equalities 
 having i for a second member by multiplying their first 
 members. 
 
 Conversely, to say that a sum is "null" [zero] is to say that 
 each of the summands is null; to say that a product is equal 
 to I is to say that each of its factors is equal to i. 
 
 Thus we have 
 
 (<7 + ^ = o)< (a = O), 
 
 {ab= i)<(<z = i), 
 and more generally (by the principle of the syllogism) 
 {a < <5) (^ = o)< (a = o), 
 {a<b) (a=i)<(^= I). 
 
 It will be noted that we can not conclude from these the 
 equalities ab ^= a and a-\-b^i. And indeed in the con- 
 ceptual interpretation the first equality denotes that the part 
 common to the classes a and b is null; it by no means 
 follows that either one or the other of these classes is null. 
 The second denotes that these two classes combined form 
 the whole; it by no means follows that either one or the 
 other is equal to the whole. 
 
 The following formulas comprising the rules for the cal- 
 culus of o and i, can be demonstrated: 
 
 aXo = o, a+i = i, 
 a -\- o = a, a X 1 = a. 
 For 
 
 (o <[ a) = (o = o X a) = (a + o = a), 
 
 {a<Ci) = (a == aXi) = (a + I = i). 
 
 Accordingly it does not change a term to add o to it or 
 to multiply it by i. We express this fact by saying that 
 
 is the modulus of addition and i the modulus of multi- 
 plication. On the other hand, the product of any term 
 whatever by o is o and the sum of any term whatever with 
 
 1 is I. 
 
 These formulas justify the following interpretation of the 
 two terms: 
 
20 DUALITY. 
 
 C. I.: The part common to any class whatever and to the 
 null class is the null class; the sum of any class whatever 
 and of the whole is the whole. The sum of the null class and 
 of any class whatever is equal to the latter; the part common 
 to the whole and any class whatever is equal to the latter. 
 
 P. I.: The simultaneous affirmation of any proposition 
 whatever and of a false proposition is equivalent to the latter 
 (i. e., it is false); while their alternative affirmation is equal 
 to the former. The simultaneous affirmation of any prop- 
 osition whatever and of a true proposition is equivalent to 
 the former; while their alternative affirmation is equivalent to 
 the latter (i. e., it is true). 
 
 Remark. — If we accept the four preceding formulas as 
 axioms, because of the proof afforded by the double inter- 
 pretation, we may deduce from them the paradoxical formulas 
 
 o <^x, and x <i_\, 
 
 by means of the equivalences established above, 
 
 {a == ab) = {a <^b) = {a ^ b = b). 
 
 14. The Law of Dusdity. — We have proved that a perfect 
 symmetry exists between the formulas relating to multiplication 
 and those relating to addition. We can pass from one class 
 to the other by interchanging the signs of addition and 
 multiplication, on condition that we also interchange the 
 terms o and i and reverse the meaning of the sign ■<^ (or 
 transpose the two members of an inclusion). This symmetry, or 
 duality as it is called, which exists in principles and definitions, 
 must also exist in all the formulas deduced from them as 
 long as no principle or definition is introduced which would 
 overthrow them. Hence a true formula may be deduced 
 from another true formula by transforming it by the principle 
 of duality; that is, by following the rule given above. In its 
 application the law of duality makes it possible to replace 
 two demonstrations by one. It is well to note that this law 
 is derived from the definitions of addition and multipli- 
 cation (the formulas for which are reciprocal by duality) 
 
DEFINITION OF NEGATION. 21 
 
 and not, as is often thought', from the laws of negation 
 which have not yet been stated. We shall see that these 
 laws possess the same property and consequently preserve 
 the duality, but they do not originate it; and duality would 
 exist even if the idea of negation were not introduced. For 
 instance, the equality (S 12) 
 
 ab-\-ac^bc = {a-V b) (a + c) (b + c) 
 is its own reciprocal by duality, for its two members are 
 transformed into each other by duality. 
 
 It is worth remarking that the law of duality is only 
 applicable to primary propositions. We call [after Boole] 
 those propositions primary which contain but one copula 
 (<C or =)• We call those propositions secondary of which 
 both members (connected by the copula <; or ^) are primary 
 propositions, and so on. For instance, the principle of 
 identity and the principle of simplification are primary pro- 
 positions, while the principle of the syllogism and the principle 
 of composition are secondary propositions. 
 
 15. Definition of Negation.— The introduction of the terms 
 o and 1 makes it possible for us to define negation. This 
 is a "uni-naiy"' operation which transforms a single term into 
 another term called its negative.' The negative of a is called 
 not-a and is written d.^ Its formal definition implies the 
 following postulate of existence^: 
 
 I [Boole thus derives it (Laws of Thought, London 1854, Chap. Ill, 
 Prop. IV).] 
 
 ' [In French] the same word negation denotes both the operation 
 and its result, which becomes equivocal. The result ought to be denoted 
 by another word, like [the English] "negative". Some authors say, "supple- 
 mentary" or "supplement", [e. g. Boole and Huntington]. Classical 
 logic makes use of the term "contradictory" especially for propositions. 
 
 3 We adopt here the notation of Mac Coll ; Schr6der indicates 
 not-a by a I which prevents the use of indices and obliges us to express 
 them as exponents. The notation a' has the advantage of excluding 
 neither indices nor exponents. The notation a employed by many 
 authors is inconvenient for typographical reasons. When the negative 
 affects a proposition written in an explicit form (with a copula) it is 
 applied to the copula « or =) by a vertical bar (<j;; or =[=). The 
 accent can be considered as the indication of a vertical bar applied to letters. 
 
 4 [Boole follows Aristotle in usually calling the law of duality the 
 
22 DEFINITION OF NEGATION. 
 
 (Ax. VIU.) Whatever the term a may be, there is also a 
 term a such that we have at the same time 
 
 aa = o, a + a ^ I. 
 
 It can be proved by means of the following lemma that if 
 a term so defined exists it is unique: 
 
 If at the same time 
 
 ac = bc, a + c = d + c, 
 then 
 
 Demonstration. — Multiplying both members of the second 
 premise by a, we have 
 
 a ■\- ac= ab -\- ac. 
 Multiplying both members by b, 
 
 ab ■¥ bc = b + be. 
 Ey the first premise, 
 
 ab + ac= ab + be. 
 Hence 
 
 a-\- ae = b ■\- be, 
 
 which by the law of absorption may be reduced to 
 
 a = b. 
 
 Remark. — This demonstration rests upon the direct dis- 
 tributive law. This law cannot, then, be demonstrated by means 
 of negation, at least in the system of principles which we are 
 adopting, without reasoning in a circle- 
 
 This lemma being established, let us suppose that the same 
 term a has two negatives; in other words, let a'l and a\ be 
 two terms each of which by itself satisfies the conditions of 
 
 principle of contradiction "which affirms that it is impossible for any 
 being to possess a quality and at the same time not to possess it". He 
 writes it in the form of an equation of the second degree, x — x'^^o, 
 or ^ (I — ;ir) = o in which I • — x expresses the universe less x, or not 
 • X. Thus he regards the law of duality as derived from negation as 
 stated in note I above.] 
 
CONTRADICTION AND EXCLUDED MIDDLE. 23 
 
 the definition. We will prove that they are equal. Since, 
 by hypothesis, 
 
 aa'i^o, a + a'i^i, 
 
 aa'j ^ o, a + a'2 = I , 
 we have 
 
 aa^^aa-i, a + a i^= a -^ a^; 
 
 whence we conclude, by the preceding lemma, that 
 
 a\ = a 2. 
 
 We can now speak of (Ae negative of a term as of a unique 
 and well-defined term. 
 
 The uniformity of the operation of negation may be ex- 
 pressed in the following manner: 
 
 If a^b, then also a = b' . By this proposition, both 
 members of an equality in the logical calculus may be 
 "denied". 
 
 16. The Principles of Contradiction and of Excluded 
 Middle. — By definition, a term and its negative verify the 
 two formulas 
 
 aa = o, a +a = 1 , 
 
 which represent respectively the principle of contradiction and 
 the prificiple of excluded middle.^ 
 
 C. I. : I. The classes a and a have nothing in common; 
 in other words, no element can be at the same time both a 
 and not-(z. 
 
 2. The classes a and a combined form the whole; in 
 other words, every element is either a or not-a. 
 
 I As Mrs. Ladd- Franklin has truly remarked (Baldwin, Dictionaxy 
 of Philosophy and Psychology, article "Laws of Thought"), the principle of 
 contradiction is not sufficient to define contradictories; the principle of 
 excluded middle must be added which equally deserves the name of 
 principle of contradiction. This is why Mrs. Ladd-Franklin proposes 
 to call them respectively the principle of exclusion and the principle of 
 exhaustion, inasmuch as, according to the first, two contradictory terms 
 are exclusive (the one of the other); and, according to the second, they 
 are exhaustive (of the universe of discourse). 
 
24 DOUBLE NEGATION. 
 
 P. I.: I. The simultaneous affirmation of the propositions 
 a and not-a is false; in other words, these two propositions 
 cannot both be true at the same time. 
 
 2. The alternative affirmation of the propositions a and 
 not-a is true; in other words, one of these two propositions 
 must be true. 
 
 Two propositions are said to be contradictory when one is 
 the negative of the other; they cannot both be true or false 
 at the same time. If one is true the other is false; if one 
 is false the other is true. 
 
 This is in agreement with the fact that the terms o and i 
 are the negatives of each other; thus we have 
 
 0X1"= o, o+i = i. 
 
 Generally speaking, we say that two terms are contradictory 
 when one is the negative of the other. 
 
 17. La^v of Double Negation. — Moreover this reciprocity 
 is general: if a term b is the negative of the term a, then the 
 term a is the negative of the term h. These two statements 
 are expressed by the same formulas 
 
 ab = o, a + b = 1, 
 
 and, while they unequivocally determine b in terms of a, they 
 likewise determine a in terms of b. This is due to the 
 symmetry of these relations, that is to say, to the commu- 
 tativity of multiplication and addition. This reciprocity is 
 expressed by the law of double negation 
 
 which may be formally proved as follows: a' being by hy- 
 pothesis the negative of a, we have 
 
 aa = o, a -{- a' = 1. 
 On the other hand, let a" be the negative of a'; we have, 
 in the same way, 
 
 a' a" = o, a' + a" = I. 
 But, by the preceding lemma, these four equalities involve 
 the equality 
 
 a = a". Q. E. D. 
 
TRANSFORMING INCLUSIONS. 25 
 
 This law may be expressed in the following manner: 
 
 If ^ = a , we have a == b\ and conversely, by symmetry. 
 
 This proposition makes it possible, in calculations, to 
 transpose the negative from one member of an equality to 
 the other. 
 
 The law of double negation makes it possible to conclude 
 the equality of two terms from the equality of their negatives 
 (if a = b' then a = b), and therefore to cancel the negation 
 of both members of an equality. 
 
 From the characteristic formulas of negation together with 
 the fundamental properties of o and i, it results that every 
 product which contains two contradictory factors is null, and 
 that every sum which contains two contradictory summands 
 is equal to i. 
 
 In particular, we have the following formulas: 
 
 a = ab + ab' , a = (a + b) (a + b'), 
 
 which may be demonstrated as follows by means of the 
 distributive law: 
 
 a^ axi = a{b + b') ^ab + ab', 
 a'=a + o = a + bb' ^ (a + b) (a + b'). 
 
 These formulas indicate the principle of the method of 
 development which we shall explain in detail later (§§ 2 1 sqq.) 
 
 18. Second Formula for Transforming Inclusions 
 into Equalities: — We can now establish two very important 
 equivalences between inclusions and equalities: 
 
 ia<:b) = {ab'=o), {a<:b) = (a +b=i). 
 
 Demonstration. — i. If we multiply the two members of the 
 inclusion a<^b by b' we have 
 
 {ab' < bb') = {ab'<, o) = {ab^ = o). 
 
 2. Again, we know that 
 
 a = ab •{■ ab'. 
 Now if ab' = o, 
 
 a=ab+o==ab. 
 
26 LAW OF CONTRAPOSITION. 
 
 On the other hand: i. Add a to each of the two members 
 of the inclusion a<i_b\ we have 
 
 (a' + a < a' + ^) = (i < a' + 3) = (a' + ^ = i). 
 
 2. We know that 
 
 b = {a-\-b) {a +b). 
 
 Now, if a' + (5= I, 
 
 b = {a -V b)xi = a + b. 
 
 By the preceding formulas, an inclusion can be transformed 
 at will into an equality whose second member is either o or i. 
 Any equality may also be transformed into an equality of 
 this form by means of the following formulas: 
 
 {a = b)-={ab' + db = 6), {a = b)=[{a + b') (a' + ^) = i]. 
 
 Demonstration : 
 (a==b)^ (a <^b) {b<ia) = {ab'= o) {db = o) = {aH + a'/5 = o), 
 (<i = ^) = (a<^) (<5<a) = (a' + 3==i) (a + ^'=i) = 
 
 [(<z' + 3')(a' + ^) = i]. 
 Again, we have the two formulas 
 (a=b)'= \(a + b) {a + b') = o], {a = b) -= {ab + a b' = i) , 
 
 which can be deduced from the preceding formulas by per- 
 forming the indicated multiplications (or the indicated additions) 
 by means of the distributive law. 
 
 ig. Law of Contraposition. — We are now able to demon- 
 strate the law of contraposition, 
 
 {a<:b)^{b'<d). 
 
 Demonstration. — By the preceding formulas, we have 
 
 (a<C.b) = {ab' = o) = {b' < a). 
 
 Again, the law of contraposition may take the form 
 
 {,a<:b') = {b<.d), 
 
 which presupposes the .law of double negation. It may be 
 expressed verbally as follows: "Two members of an inclusion 
 may be interchanged on condition that both are denied". 
 
POSTULATE OF EXISTENCE. 2"] 
 
 C. I.: "If all a is b, then all not-^ is not-a, and conversely". 
 
 P. L: "If a implies b, not-3 implies not-a and conversely"; 
 in other words, "If a is true b is true", is equivalent to 
 saying, "If b is false, a is false". 
 
 This equivalence is the principle of the reductio ad absurdum 
 (see hypothetical arguments, modus tollens, % 58). 
 
 20. Postulate of Existence. — One final axiom may be 
 formulated here, which we will call the postulate of existence: 
 
 (Ax. IX) i<o, 
 
 whence may be also deduced i 4= ° • 
 
 In the conceptual interpretation (C. I.) this axiom means 
 that the universe of discourse is not null, that is to say, that 
 it contains some elements, at least one. If it contains but 
 one, there are only two classes possible, i and o. But even 
 then they would be distinct, and the preceding axiom would 
 be verified. 
 
 In the propositional interpretation (P. I.) this axiom signifies 
 that the true and the false are distinct ; in this case, it bears 
 the mark of evidence and of necessity. The contrary 
 proposition, 1 = 0, is, consequently, the type of absurdity 
 (of the formally false proposition) while the propositions = 0, 
 and 1 = 1 are types of identity (pi the formally true pro- 
 position). Accordingly we put 
 
 (i=o) = o, (o=o) = (i= i) = i. 
 
 More generally, every equality of the form 
 
 X = X 
 
 is equivalent to one of the identity-types; for, if we reduce 
 this equality so that its second member will be o or i, we find 
 
 (xx + XX = o) = (o = o), {xx -V x' x = i) = (i = 1). 
 
 On the other hand, every equality of the form 
 
 X = X 
 
 is equivalent to the absurdity-type, for we find by the same 
 process, 
 
 {xx -t- jc'a:'= o) = (i = o), {xx -I- XX = i) = (o = i). 
 
28 DEVELOPMENTS OF O AND OF I. 
 
 21. The Developments of o and of i. — Hitherto we 
 have met only such formulas as directly express customary 
 modes of reasoning and consequently offer direct evidence. 
 
 We shall now expound theories and methods which depart 
 from the usual modes of thought and which constitute more 
 particularly the algebra of logic in so far as it is a formal 
 and, so to speak, automatic method of an absolute universality 
 and an infallible certainty, replacing reasoning by cal- 
 culation. 
 
 The fundamental process of this method is development. 
 Given the terms a, b, c . . (to any finite number), we can 
 develop o or i with respect to these terms (and their negatives) 
 by the following formulas derived from the distributive law: 
 
 o = aa', 
 
 = aa'+ bb'^ {a -\- b) {a -\- b') {a ^ b) («'+ b'), 
 
 Q ^ ad -\- bb' + cc ^ {a + b -\- c) (a + ^ + /) {a + b' -y c) 
 x(a + 3' + <r') {d+b + c) 
 x(a'+^ + <r') {d-Vb'+c){a:+b + c'); 
 
 1 = a + a', 
 
 1 •= (a + a) {b + b') = ab + ab' + a b + db', 
 
 1 = (a + d) {b 4- b') {c + /) = abc + a^/-t- ab' c + ab' c 
 
 + d be -^ dbc + d b'c + d He ; 
 
 and so on. In general, for any number n of simple terms 
 
 will be developed in a product containing 2" factors, and 
 
 1 in a sum containing 2" summands. The factors of zero 
 comprise all possible additive combinations, and the summands 
 of I all possible multiplicative combinations of the n given 
 terms and their negatives, each combination comprising n 
 different terms and never containing a term and its negative 
 at the same time. 
 
 The summands of the development of i are what Boole 
 called the constituents (of the universe of discourse). We may 
 equally well call them, with Poretskv," the minima of dis- 
 course, because they are the smallest classes into which the 
 
 ' See the Bibliography, page xiv. 
 
PROPERTIES OF THE CONSTITUENTS. 29 
 
 universe of discourse is divided with reference to the n given 
 terms. In the same way we shall call the factors of the 
 development of o the maxima of discourse, because they 
 are the largest classes that can be determined in the universe 
 of discourse by means of the n given terms. 
 
 22. Properties of the Constituents. — The constituents 
 or minima of discourse possess two properties characteristic 
 of contradictory terms (of which they are a generalization); 
 they are mutually exclusive, i. e., the product of any two of 
 them is o; and they are collectively exhaustive, i. e., the sum 
 of all ''exhausts" the universe of discourse. The latter prop- 
 erty is evident from the preceding formulas. The other 
 results from the fact that any two constituents differ at least 
 in the "sign" of one of the terms which serve as factors, i. e., 
 one contains this term as a factor and the other the negative 
 of this term. This is enough, as we know, to ensure that 
 their product be null. 
 
 The maxima of discourse possess analogous and correlative 
 properties; their combined product is equal to o, as we have 
 seen; and the sum of any two of them is equal to i, inasmuch 
 as they differ in the sign of at least one of the terms which 
 enter into them as sumniands. 
 
 For the sake of simplicity, we shall confine ourselves, with 
 Boole and Schroder, to the study of the constituents or 
 minima of discourse, i. e., the developments of i. We shall 
 leave to the reader the task of finding and demonstrating 
 the corresponding theorems which concern the maxima of 
 discourse or the developments of o. 
 
 23. Logical Functions. — We shall call a logical function 
 any term whose expression is complex, that is, formed of 
 letters which denote simple terms together with the signs of 
 the three logical operations.' 
 
 I In this algebra the logical function is analogous to the integral 
 function of ordinary algebra, except that it has no powers beyond 
 the first. 
 
30 LOGICAL FUNCTIONS AND THEIR DEVELOPMENT. 
 
 A logical function may be considered as a function of all 
 the terms of discourse, or only of some of them which may 
 be regarded as unknown or variable and which in this case 
 are denoted by the letters x, y, z. We shall represent a 
 function of the variables or unknown quantities, x, y, z, by 
 the symbol / {x, y, z) or by other analogous symbols, as in 
 ordinary algebra. Once for all, a logical function may be 
 considered as a function of any term of the universe of dis- 
 course, whether or not the term appears in the explicit ex- 
 pression of the function. 
 
 24. The Law of Development. — This being established, 
 we shall proceed to develop a function f{x) with respect to x. 
 Suppose the problem solved, and let 
 
 ax + bx 
 
 be the development sought. By hypothesis we have the 
 equality 
 
 f{x) = ax + bx 
 
 for all possible values of x. Make x = \ and consequently 
 X = o. We have 
 
 /(i) = a. 
 
 Then put x = o and x = \; we have 
 
 /(o) = b. 
 
 These two equalities determine the coefficients a and b of 
 the development which may then be written as follows: 
 
 in which /(i), f{o) represent the value of the function /(:r) 
 when we let a: = i and ;i: = o respectively. 
 
 Corollary. — Multiplying both members of the preceding 
 equalities by x and x' in turn, we have the following pairs 
 of equalities (Mac Coll): 
 
 ^f{^) = (ix x'/{x) = bx' 
 
 xf(x) = xf{ I ) , xfiyx) = x'f{o) . 
 Now let a function of two (or more) variables be developed 
 
LAW OF DEVELOPMENT. 31 
 
 with respect to the two variables x and y. Developing 
 /{x, y) first with respect to x, we find 
 
 Ax, y) =/(i, y)x +/(o, y)x. 
 
 Then, developing the second member with respect to y, 
 we have 
 
 Ax, y) =/(i, i.)xy +/(i, o)xy'-^/{o, i)x'y +/(o, o)x'y. 
 
 This result is symmetrical with respect to the two variables, 
 and therefore independent of the order in which the develop- 
 ments with respect to each of them are performed. 
 
 In the same way we can obtain progressively the develop- 
 ment of a function of 3, 4, , variables. 
 
 The general law of these developments is as follows: 
 
 To develop a function with respect to n variables, form all 
 the constituents of these n variables and multiply each of 
 them by the value assumed by the function when each of 
 the simple factors of the corresponding constituent is equated 
 to I (which is the same thing as equating to o those factors 
 whose negatives appear in the constituent). 
 
 When a variable with respect to which the development is 
 made, y for instance, does not appear explicitly in the 
 function (/(*) for instance), we have, according to the 
 general law, 
 
 Ax)=Ax)y+Ax)y'. 
 
 In particular, if a is a constant term, independent of the 
 variables with respect to which the development is made, 
 we have for its successive developments, 
 a = ax + ax, 
 
 a = axy -t- axy + ax' y + ax'y, 
 a = axyz + axyz + axy z + axy z' ■{■ ax'yz + ax'yz -\- ax y z 
 
 + ax'yz ' 
 and so on. Moreover these formulas may be directly obtained 
 by multiplying by a both members of each development of i. 
 
 Cor. I. We have the equivalence 
 
 {a + x) {b + x) = ax -f bx + ab = ax + bx . 
 
 I These fonnulas express the method of classification by dichotomy. 
 
32 FORMULAS OF DE MORGAN. 
 
 For, if we develop with respect to x, we have 
 ax + dx' + adx + abx' = (a + ab)x + (^ + ab)x ^ ax + bx'. 
 
 Cor. 2. We have the equivalence 
 
 ax + bx + f == (a + c)x + (^ + c)x' . 
 
 For if we develop the term c with respect to x, we find 
 ax-^ bx -^^ cx-V ex = (a + tf)^ + (^ + ^) x . 
 
 Thus, when a function contains terms (whose sum is 
 represented by c) independent of x, we can always reduce it 
 to the developed form ax-\- bx by adding c to the coefficients 
 of both X and x. Therefore we can always consider a 
 function to be reduced to this form. 
 
 In practice, we perform the development by multiplying 
 each term which does not contain a certain letter {x for 
 instance) by {x + x) and by developing the product according 
 to the distributive law. Then, when desired, like terms may 
 be reduced to a single term. 
 
 25. The Formulas of De Morgan. — In any development 
 of I, the sum of a certain number of constituents is the negative 
 of the sum of all the others. 
 
 For, by hypothesis, the sum of these two sums is equal 
 to I, and their product is equal to o, since the product of 
 two different constituents is zero. 
 
 From this proposition may be deduced the formulas of 
 De Morgan: 
 
 {a + b)' ^ a'b', {ab)'= a' + b'. 
 
 Demonstration. — Let us develop the sum {a-\-b): 
 
 a + b ^= ab ■\- ab + ab + a b = ab -^ ab' ■\- a b . 
 
 Now the development of i with respect to a and b contains 
 the three terms of this development plus a fourth term a b' . 
 This fourth term, therefore, is the negative of the sum of the 
 other three. 
 
 We can demonstrate the second formula either by a correl- 
 ative argument {i. e., considering the development of o by 
 factors) or by observing that the development of {a'-\-b'). 
 
FORMULAS OF DE MORGAN. 33 
 
 a b + ab' ■\- a b\ 
 differs from the development of i only by the sumraand ab. 
 How De Morgan's formulas may be generalized is now 
 clear; for instance we have for a sum of three terms, 
 
 a + ^ + f = abc -\r abc -\- all c + ab' c + a bc+ a be ■\- a' b' c . 
 
 This development dififers from the development of i only 
 by the term a b' c . Thus we can demonstrate the formulas 
 
 (a + 3 + f)' = db' c , (abc)' = a' + b' + c, 
 
 which are generalizations of De Morgan's formulas. 
 
 The formulas of De Morgan are in very frequent use in 
 calculation, for they make it possible to perform the negation 
 of a sum or a product by transferring the negation to the 
 simple terms: the negative of a sum is the product of the 
 negatives of its summands; the negative of a product is the 
 sum of the negatives of its factors. 
 
 These formulas, again, make it possible to pass from a 
 primary proposition to its correlative proposition by duality, 
 and to demonstrate their equivalence. For this purpose it 
 is only necessary to apply the law of contraposition to the 
 given proposition, and then to perform the negation of both 
 members. 
 
 Example: 
 
 ab + ae+ be = {a-\- b) {a + f) (^ + c). 
 
 Demonstration : 
 
 {ab -V ac^ be)' '^ [{a + b) {a + e) {b + e)], 
 {ab)'{ae)'{be)' = (a + b)'+(a + f)'+ (b + c)' , 
 {a -\- b') (a'+ c) (^'+ e) = a b' -^ a'e'+ b'e. 
 
 Since the simple terms, a, b, e, may be any terms, we may 
 suppress the sign of negation by which they are affected, and 
 obtain the given formula. 
 
 Thus De Morgan's formulas furnish a means by which to 
 find or to demonstrate the formula correlative to another; 
 but, as we have said above (§14), they are not the basis of 
 
 this correlation. 
 
 3 
 
34 DISJUNCTIVE SUMS. 
 
 26. Disjunctive Sums. — By means of development we can 
 transform any sum into a disjunctive sum, /. e., one in which 
 each product of its summands taken two by two is zero. 
 For, let ((7 + ^ 4- ^) be a sum of which we do not know 
 whether or not the three terms are disjunctive; let us assume 
 that they are not. Developing, we have: 
 
 a-\- b ■\- c = abc + abc +ab' c + ab' c + a be + a'bc ■{■ a b'c. 
 
 Now, the first four terms of this development constitute 
 the development of a with respect to b and c; the two 
 following are the development of a b with respect to c. The 
 above sum, therefore, reduces to 
 
 a + a b + a b' c , 
 
 and the terms of this sum are disjunctive like those of the 
 preceding, as may be verified. This process is general and, 
 moreover, obvious. To enumerate without repetition all the 
 a's, all the ^'s, and all the c's, etc., it is clearly sufficient to 
 enumerate all the a's, then all the ^'s which are not a's, and 
 then all the c's which are neither a's nor ^'s, and so on. 
 
 It will be noted that the expression thus obtained is not 
 symmetrical, since it depends on the order assigned to the 
 original summands. Thus the same sum may be written: 
 
 b + ab'+a'b'c, c + ac + a be, . . . . 
 
 Conversely, in order to simplify the expression of a sum, 
 we may suppress as factors in each of the summands (arranged 
 in any suitable order) the negatives of each preceding sum- 
 mand. Thus, we may find a symmetrical expression for a 
 sum. For instance, 
 
 a + a b = ^ + ab' = a + b. 
 
 27. Properties of Developed Functions. — The practical 
 utility of the process of development in the algebra of logic 
 lies in the fact that developed functions possess the following 
 property: 
 
 The sum or the product of two functions developed with 
 respect to the same letters is obtained simply by finding the 
 sum or the product of their coefficients. The negative of a 
 
PROPERTIES OF DEVELOPED FUNCTIONS. 35 
 
 developed function is obtained simply by replacing the 
 coefficients of its development by their negatives. 
 
 We shall now demonstrate these propositions in the case 
 of two variables; this demonstration will of course be of 
 universal application. 
 
 Let the developed functions be 
 
 a^xy + b^xy + c-^x y + d^x'y, 
 a^xy + b^xy ■'r c^x y + d^x y . 
 
 1. I say that their sum is 
 
 (a^ + a^xy + {b^ + b^xy + (f, + c^xy + {d.^ + d^xy. 
 This result is derived directly from the distributive law. 
 
 2. I say that their product is 
 
 aj,a:!,xy + b-^b^xy -V c^c^x y + d^d^x'y, 
 
 for if we find their product according to the general rule 
 (by applying the distributive law), the products of two terms 
 of difiierent constituents will be zero; therefore there will remain 
 only the products of the terms of the same constituent, and, 
 as (by the law of tautology) the product of this constituent 
 multiplied by itself is equal to itself, it is only necessary to 
 obtain the product of the coefficients. 
 
 3. Finally, I say that the negative of 
 
 axy + bxy -^^ cxy-\r dx'y 
 is 
 
 axy + b' xy + c x'y + d' x'y. 
 
 In order to verify this statement, it is sufficient to prove 
 that the product of these two functions is zero and that their 
 sum is equal to i. Thus 
 
 {axy + bxy ■¥ cxy + dxy) {a xy + b'xy + c'x'y + d'x'y) 
 = {ad xy + bb' xy + cc xy + dd'x'y) 
 
 '={o-xy + o-xy'-\- o-x'y + o-x'y)=o^ 
 
 {axy + bxy' + cx'y + dxy) + {a xy + b' xy + c' x y + d' x y) 
 = [(a + a) xy+{b + b') xy + (f + c) xy + {d + d') x'y] 
 
 = {i-xy+ i-xy + 1 -x'y + i • *'/) = i. 
 
36 PROPERTIES OF DEVELOPED FUNCTIONS. 
 
 Special Case. — We have the equalities: 
 
 (ab + a b')' = ab' + a b, 
 {ab' + a b)' = ab + a'b' , 
 
 which may easily be demonstrated in many ways; for instance, 
 by observing that the two sums {ab + a'b') and (ab'+a'b) 
 combined form the development of i ; or again by performing 
 the negation {ab-\-a'b')' by means of De Morgan's formulas 
 (§ 2 5)- 
 
 From these equalities we can deduce the following equality: 
 (ff^'+ a b = o) = {ab + a' b' = i), 
 
 which result might also have been obtained in another way 
 by observing that (S 18) 
 
 (a = 3) = {ab'+ ab = o) = [(a + b') (a'+ 3) = i], 
 
 and by performing the multiplication indicated in the last 
 equality. 
 
 Theorem. — We have the following equivalences:^ 
 
 (a = be + b'c) = {b = ac'+a'c) = (c = ab'+ a'b). 
 
 For, reducing the first of these equalities so that its second 
 member will be o, 
 
 a{bc + b' c) + a'{bc' +b'c) = o, 
 
 abc + ab' c + a' be + a' b'c = o. 
 
 Now it is clear that the first member of this equality is 
 symmetrical with respect to the three terms a, b, c. We may 
 therefore conclude that, if the two other equalities which difiFer 
 from the first only in the permutation of these three letters 
 be similarly transformed, the same result will be obtained, 
 which proves the proposed equivalence. 
 
 Corollary.— \i we have at the same time the three inclusions: 
 a<ibc'-{- b'c, b<^ac' ■\- dc, c<^ab'+a'b, 
 
 we have also the converse inclusions, and therefore the 
 corresponding equalities 
 
 a ^ be -\- b'c, b = ac ■\- a'c, c ^ ab' -\-a'b. 
 ' W. Stanley Jevons, Pure Logic, 1864, p. 61. 
 
LIMITS OF A FUNCTION. 3/ 
 
 For if we txansform the given inclusions into equalities, we 
 shall have 
 
 abc ■\- ab' c == o, abc ■\- a be = o, abc -k- a b' c =^ o, 
 
 whence, by combining them into a single equality, 
 
 abc + a.b' c ■\- a be ■\- a b' e = o. 
 
 Now this equality, as we see, is equivalent to any one of 
 the three equalities to be demonstrated. 
 
 28. The Limits of a Function. — A term x is said to be 
 comprised between two given terms, a and b, when it contains 
 one and is contained in the other; that is to say, if we have, 
 for instance, 
 
 a<.x, x<.b, 
 
 which we may write more briefly as 
 a<^x<Cb. 
 
 Such a formula is called a double inclusion. When the 
 term x is variable and always comprised between two 
 constant terms a and b, these terms are called the limits 
 of X. The first (contained in sc) is called inferior limit; the 
 second (which contains' oc) is called the superior limit. 
 
 Theorem. — A developed function is comprised between the sum 
 and the product of its coefficients. 
 
 We shall first demonstrate this theorem for a function of 
 one variable, 
 
 ax + bx . 
 
 We have, on the one hand, 
 
 {ab <Zt) <i {abx <C ax), 
 
 {ab <.b)<,{abx' <bx'). 
 
 Therefore 
 
 abx + abx <C ax + bx', 
 
 ab <Cax -^ bx'. 
 On the other hand, 
 
 {a<a + bX{ax<i{a + b)x], 
 {b<a +.3) < [bx< {a + b)x']. 
 
 or 
 
38 FORMULA OF PORETSKY. 
 
 Therefore 
 
 or 
 
 ax + bx<^ (a + d) (x + x'), 
 ax + dx'<lia + 6. 
 
 To sum up, 
 
 ab<Cax + bx! <ia + ^. q. e. d. 
 
 Remark i. This double inclusion may be expressed in the 
 following form: ' 
 
 For 
 
 f{a) = aa + 5a' = a + b, 
 
 f{b)=^ab-\-bb'^ab. 
 
 But this form, pertaining as it does to an equation of one 
 unknown quantity, does not appear susceptible of generalization, 
 whereas the other one does so appear, for it is readily seen 
 that the former demonstration is of general application. 
 Whatever the number of variables n (and consequently the 
 number of constituents 2") it may be demonstrated in exactly 
 the same manner that the function contains the product of 
 its coefficients and is contained in their sum. Hence the 
 theorem is of general application. 
 
 Remark 2. — This theorem assumes that all the constituents 
 appear in the development, consequently those that are wanting 
 must really be present with the coefficient o. In this case, 
 the product of all the coefficients is evidently o. Likewise 
 when one coefficient has the value i, the sum of all the 
 coefficients is equal to 1. 
 
 It will be shown later (S 38) that a fiinction may reach 
 both its limits, and consequently that they are its extreme 
 values. As yet, however, we know only that it is always 
 comprised between them. 
 
 29. Formula of Poretsky." — We have the equivalence 
 {x =^ ax + bx) = {b<^x<^a). 
 
 ' EuGEN M&LLER, Aus der Algebra der Logik, Art. II. 
 2 PORETSKY, "Sur les m^thodes pour resoudre les ^galit^s logiques". 
 (Bull, de la Soc. phys.-math. de Kazan, Vol. II, 1884). 
 
Schroder's theorem. 39 
 
 Demonstration.— Vast multiplying by x both members of 
 the given equality [which is the first member of the entire 
 secondary equality], we have 
 
 X = ax, 
 
 which, as we know, is equivalent to the inclusion 
 
 x<^a. 
 
 Now multiplying both members by x', we have' 
 o = bx , 
 which, as we know, is equivalent to the inclusion 
 
 b<^x. 
 Summing up, we have 
 
 (« = a« + bx) <.{b<,x<ia). 
 Conversely, 
 
 (^ < * < a)< (a; = aa; + bx). 
 For 
 
 {x<^a) = {x = ax), 
 
 ib<^x).= (bx = o). 
 
 Adding these two equalities member to member [the second 
 members of the two larger equalities], 
 
 (x = ax) (o = bx) <Z (x = ax + bx). 
 Therefore 
 
 (b<x<aXix = ax + bx'), 
 and thus the equivalence is proved. 
 
 30. Schroder's Theorem.' — The equality 
 ax + bx'= o 
 signifies that x lies between a' and b. 
 Demonstration : 
 
 (ax + bx' = o) = (ax = o) (bx' = o), 
 (ax^ o) = (*<«'), 
 (^ix'-^o) = (b<x). 
 
 • Schr6der, Optrationskreis dcs Logiikalkuls (1877), Theorem 20. 
 
40 Schroder's theorem. 
 
 Hence 
 
 (ax + ix ^ 6) = {b <ix<^a'). 
 
 Comparing this theorem with the formula of Poretsky, we 
 obtain at once the equality 
 
 (ax+ bx' •= <i) = {x = ax + bx), 
 
 which may be directly proved by reducing the formula of 
 Poretsky -to an equality whose second member is o, thus: 
 
 {x = ax + bx) = [x{ax + b' x) + x {a x + bx') = o] 
 = (ax + bx = o). 
 
 If we consider the given equality as an equation in which 
 X is the unknown quantity, Poretsky's formula will be its 
 solution. 
 * From the double inclusion 
 
 b <^x<Ca' 
 we conclude, by the principle of the syllogism, that 
 
 b<a. 
 
 This is a consequence of the given equality and is in- 
 dependent of the unknown quantity x. It is called the 
 resultant of the elimination of x in the given equation. It is 
 equivalent to the equality 
 
 ab = o. 
 Therefore we have the implication 
 
 (ax-{- bx = o) <^ (ab = o). 
 
 Taking this consequence into consideration, the solution 
 may be simphfied, for 
 
 (ab = o) ^ (b = a'b). 
 Therefore 
 
 X = a'x + bx = ax + a bx 
 = a bx + a b' X + a bx = a b + a b' x 
 = b-\- a b' X = ^ + a'x. 
 
 This form of the solution conforms most closely to common 
 sense: since x contains b and is contained in a', it is natural 
 that X should be equal to the sum of b and a part of a' 
 
REStJLTANT OF ELIMINATION. 4 1 
 
 (that is to say, the part common to a and x). The solution 
 is generally indeterminate (between the limits a and h); it is 
 determinate only when the limits are equal, 
 
 a = b, 
 for then 
 
 X = b-\- a X = b + bx =- b = a . 
 
 Then the equation assumes the form 
 
 {ax + a x' = o) = {a = x) 
 
 and is equivalent to the double inclusion 
 
 (a' •<[ « <! a) = (x ^ a) . 
 
 31. The Resultant of Elimination. — When ab is not 
 zero, the equation is impossible (always false), because it has 
 a false consequence. It is for this reason that Schroder 
 considers the resultant of the elimination as a condition of 
 the equation. But we must not be misled by this equivocal 
 word. The resultant of the elimination of x is not a cause of 
 the equation, it is a consequence of it; it is not a sufficient but 
 a necessary condition. 
 
 The same conclusion may be reached by observing that 
 ab is the inferior limit of the function ax -\- bx\ and that 
 consequently the function can not vanish unless this limit is o. 
 
 {ab <^ax-'r bx) {ax + bx >= o) <^ {ab = o). 
 
 We can express the resultant of elimination in other equiv- 
 alent forms; for instance, if we write the equation in the form 
 
 {a + x) {b->fx) = o, 
 we observe that the resultant 
 
 ab = o 
 
 is obtained simply by dropping the unknown quantity (by 
 suppressing the terms x and x'). Again the equation may be 
 written: 
 
 a x-^ b X = 1 
 
 and the resultant of elimination: 
 
 ci -^-h = I . 
 
42 RESULTANT OF ELIMINATION. 
 
 Here again it is obtained simply by dropping the unknown 
 quantity.' 
 
 Remark. If in the equation 
 
 ax + bx = o 
 
 we substitute for the unknown quantity x its value derived 
 from the equations, 
 
 X == a'x + bx , X = ax + H x , 
 we find 
 
 (abx + abx = o) = (ab = o), 
 
 that is to say, the resultant of the elimination of x which, as 
 we have seen, is a consequence of the equation itself. Thus 
 we are assured that the value of x verifies this equation. 
 Therefore we can, with Voigt, define the solution of an equation 
 as that value which, when substituted for x in the equation, 
 reduces it to the resultant of the elimination of x. 
 
 Special Case. — When the equation contains a term independent 
 of X, i. e-, when it is of the form 
 
 ax + bx' -\- c = o 
 it is equivalent to 
 
 (a + c)x + (iJ + c)x' = o, 
 and the resultant of elimination is 
 
 {a + c) (b + c) ^ ab + c ^ o, 
 
 " This is the method of elimination of Mrs. Ladd- Franklin and 
 Mr. Mitchell, but this rule is deceptive in its apparent simplicity, for it 
 cannot be applied to the same equation when put in either of the forms 
 
 ax + ix' = o, {a + x') (i'-\- x) = I. 
 
 Now, on the other hand, as we shall see (S 54), for inequalities it 
 may be applied to the forms 
 
 ax + ix'^ o, (a + x') (*' -f- ar) =(= I . 
 
 and not to the equivalent forms 
 
 (a + x") {i + x)^ o, a'x + d'x''^ I. 
 
 Consequently, it has not the mnemonic property attributed to it, for, to 
 use it correctly, it is necessary to recall to which forms it is applicable. 
 
CASE OF INDETERMINATION. 43 
 
 whence we derive this practical rule: To obtain the resultant 
 of the elimination of x in this case, it is sufficient to equate 
 to zero the product of the coefficients of x and x', and add 
 to them the term independent of x. 
 
 32. The Case of Indetermination. — Just as the resultant 
 
 a/> = o 
 
 corresponds to the case when the equation is possible, so the 
 equality 
 
 a + l> ^ o 
 
 corresponds to the case of absolute indetermination. For in 
 this case the equation both of whose coefficients are zero 
 (a = o), (^ ^ o), is reduced to an identity (0 = 0), and 
 therefore is "identically" verified, whatever the value of x may 
 be; it does not determine the value of x at all, since the 
 double inclusion 
 
 ^ < a: < a' 
 then becomes 
 
 o<«<i, 
 
 which does not limit in any way the variability of x. In this 
 case we say that the equation is indeterminate. 
 
 We shall reach the same conclusion if we observe that 
 {a + b) is the superior limit of the function ax + bx and that, 
 if this limit is o, the function is necessarily zero for all 
 values of x, 
 
 {ax + bx'<i a -\- b) {a ■\- b = 0) <^{ax ->r bx = 0). 
 
 Special Case. — When the equation contains a tenn in- 
 dependent of X, 
 
 ax + tx' -^ c = o, 
 
 ihe condition of absolute indetermination takes the form 
 
 a •'r b-\- c = o. 
 For 
 
 ax -\- bx' + c = {a -^ c)x -^ {b + c)x', 
 
 (a + f) + (;J + <.-) = a + I* + c -= o. 
 
44 SUMS AND PRODUCTS OF FUNCTIONS. 
 
 33. Sums and Products of Functions. — It is desirable 
 at this point to introduce a notation borrowed from mathe- 
 matics, which is very useful in the algebra of logic. Let/(«) 
 be an expression containing one variable; suppose that the 
 class of all the possible values of x is determined; then the 
 class of all the values which the function /(x) can assume 
 in consequence will also be determined. Their sum will be 
 
 represented by ^/{x) and their product by fJ/(*). This 
 
 is a new notation and not a new notion, for it is merely the 
 idea of sum and product applied to the values of a function. 
 
 When the symbols ^ ^^d Y[ ^re applied to propositions, 
 they assume an interesting significance; 
 
 n[/w = °] 
 
 X 
 
 means that f{x) = o is true for every value of x\ and 
 
 X 
 
 that /(ar) = o is true for some value of x. For, in order 
 that a product may be equal to i (»'. e., be true), all its factors 
 must be equal to i {i. e., be true); but, in order that a sum 
 may be equal to i {i. e., be true), it is sufficient that only 
 one of its summands be equal to i (i. e., be true). Thus we 
 have a means of expressing universal and particular propositions 
 when they are applied to variables, especially those in the 
 form: "For every value of x such and such a proposition is 
 true", and "For some value of x, such and such a proposition 
 is true'', etc. 
 
 For instance, the equivalence 
 
 (a = i) = (ac = be) {a ■\- c ^ b + c) 
 
 is somewhat paradoxical because the second member contains 
 a term {c) which does not appear in the first. This equivalence 
 is independent of c, so that we can write it as follows, 
 considering c 3S a. variable x 
 
 Y\[{a = b)^(ax^bx) {a + x = b + x}], 
 
SUMS AND PRODUCTS OF FUNCTIONS. 45 
 
 or, the first member being independent of x, 
 
 (a = ^) = ]^[(aa; = 6x) (a + x = i + x)]. 
 
 In general, when a proposition contains a variable term, 
 great care is necessary to distinguish the case in which it is 
 true for every value of the variable, from the case in which 
 it is true only for some value of the variable.' This is the 
 purpose that the symbols J~J and 2 serve. 
 
 Thus when we say for instance that the equation 
 
 ax+ bx' = o 
 
 is possible, we are stating that it can be verified by some 
 value of x; that is to say, 
 
 ^'^ax + bx = o), 
 
 X 
 
 and, 'since the necessary and sufficient condition for this is 
 that the resultant {ab = o) is true, we must write 
 
 2(<2'*+ bx ^ o) = (ab = o), 
 
 X 
 
 although we have only the implication 
 
 (ax + bx' = o)<^ (ab = o). 
 
 On the other hand, the necessary and sufficient condition 
 for the equation to be verified by every value of x is that 
 
 a + b^ o. 
 Demonstration. — i. The condition is sufficient, for if 
 (a + ^ = o) = (3 = o) (i5 = o) , 
 we obviously have 
 
 ax + bx' = o 
 whatever the value of x; that is to say, 
 f][(a^ + bx' = o). 
 
 I This is the same as the distinction made in mathematics between 
 ideniilies and equations, except that an equation may not be verified by 
 any value of the variable. 
 
46 INCLUSION AND INDETERMINATES. 
 
 2. The condition is necessary, for if 
 Yliax + 6x') = o, 
 
 the equation is true, in particular, for the value x = a; hence 
 a + 6 = o. 
 Therefore the equivalence 
 
 Yliax + 6x'^ o) = ia + li = o} 
 
 is proved.' In this instance, the equation reduces to an 
 identity: its first member is "identically" null. 
 
 34. The Expression of an Inclusion by Means of an 
 Indeterminate. — The foregoing notation is indispensable in 
 almost every case where variables or indeterminates occur in 
 one member of an equivalence, which are not present in the 
 other. For instance, certain authors predicate the two following 
 equivalences 
 
 (a <i d) = (a = 6u) =- (a + p =■ b), 
 
 in which t(, v are two "indeterminates". Now, each of the 
 two equalities has the inclusion {a <C b) as its consequence, 
 as we may assure ourselves by eliminating u and v respectively 
 from the following equalities: 
 
 1. [a{b' + u) + a bu = o] ^ [{ab' + a b)u + au = o]. 
 Resultant: 
 
 [{ab'+db) a = o] = {ab' = o) = (<z < b). 
 
 2. [(a + v)b'+ abv = o] = \b' v + (ab' ■\- a b)v' = o]. 
 Resultant : 
 
 [b'(ab'+ ab) = o] = (ab' =0) + (a < i5). 
 
 But we cannot say, conversely, that the inclusion implies 
 the two equalities for any values of u and v; and, in fact, we 
 restrict ourselves to the proof that this implication holds for 
 some value of u and v, namely for the particular values 
 
 I EUGEN MULLER, Op. cU. 
 
INCLUSION AND INDETERMINATES. 4/ 
 
 u == a, b •= v; 
 for we have 
 
 (a = ab)^{a<ih) = {a + b = b). 
 
 But we cannot conclude, from the fact that the impHcation 
 (and therefore also the equivalence) is true for some value of 
 the indeterminates, that it is true for all; in particular, it is 
 not true for the values 
 
 for then {a = bu) and {a -^v = b) become {a = b), which 
 obviously asserts more than the given inclusion (a<ib)^ 
 Therefore we can write only the equivalences 
 
 (a<b)^ 2(a = bu) = ^(a + v = b), 
 
 but the three expressions 
 
 {<^<i), Yl{a = bu), Y[(a + v = b) 
 
 are not equivalent.' 
 
 ' Likewise if we make 
 
 K = o, v^ I, 
 we obtain the equalities 
 
 (a = 0), (*=I), 
 
 which assert still more than the given inclusion. 
 
 ' According to the remark in the preceding note, it is clear that 
 we have 
 
 Y[ {a = iu) = {a = 6 = o), ]~[ (a -f y == i) = (a = * = i), 
 
 since the equalities affected by the sign I I may be likewise verified 
 by the values 
 
 « = o, » = I and V = o, v = I. 
 
 If we wish to know within what limits the indeterminates u and 7j are 
 variable, it is sufficient to solve with respect to them the equations 
 
 (a<i5) = (a = ^a), (a <*) = (a + z/ = ^), 
 or 
 
 ab'=abu \- ab' -\- au , ab'=ab' ^r b'v + a'bv , 
 or 
 
 dbu-\-abu^<:>, a'b'v -^ a'bv' ^o, 
 
48 DOUBLE INCLUSION AND INDETERMINATES. 
 
 35. The Expression of a Double Inclusion by Means 
 of an Indeterminate. — Theorem. The double inclusion 
 
 b<ix<ia 
 
 is equivalent to the equality x =^ au-\- bu together with the 
 condition {b<^d), u being a term absolutely indeterminate. 
 
 Demonstration. — Let us develop the equality in question, 
 x{du + })u) + x'{au + bu) = o, 
 {ci X + ax')u + (b' X + bx)u = o. 
 
 Eliminating u from it, 
 
 ci b' X + abx = o. 
 This equality is equivalent to the double inclusion 
 
 ab<ix<^a-^b. 
 But, by hypothesis, we have 
 
 {b <,a) = {ab ^ b) = {a ^ b ^ a). 
 The double inclusion is therefore reduced to 
 
 b<Cx<^a. 
 So, whatever the value of u, the equality under consideration 
 involves the double inclusion. Conversely, the double in- 
 clusion involves the equality, whatever the value of x may be, 
 for it is equivalent to 
 
 a a; + bx = o, 
 and then the equality is simplified and reduced to 
 ax u + b' xu = o. 
 
 from which (by a formula to be demonstrated later on) we derive the 
 solutions 
 
 u^ab -^-w (3 + b'), V = ab -^w (a.-\- b), 
 or simply 
 
 u = ab •\- wb' y v= a b -\- w a, 
 w being absolutely indeterminate. We would arrive at these solutions 
 simply by asking ; By what term must we multiply b in order to obtain 
 a? By a term which contains ab plus any part of b' . What term must 
 we add to a in order to obtain bl A term which 'contains a'b plus 
 any part of o. In short, k can vary between ab and a-^b', v between 
 ab and a-\- b. 
 
DOXJBLE INCLUSION AND INDETERMINATES. 49 
 
 We can always derive from this the value of u in terms 
 of X, for the resultant {ab' xx = o) is identically verified. 
 The solution is given by the double inclusion 
 
 b' x<C u<i.d -^ X. 
 
 Remark. — There is no contradiction between this result, 
 which shows that the value of u hes between certain limits, 
 and the previous assertion that u is absolutely indeterminate; 
 for the latter assumes that x is any value that will verify the 
 double inclusion, while when we evaluate u in terms of x the 
 value of x is supposed to be determinate, and it is with 
 respect to this particular value of x that the value of u is 
 subjected to limits.' 
 
 In order that the value of u should be completely deter- 
 mined, it is necessary and sufficient that we should have 
 
 b' X = a' + X, 
 that is to say, 
 
 b' xax ■\- {b -V x) {a -\- x) = o 
 or 
 
 bx + a X = o. 
 
 Now, by hypothesis, we already have 
 
 a X + bx' = o. 
 
 If we combine these two equalities, we find 
 
 (a'+ (5 = o) = (a = i) {b = o). 
 
 This is the case when the value of x is absolutely in- 
 determinate, since it lies between the limits o and i. 
 
 In this case we have 
 
 u = b' X ^ a + X = X. 
 
 In order that the value of u be absolutely indeterminate, 
 it is necessary and sufficient that we have at the same time 
 
 I Moreover, if we substitute for x its inferior limit 6 in the inferior 
 limit of u, this limit becomes 6i' = o; and, if we substitute for x its 
 superior limit a in the superior limit of «, this limit becomes a -\- a =1. 
 
 4 
 
JO EQUATION INVOLVING ONE UNKNOWN. 
 
 b'x -^ o, a'+ X = 1, 
 
 or 
 
 i' X + ax' = o, 
 
 that is 
 
 a <^x <il>. 
 
 Now we already have, by hypothesis, 
 
 d<Cx<ia; 
 
 so we may infer 
 
 5 = X = a. 
 
 This is the case in which the value of x is completely 
 determinate. 
 
 36. Solution of an Equation Involving One Unknown 
 Quantity. — The solution of the equation 
 
 ax + bx = o 
 
 may be expressed in the form 
 
 X = du + bii , 
 
 u being an indeterminate, on condition that the resultant of 
 the equation be verified; for we can prove that this equality 
 implies the equality 
 
 ab' X + a' bx' = o, 
 
 which is equivalent to the double inclusion 
 a'b <Cx<ia' + b. 
 Now, by hypothesis, we have 
 
 (ab = o) ^ (a'b = b) = (a + b = a') . 
 
 Therefore, in this hypothesis, the proposed solution implies 
 the double inclusion 
 
 b<Cx<Ca'; 
 which is equivalent to the given equation. 
 
 Remark. — In the same hypothesis in which we have 
 {ab=o) = {b<a'), 
 
 we can always put this solution in the simpler but less sym- 
 metrical forms 
 
 x^ b ■\- a u, x = a'(b + u). 
 
EQUATION INVOLVING ONE UNKNOWN. 5 1 
 
 For 
 
 1. We have identically 
 
 b = bu ■\- bit . 
 Now 
 
 (,5<a')<(^«<a'a). 
 Therefore 
 
 {x ■= bu Ar du) = {x = b + a u). 
 
 2. Let us now demonstrate the formula 
 
 X ^ d b -\- a u . 
 Now 
 
 a b = b. 
 Therefore 
 
 X = b + a u 
 
 which may be reduced to the preceding form. 
 Again, we can put the same solution in the form 
 X = a b + u(ab + a ll), 
 which follows from the equation put in the form 
 
 ab' X + a bx = o, 
 if we note that 
 
 a'+ b = ab -\- a b + a b' 
 and that 
 
 ud b<Z. d b. 
 
 This last form is needlessly complicated, since, by hypothesis, 
 ab = o. 
 Therefore there remains 
 
 X = d b + ud b' 
 which again is equivalent to 
 
 X ^ b + ud, 
 since 
 
 d b = b and d = db + db'. 
 
 Whatever form we give to the solution, the parameter u 
 in it is absolutely indeterminate, /. e., it can receive all possible 
 values, including o and i ; for when u = o we have 
 
 4* 
 
52 EQUATION INVOLVING ONE UNKNOWN. 
 
 and when a = i we have 
 
 X = a, 
 and these are the two extreme values of x. 
 
 Now we understand that x is determinate in the particular 
 case in which a = i, and that, on the other hand, it is 
 absolutely indeterminate when 
 
 ^ = 0, a = I, (or a = o). 
 
 Summing up, the formula 
 
 x = a'u + bu 
 
 replaces the "limited" variable x (lying between the limits a 
 and b) by the "unlimited" variable u which can receive all 
 possible values, including o and i. 
 
 Remarks — The formula of solution 
 X = a X + bx 
 
 is indeed equivalent to the given equation, but not so the 
 formula of solution 
 
 X = a ti + bu 
 as a function of the indeterminate u. For if we develop the 
 latter we find 
 
 ab' X + a bx ■\- ab(xu + x u^ + a' b' {xu + x'u) = o, 
 and if we compare it with the developed equation 
 
 ab + ab' X + dbx' = o, 
 we ascertain that it contains, besides the solution, the equality 
 
 ab{xu' + X u) = o, 
 and lacks of the same solution the equality 
 
 a' b' {xu -\- x' u) == o. 
 Moreover these two terms disappear if we make 
 u == X 
 and this reduces the formula to 
 
 X = a' X + bx'. 
 
 I PoRETSKY. Sept his. Chaps. XXXHI and XXXIV. 
 
ELIMINATION OF SEVERAL XJNKNOWNS. S3 
 
 From this remark, Poretsky concluded that, in general, the 
 solution of an equation is neither a consequence nor a cause 
 of the equation. It is a cause of it in the particular case in which 
 
 ab = o, 
 
 and it is a consequence of it in the particular case in which 
 
 {db' = o)= (a + i5 = i). 
 
 But if ab is not equal to o, the equation is unsolvable and 
 the formula of solution absurd, which fact explains the 
 preceding paradox. If we have at the same time 
 
 ab = o and a + ^^i, 
 
 the solution is both consequence and cause at the same time, 
 that is to say, it is equivalent to the equation. For when 
 a = b the equation is determinate and has only the one 
 solution 
 
 X = a = b. 
 
 Thus, whenever an equation is solvable, its solution is one 
 of its causes; and, in fact, the problem consists in finding a 
 value of X which wiU verify it, /". e., which is a cause of it. 
 
 To sum up, we have the following equivalence: 
 
 {ax + bx' = o) = {ab = oi'^S-X = a u + bu) 
 
 which includes the following implications: 
 
 {ax + bx' = oX {ab = o), 
 {ax + bx' = o) <[ ^{x = a'u + bu), 
 
 {ab = o)^{x = a'u-'rbu') <C {ax-\- bx = o). 
 
 37. Elimination of Several Unknown Quantities. — 
 
 We shall now consider an equation involving several unknown 
 quantities and suppose it reduced to the normal form, i. e., 
 its first member developed with respect to the unknown 
 quantities, and its second member zero. Let us first concern 
 ourselves with the problem of elimination. We can eliminate 
 the unknown quantities either one by one or all at once. 
 
54 ELIMINATION OF SEVERAL UNKNOWNS. 
 
 For instance, let 
 (i) (p{x,y, z) = axyz + bxyz + cxy z + dxy z' 
 
 -\-fxyz + gx'yz + hx'y'z + kx'y'z' = o 
 be an equation involving three unknown quantities. 
 
 We can eliminate z by considering it as the only unknown 
 quantity, and we obtain as resultant 
 {axy + cxy'+fx'y + hx'y) {bxy + dxy'-\-gxy + kx'y) = o 
 
 or 
 
 (2) abxy + cdxy'+fgxy-\-hkx'y' = o. 
 
 If equation (i) is possible, equation (2) is possible as well; 
 that is, it is verified by some values of x and y. Accordingly 
 we can eliminate y from the equation by considering it as 
 the only unknown quantity, and we obtain as resultant 
 
 {abx-\-/gx') {cdx+ hkx) = o 
 or 
 
 (3) abcdxArfghkx ^ o. 
 
 If equation (i) is possible, equation (3) is also possible; 
 that is, it is verified by some values of x. Hence we can 
 eliminate x from it and obtain as the final resultant, 
 
 abed -fghk = o 
 
 which is a consequence of (i), independent of the unknown 
 quantities. It is evident, by the principle of symmetry, that 
 the same resultant would be obtained if we were to eliminate 
 the unknown quantities in a different order. Moreover this 
 result might have been foreseen, for since we have (S 28) 
 
 abcdfghk < (p {x, y, z) , 
 
 (p(x, y, z) can vanish only if the product of its coefficients 
 
 is zero: 
 
 [fpix,y, 2) = o] < {abcdfghk = o). 
 
 Hence we can eliminate all the unknown quantities at once 
 by equating to o the product of the coefficients of the 
 function developed with respect to all these unknown quantities. 
 
 We can also eliminate some only of the unknown quantities 
 at one time. To do this, it is sufficient to develop the first 
 
VALUES OF A FUNCTION. 55 
 
 member with respect to these unknown quantities and to 
 equate the product of the coefficients of this development 
 to o. This product will generally contain the other unknown 
 quantities. Thus the resultant of the elimination of z alone, 
 as we have seen, is 
 
 abxy + cdxy -\-fgxy + hkx y = o 
 and the resultant of the elimination of y and z is 
 abcdx -\- fghkx = o. 
 
 These partial resultants can be obtained by means of the 
 following practical rule: Form the constituents relating to the 
 unknown quantities to be retained; give each of them, for a 
 coefficient, the product of the coefficients of the constituents 
 of the general development of which it is a factor, and equate 
 the sum to o. 
 
 38. Theorem Concerning the Values of a Function: — 
 
 All the values which can be assumed by a function of any number 
 of variables f{x,y, z . . .) are given by the formula 
 
 abc ...k + u{aJrb-\-c+... + k), 
 
 in which u is absolutely indeterminate, and a, b- c . . ., k are 
 the coefficients of the development of f 
 
 Demonstration. — It is sufficient to prove that in the equahty 
 
 f{x, y,z ..:) ^ abc...k-^u{a-Vb-Vc^- ...^k) 
 
 u can assume all possible values, that is to say, that this 
 equality, considered as an equation in terms of «, is in- 
 determinate. 
 
 In the first place, for the sake of greater homogeneity, we 
 may put the second member in the form 
 
 u abc .. .k\ u{a + b+ c+ .. .-\-k), 
 for 
 
 abc . . . i = uabc . . .i + u abc . . . k, 
 and 
 
 uabc ...h<::u(a + b + c + ... + i). 
 
 Reducing the second member to o (assuming there are 
 only three variables x, y, z) 
 
56 VALUES OF A FUNCTION. 
 
 (axyz + bxyz + cxy z + ... + kx y z) 
 
 X [ua'i'c' ...i'+u(a' + d'+c' + ... + k')] 
 + (axyz + ^ xyz + c xy z-\- ... + k'x'y z) 
 x[«(a + ^ + c + . . . + >J) + u abc . . . k] = o, 
 or more simply 
 
 u{a-\-b + c + ...-\-k){a'xyz + ixyz +c' xy z-\- ...->rk' x'y z) 
 -\-u {a -k-li -Vc +...-\-k'){axyz + bxyz +...-{- kx'y z') = o. 
 
 If we eliminate all the variables x, y, z, but not the in- 
 determinate u, we get the resultant 
 
 u{a -\- b -\- c + . . . + k) a b' c . . . k 
 
 + u {a + b' + c + . .. + k')abc .. .k = o. 
 
 Now the two coefficients of « and u are identically zero; 
 it follows that » is absolutely indeterminate, which was to be 
 proved.' 
 
 From this theorem follows the very important consequence 
 that a function of any number of variables can be changed 
 into a function of a single variable without diminishing or 
 altering its "variability". 
 
 Corollary. — A function of any number of variables can 
 become equal to either of its limits. 
 
 For, if this function is expressed in the equivalent form 
 
 abc...k + u{a + b-^c + ... + k), 
 
 it will be equal to its minimum {abc . . . k) when « = o, and 
 to its maximum {a + b -\- c -\- . . . + k) when « = i. 
 
 Moreover we can verify this proposition on the primitive 
 form of the function by giving suitable values to the 
 variables. 
 
 Thus a function can assume all values comprised between 
 its two limits, including the limits themselves. Consequently, 
 it is absolutely indeterminate when 
 
 abc . . . k = o and a + b-\-c-'r...-\-k^i 
 at the same time, or 
 
 abc ... k = o = a be . . .k . 
 
 I Whitehead, Universal Algebra, Vol. I, § 33 (4). 
 
IMPOSSIBILITY AND INDETERMINATION. 57 
 
 39. Conditions of Impossibility and Indetermination. — 
 
 The preceding theorem enables us to find the conditions 
 under which an equation of several unknown quantities is 
 impossible or indeterminate. Let f{x, y, z . . .) be the first 
 member supposed to be developed, and a, b, c . . ., k its 
 coefficients. The necessary and sufficient condition for the 
 equation to be possible is 
 
 abc . . . k = o. 
 
 For, (i) if / vanishes for some value of the unknowns, 
 its inferior limit abc .. .k must be zero; (2) if abc . . . i is zero, 
 /"may become equal to it, and therefore may vanish for certain 
 values of the unknowns. 
 
 The necessary and sufficient condition for the equation to 
 be indeterminate (identically verified) is 
 
 a-^rb-^c...-\-k = o. 
 
 For, (i) if a + i5 + c + . . . + i is zero, since it is the 
 superior limit of f, this function will always and necessarily 
 be zero; (2) if / is zero for all values of the unknowns, 
 a ■>r b-'r c-'r ...-{■ k will be zero, for it is one of the values 
 of/. 
 
 Summing up, therefore, we have the two equivalences 
 
 2 [/(^' y^ z,..^ =o\ = {abc...k=o). 
 
 W\f{x,y,z..) = o'\ = {a^b\c...^k = o). 
 
 The equality abc . . .k = o is, as we know, the resultant 
 of the elimination of all the unknowns; it is the consequence 
 that can be derived fi-om the equation (assumed to be veri- 
 fied) independently of all the unknowns. 
 
 40. Solution of Equations Containing Several Un- 
 known Quantities. — On the other hand, let us see how 
 we can solve an equation with respect to its various un- 
 knowns, and, to this end, we shall limit ourselves to the 
 case of two unknowns 
 
 axy + bxy-\- cxy -^ dx y = o. 
 
58 EQUATIONS CONTAINING SEVERAL UNKNOWNS. 
 
 First solving with respect to x, 
 
 X = {ay + b'y) x + (cy + dy) x . 
 
 The resultant of the eHmination of x is 
 
 acy ^ bdy = o. 
 
 If the given equation is true, this resultant is true. 
 Now it is an equation involving jj' only; solving it, 
 
 y = (a \ c) y + bdy . 
 
 Had we eliminated y first and then x, we would have 
 obtained the solution 
 
 y = {ax + ex) y + {bx + dx ) y 
 and the equation in x 
 
 abx + cdx = o, 
 whence the solution 
 
 x=^ {a -\-b) X ■\- cdx . 
 
 We see that the solution of an equation involving two 
 unknown quantities is not symmetrical with respect to these 
 unknowns; according to the order in which they were elim- 
 inated, we have the solution 
 
 X = {a y + b'y) x + {cy-\-dy') x , 
 y = {a'+ c) y + bdy , 
 or the solution 
 
 x = {a -^ b) X -\- cdx, 
 
 y = {a'x + c x) y + {bx + dx) y . 
 
 If we replace the terms x, y, in the second members by 
 indeterminates u, v, one of the unknowns will depend on only 
 one indeterminate, while the other will depend on two. We 
 shall have a symmetrical solution by combining the two formulas, 
 
 x= {a + I)) u + cdu , 
 
 y = {a + c) V + bdv , 
 
 but the two indeterminates u and v will no longer be inde- 
 pendent of each other. For if we bring these solutions in- 
 to the given equation, it becomes 
 
PROBLEM OF BOOLE. 59 
 
 abed + ab c uv + abduv + a'ci/uv + b'c dii'v = o 
 
 or since, by hypothesis, the resultant abcd= o is verified, 
 
 ab c uv + a bd'uv + a cdu v + b' c du'v = o. 
 
 This is an "equation of condition" which the indeterminates 
 u and V must verify; it can always be verified, since its 
 resultant is identically true, 
 
 ab c . a ba . a cd . b c d = aa , bb' . cc . dct =■ o, 
 
 but it is not verified by any pair of values attributed to u 
 and V. 
 
 Some general symmetrical solutions, i. e., symmetrical 
 solutions in which the unknowns are expressed in terms of 
 several independent indeterminates, can however be found. 
 This problem has been treated by Schroder', by White- 
 head ^ and by Johnson. ^ 
 
 This investigation has only a purely technical interest; for, 
 from the practical point of view, we either wish to eliminate 
 one or more unknown quantities (or even all), or else we seek 
 to solve the equation with respect to one particular unknown. 
 In the first case, we develop the first member with respect 
 to the unknowns to be eliminated and equate the product of 
 its coefficients to o. In the second case we develop with 
 respect to the unknown that is to be extricated and apply 
 the formula for the solution of the equation of one unknown 
 quantity. If it is desired to have the solution in terms of 
 some unknown quantities or in terms of the known only, the 
 other unknowns (or all the unknowns) must first be eliminated 
 before performing the solution. 
 
 41. The Problem of Boole. — According to Boole the 
 most general problem of the algebra of logic is the follow- 
 ing <: 
 
 1 Algebra der Logik, Vol. I, S 24. 
 
 2 Universal Algebra, Vol. I, SS 35—37- 
 
 ^ "Sur la th^orie des igaliles logiques", Bibl. du Cong, intern, de Phil., 
 Vol. Ill, p. 18s (Paris, 1 901). 
 
 4 Laws of Thought, Chap. IX, S 8. 
 
6o PROBLEM OF BOOLE. 
 
 Given any equation (which is assumed to be possible) 
 f{x, y, 2, . . .) = o, 
 and, on the other hand, the expression of a terna t in terms 
 of the variables contained in the preceding equation 
 
 t==<p {x,y,z,...), 
 to determine the expression of t in terms of the constants 
 contained in / and in (f. 
 
 Suppose / and qo developed with respect to the variables 
 x,y,z... and let /i, j^^./j, . . . be their constituents: 
 
 f{x,y, z,...) = A/, + BA + C/3 + . . ., 
 cp (x,y, «,...) = «/i + *A + c/>^ + .... 
 Then reduce the equation which expresses ( so that its 
 second member will be o: 
 
 (tf + i'(p = o) = \_{dpi + b p:, + c'p^ +...)' 
 
 + (a/, + bp^ + f/3 + ...)/'= o]. 
 Combining the two equations into a single equation and 
 developing it with respect to /: 
 
 \{A + <z')/x + (^ + ^)P^ + (C+ /)/3 + . . .J / 
 + \{A + d)p^ + (^ + b)p, + (C + c)p^ + . . .] /'= o. 
 
 This is the equation which gives the desired expression 
 of /. Eliminating /, we obtain the resultant 
 Ap^ + Bp^ + C/3 + . . . = o, 
 as we might expect. If, on the other hand, we wish to 
 eliminate x,y,z,... (i. e., the constituents p^, Pz> Pi- • •)) 'we 
 put the equation in the form 
 
 (^ + a/+ at)p^ + {£+ b't+ bt)p^+{C+ci-^ct)p^ + ... = o, 
 and the resultant will be 
 
 (A + a't+at) {B + b't+bt') {C + c t + ct) ...= o, 
 
 an equation that contains only the unknown quantity t and 
 the constants of the problem (the coefficients of / and of (p). 
 From this may be derived the expression of t in terms of 
 these constants. Developing the first member of this equation 
 
 {A + d){B + b){C+c)...xt+{A + a}{B+b){C+c)...-xt' = 0. 
 
METHOD OF PORETSKY. 6 1 
 
 The solution is 
 
 t={A + a) (B+i) {C+c)...+u{j'a + B'l>+ C'c + ...). 
 
 The resultant is verified by hypothesis since it is 
 
 A£C... = o, 
 
 which is the resultant of the given equation 
 
 / (^, y,z,...) = o. 
 
 We can see how this equation contributes to restrict the 
 
 variability of t. Since / was defined only by the function fp, 
 
 it was determined by the double inclusion 
 
 a,Jc...</<a + i5 + <:+ 
 
 Now that we take into account the condition /= o, t is 
 
 determined by the double inclusion 
 
 (A + a) (B + <J) {C+c)...<i< {A' a-\-^b + C'c +...).' 
 
 The inferior limit can only have increased and the superior 
 limit diminished, for 
 
 abc...<i {A + a) (£+f) (C + c) . . . 
 and 
 
 a' a + B'6+ Cfc...<^a + b-\-c 
 
 The limits do not change if^ = ^=C=...==o, that 
 is, if the equation y = o is reduced to an identity, and this 
 was evident a priori. 
 
 42. The Method of Poretsky. — The method of Boole 
 and Schroder which we have heretofore discussed is clearly 
 inspired by the example of ordinary algebra, and it is summed 
 up in two processes analogous to those of algebra, namely 
 the solution of equations with reference to unknown quantities 
 and elimination of the unknowns. Of these processes the 
 second is much the more important from a logical point of 
 view, and Boole was even on the point of considering de- 
 duction as essentially consisting in the dimination of middle 
 
 I "Whitehead, Universal Algebra, p. 63. 
 
62 LAW OF FORMS. 
 
 terms. This notion, which is too restricted, was suggested 
 by the example of the syllogism, in which the conclusion 
 results from the elimination of the middle term, and which 
 for a long time was wrongly considered as the only type 
 of mediate deduction.' 
 
 However this may be, Boole and Schroder have exag- 
 gerated the analogy between the algebra of logic and ordi- 
 nary algebra. In logic, the distinction of known and unknown 
 terms is artificial and almost useless. All the terms are — in 
 principle at least — known, and it is simply a question, certain 
 relations between them being given, of deducing new 
 relations (unknown or not explicitly known) from these known 
 relations. This is the purpose of PorEtsky's method which 
 we shall now expound. It may be summed up in three 
 laws, the law of forms, the law of consequences and the 
 law of causes. 
 
 43. The Law of Forms. — This law answers the following 
 problem: An equality being given, to find for any term 
 (simple or complex) a determination equivalent to this equal- 
 ity. In other words, the question is to find all the forms 
 equivalent to this equality, any term at all being given as 
 its first member. 
 
 We know that any equality can be reduced to a form in 
 which the second member is o or i; i. e., to one of the 
 two equivalent forms 
 
 N^o, iV^= I. 
 
 The function N is what Poretsky calls the logical zero 
 of the given equality; N" is its logical 7t)hole.^ 
 
 1 In fact, the fundamental formula of elimination 
 
 {ax -\- bx'^ o) < (3* = o) 
 is, as we have seen, only another form and a consequence of the prin- 
 ciple of the syllogism 
 
 {b<ix<dX{b<ia). 
 
 2 They are called "logical" to distinguish them from the identical 
 zero and whole, i. e., to indicate that these two terms are not equal to o 
 and I respectively except by virtue of the data of the problem. 
 
LAW OF CONSEQUENCES. 6$ 
 
 Let U be any term; then the determination of U: 
 
 U== N' U ^ NV' 
 
 is equivalent to the proposed equality; for we know it is 
 equivalent to the equality 
 
 {NU + Nlf = o) = {N= o). 
 
 Let us recall the signification of the determination 
 
 U= 1/ U + NU'. 
 
 It denotes that the term U is contained in N" and con- 
 tains N. This is easily understood, since, by hypothesis, 
 N is equal to o and N" to i. Therefore we can formulate 
 the law of forms in the following way: 
 
 To obtain all the forms equivalent to a given equality, it 
 is sufficient to express that any term contains the logical zero 
 of this equality and is contained in its logical whole. 
 
 The number of forms of a given equality is unlimited; for 
 any term gives rise to a form, and to a form different from 
 the others, since it has a different first member. But if we 
 are limited to the universe of discourse determined by n 
 simple terms, the number of forms becomes finite and de- 
 terminate. For, in this limited universe, there are 2« con- 
 stituents. Now, all the terms in this universe that can be 
 conceived and defined are sums of some of these con- 
 stituents. Their number is, therefore, equal to the number 
 of combinations that can be made with 2« constituents, 
 namely 2^* (including o, the combination of o constituent, 
 and I, the combination of all the constituents). This will 
 also be the number of different forms of any equality in the 
 universe in question. 
 
 44. The La^v of Consequences. — We shall now pass to 
 the law of consequences. Generalizing the conception of 
 Boole, who made deduction consist in the elimination of 
 middle terms, Poretsky makes it consist in the elimination 
 of known terms (connaissances). This conception is explained 
 and justified as follows. 
 
64 LAW OF CONSEQUENCES. 
 
 All problems in which the data are expressed by logical 
 equalities or inclusions can be reduced to a single logical 
 equality by means of the formula ' 
 
 iA = o) (B= o) (C^o)... = iA + £+ C.^o). 
 
 In this logical equality, which sums up all the data of the 
 problem, we develop the first member with respect to all 
 the simple terms which appear in it (and not with respect 
 to the unknown quantities). Let « be the number of simple 
 terms; then the number of the constituents of the develop- 
 ment of I is 2". Let m (< 2") be the number of those 
 constituents appearing in the first member of the equality. 
 All possible consequences of this equality (in the universe 
 of the n terms in question) may be obtained by forming all 
 the additive combinations of these m constituents, and equat- 
 ing them to o; and this is done in virtue of the formula 
 
 (A + £=o)<(A = o). 
 
 We see that we pass from the equality to any one of its 
 consequences by suppressing some of the constituents in its 
 first member, which correspond to as many elementary equal- 
 ities (having o for second member), i. e., as many as there are 
 data in the problem. This is what is meant by "eliminating 
 the known terms". 
 
 The number of consequences that can be derived from 
 an equality (in the universe of n terms with respect to which 
 it is developed) is equal to the number of additive com- 
 binations that may be formed with its m constituents; i. e., 
 2'". This number includes the combination of o constituents, 
 which gives rise to the identity 0^0, and the combination 
 of the m constituents, which reproduces the given equality. 
 
 Let us apply this method to the equation with one un- 
 known quantity 
 
 ax + 6x = o. 
 
 I We employ capitals to denote complex terms (logical functions) in 
 contrast to simple terms denoted by small letters (a, i, <-, . . .) 
 
LAW OF CONSEQUENCES. 65 
 
 Developing it with respect to the three terms a, b, x: 
 {abx + ab X -{- abx +a bx = o) 
 ^[ab{x-\-x)-\-abx-\-abx ^o] 
 = {ab = o) (ab x = o) {a bx' = o). 
 
 Thus we find, on the one hand, the resultant ab = o, 
 and, on the other hand, two equalities which may be trans- 
 formed into the inclusions 
 
 X <^ a + d, a b <^x. 
 
 But by the resultant which is equivalent to b <ia , we have 
 
 a + b = a , a b = b. 
 
 This consequence may therefore be reduced to the double 
 inclusion 
 
 x<^ a , b <ix, 
 
 that is, to the known solution. 
 
 Let us apply the same method to the premises of the 
 syllogism 
 
 {a<b) {b<:c). 
 
 Reduce them to a single equality 
 
 (a < ^) = {a^ = o), {b<ic) = {bc'= o), {all + be = o), 
 
 and seek all of its consequences. 
 
 Developing with respect to the three terms a, b, e: 
 
 abc + ab c -r ab c + a be = o. 
 
 The consequences of this equality, which contains four 
 constituents, are i6 (2'*) in number as follows: 
 
 1. {abc' = 0) = {ab<^c); 
 
 2. {ab c = o) = {ac <^ b); 
 
 3. {ab c ^ o) ^ {a<^b + c); 
 
 4. {a'bc =^ o) = {b<Z.a + c); 
 
 5. {abc' + ab' c = o) = {a <^ be + b c ) ; 
 
 6. {abc + ab'e'= o) = {ae = o) = (<z <[ c). 
 
66 LAW OF CONSEQUENCES. 
 
 This is the traditional conclusion of the syllogism.* 
 
 7. (ait:' + a be == o) = (be = o) = (b<ie). 
 This is the second premise. 
 
 8. (ab' c-\- ab'c = 6) = (ab' = o) = (a<^b). 
 
 This is the first premise. 
 
 9. (ab'c + a be = o) = (ae <ib<ia + c); 
 
 10. (ab'c + a'l^/ = o) = (ab' + a'^ < f); 
 
 11. (abc' + ab c-\- ab'e' ^0)= (ab + ac = o)'=(a<ibc); 
 
 12. (a^c -\- ab c + a be = o) = (a^ ^ + ^<r =0) 
 
 = (ac<b<c); 
 
 13. (abc' + ab'c -\- a be = o) = (a^ + 3^ = o) 
 
 = (a + ^<f); 
 
 14. (ab e -\- ab c + a be =^ 6) = (ab + a be = o) 
 
 = (a<ib<ia + c). 
 
 The last two consequences (15 and 16) are those ob- 
 tained by combining o constituent and by combining all; the 
 first is the identity 
 
 15. = 0, 
 
 which confirms the paradoxical proposition that the true 
 (identity) is implied by any proposition (is a consequence 
 of it); the second is the given equality itself 
 
 16. ab + be = o, 
 
 which is, in fact, its own consequence by virtue of the 
 principle of identity. These two consequences may be called 
 the "extreme consequences" of the proposed equality. If 
 we wish to exclude them, we must say that the number of 
 the consequences properly so called of an equality of m 
 constituents is 2"' — 2. 
 
 I It will be observed that this is the only consequence (except the 
 two extreme consequences [see the text below]) independent of i; there- 
 fore it is the resultant of the elimination of that middle term. 
 
LAW OF CAUSES. 6/ 
 
 45. The Law of Causes. — The method of finding the 
 consequences of a given equality suggests directly the method 
 of finding its causes, namely, the propositions of which it is 
 the consequence. Since we pass from the cause to the 
 consequence by eliminating known terms, i. <?., by suppressing 
 constituents, we will pass conversely from the consequence 
 to the cause by adjoining known terms, i. e., by adding con- 
 stituents to the given equality. Now, the number of con- 
 stituents that may be added to it, /. e., that do not already 
 appear in it, is 2" — m. We will obtain all the possible 
 causes (in the universe of the n terms under consideration) 
 by forming all the additive combinations of these constituents, 
 and adding them to the first member of the equality in virtue 
 of the general formula 
 
 (^ + ^ = o)< (^ = o), 
 
 which means that the equality {A = 0) has as its cause the 
 equality [A -^ J3 = o), in which B is any term. The number 
 of causes thus obtained will be equal to the number of the 
 aforesaid combinations, or 2'"-m. 
 
 This method may be applied to the investigation of the 
 causes of the premises of the syllogism 
 
 (a<i>) {b<c), 
 
 which, as we have seen, is equivalent to the developed 
 equality 
 
 abc -^ ab c + ab c + a be =0. 
 
 This equality contains four of the eight (23) constituents 
 of the universe of three terms, the four others being 
 
 abc, abc, abc, abc. 
 
 The number of their combinations is 16 (2*), this is also 
 the number of the causes sought, which are: 
 
 1. {abc ■\- abc -\r ab c ■\- ah c -^ abc ^ 6) 
 
 = (a + be = o) = (a = o) {b<^c); 
 
 2. (abc -V ab c Ar ab c ■\- abc -V ab c= o) 
 
 = {abc -^r ab' + a b = o) = {ab<^c) {a = b); 
 
 S* 
 
68 LAW OF CAUSES. 
 
 3. {abc + ab c + a b c + abc-\-abc = o) 
 
 = {be + b'c + ab'c' =o) = (b = e) (a<b + c); 
 
 4. (abc+ab'c + ab'c+abc+abc^o) 
 
 = (/ + a^=o) = (f = i) (a<^); 
 
 5. {abc + abc -\- aic ^ ab c + abc -\- abc =0) 
 
 = {a + b = o) = {a = o) {b = o); 
 
 6. (abc + abc + ab'c + a1) c + a b c -V a' b' c = 0) 
 
 = (a + ^/ + b'c = o) = (3 == o) ((J = ^); 
 
 7. (ai^f + a^/ +a^f + a^i: -\- a be + a b c' = o) 
 
 = (a + / = o) = (a = o) (f = i) I; 
 
 8. {abc -\-abc-i-abc'+abc-\-abc'+abc^o) 
 
 = {ac -\-ac-\-abc-\-abc = o) 
 
 = {a = c) {ac <^ b <Z. a -{■ c) = {a ^ b = c); 
 
 9. {abc + a b c -\- a b c -\- a b c -\- a b c + a' b c ^ o) 
 
 == {c + ab + a b = o) '= {c = i) {a = b); 
 
 10. {abc + abc + abc +abc+abc + abc = o) 
 
 = (^ + / = o) = (^ = <; = i). 
 
 Before going any further, it may be observed that when 
 the sum of certain constituents is equal to o, the sum of 
 the rest is equal to i. Consequently, instead of examining 
 the sum of seven constituents obtained by ignoring one of 
 the four missing constituents, we can examine the equalities 
 obtained by equating each of these constituents to i : 
 
 1 1. {a b'c' = 1) = {a + b + c = o) = {a = b = c = o); 
 
 1 2. {a b'c = 1) ={a + b + c = o) = {a = b = o) {c= 1); 
 
 13. {abc = i) = (3 + b'+ /= o) = (a = o) (3 = <r = i); 
 
 14. {abc = 1} ^{a = b = c=\). 
 
 I It will be observed that this cause is the only one which is inde- 
 pendent of b; and indeed, in this case, whatever b is, it will always 
 contain a and will always be contained in c. Compare Cause 5. which 
 is independent of c, and Cause 10, which is independent of a. 
 
FORMS OF CONSEQUENCES AND CAUSES. 69 
 
 Note that the last four causes are based on the inclusion 
 
 0<I. 
 
 The last two causes (15. and 16.) are obtained either by 
 adding all the missing constituents or by not adding any. 
 In the first case, the sum of all the constituents being equal 
 to I, we find 
 
 IS- I = o, 
 
 that is, absurdity, and this confirms the paradoxical prop- 
 osition that the false (the absurd) implies any proposition 
 (is its cause). In the second case, we obtain simply the 
 given equality, which thus appears as one of its own causes 
 (by the principle of identity): 
 16. ab' + be' = o. 
 
 If we disregard these two extreme causes, the number of 
 causes properly so called will be 
 
 2^"-"' — 2. 
 
 46. Forms of Consequences and Causes. — We can 
 
 apply the law of forms to the consequences and causes of a 
 given equality so as to obtain all the forms possible to each 
 of them. Since any equality is equivalent to one of the two forms 
 
 7V^= (9, IT = 1, 
 each of its consequences has the form' 
 
 jy^= o, or JV + X' -=- 1, 
 and each of its causes has the form 
 
 N+ X=o, or N'X'= I. 
 
 ' In S 44 wc said that a consequence is obtained by taking a part 
 of the constituents of the first member N, and not by multiplying it by 
 a term X; but it is easily seen that this amounts to the same thing. * 
 For, suppose that X (like N) be developed with respect to the n terms 
 of discourse. It will be composed of a certain number of constituents. 
 To perform the multiplication of N by X, it is sufficient to multiply 
 all their constituents each by each. Now, the product of two identical 
 constituents is equal to each of them, and the product of two different 
 constituents is o. Hence the product of A' by A" becomes reduced to 
 the sum of the constituents common to N and X, which is, of course, 
 contained in AT. So, to multiply N by an arbitrary term is tantamount 
 to taking a part of its. constituents (or all, or none). 
 
•JO Venn's problem. 
 
 In fact, we have the following formal implications: 
 
 {jsTx: = i)< (i/ == i) = (iv' + ^' = I). 
 
 Applying the law of forms, the formula of the conse- 
 quences becomes 
 
 U= (iV^'+ X') U+ NX If, 
 
 and the formula of the causes 
 
 U=N'X' U \ {N+ X) U'; 
 
 or, more generally, since X and X' are indeterminate terms, 
 and consequently are not necessarily the negatives of each 
 other, the formula of the consequences will be 
 
 U={N' + X) U + NYU', 
 
 and the formula of the causes 
 
 U= N'XU-\- {N^ Y)U'. 
 
 The first denotes that U is contained in {N' + X) and 
 contains N Y; which indeed results, a fortiori, from the hypoth- 
 esis that U is contained in N' and contains N. 
 
 The second formula denotes that U is contained in N' X 
 and contains N' + Y whence results, a fortiori, that V is 
 contained in N and contains N. 
 
 We can express this rule verbally if we agree to call 
 every class contained in another a sub -class, and every 
 class that contains another a super-class. We then say: 
 To obtain all the consequences of an equality (put in the 
 form U = N' U -'r N U' ), it is sufficient to substitute for its 
 logical whole N" all its super-classes, and, for its logical 
 zero N, all its sub-classes. Conversely, to obtain all the 
 causes of the same equality, it is sufficient to substitute for 
 its logical whole all its sub- classes, and for its logical zero, 
 all its super-classes. 
 
 47. Example: Venn's Problem. — The inembers of the 
 administrative council of a financial society are either bond- 
 holders or shareholders, but not both. Now, all the bond- 
 
Venn's problem. 71 
 
 holders form a part of the council. What conclusion must 
 we draw ? 
 
 Let a be the class of the members of the council; let ^ 
 be the class of the bondholders and c that of the share- 
 holders. The data of the problem may be expressed as 
 follows : 
 
 a<C6c + ^c, b<^a. 
 
 Reducing to a single developed equality, 
 
 a{bc = bc) = o, a b = o, 
 (i) abc + ab' c + dbc-^a be =0. 
 
 This equality, which contains 4 of the constituents, is 
 equivalent to the following, which contains the four others, 
 
 (2) abc + ab' c -\- a' b' c + a' b' c = i. 
 
 This equality may be expressed in as many different forms 
 as there are classes in the universe of the three terms 
 a, b, c. 
 
 Ex. I. a=^abc'-\-abc-{-abc + abc, 
 
 that is, 
 
 b<ia<^bc -\- b'c, 
 
 Ex. 2. b = abc + ab'c = ac ; 
 
 Ex. 3. c^ab'c-{-abc-^abc+abc 
 
 that is, 
 
 ab' + a' b <^c<C^ . 
 
 These are the solutions obtained by solving equation (i) 
 with respect to a, b, and c. 
 
 From equality (i) we can derive 16 consequences as 
 follows: 
 
 1. abc = o; 
 
 2. {ab' c = o) = {a<ib + e); 
 
 3. {a' be = o) = (be <; a); 
 
 4. {ab c = o) = {b<ia + c); 
 
72 VENN S PROBLEM. 
 
 5. {abc + ai c = o) = (a <Z be + I) c) [ist premise]; 
 
 6, 
 
 7 
 8 
 
 9 
 
 10, 
 II 
 12 
 13 
 14 
 
 (abc -\- a be = o) = (be ^ o); 
 
 {abc + a'/5/ = o) = (iJ ■< a/ +<i'^); 
 
 {ab'c + a bc= o) = (be <C a <^b + e); 
 
 (ab'e + a be = o) = (a^' + a'<^ ■< e) ; 
 
 (a be + dbc= o) = (a'^ = o) [2^ premise]; 
 {abe+ ab'c + cibc = o) = {bc + alJc = o); 
 
 ai5c+ a;J f + a ^<r = o; 
 (abc + a'^iT + abc ^ o) = (^(T + a'/j/) = o; 
 
 ab e -\- a be + a be = o. 
 
 The last two consequences, as we know, are the identity 
 (o = o) and the equality (i) itself. Among the preceding 
 consequences will be especially noted the 6^ (be = o), the 
 resultant of the elimination of a, and the iotl> (a ^ = o), 
 the resultant of the elimination of c. When b is eliminated 
 the resultant is the identity 
 
 [(a + c) ae ^ oj = (o = o). 
 
 Finally, we can deduce from the equality (i) or its equiv- 
 alent (2) the following 16 causes: 
 
 I 
 2 
 3 
 4 
 
 5 
 6 
 
 7 
 8 
 
 9 
 10 
 
 (abc' = i) = (a = ,) (^ = I) (e = o); 
 (ab'e = i) = (a = i) (^ = o) (c = i); 
 (a'b'e = i) = (a = o) (^ = o) (^ = l); 
 (ab'c = i) = (a = o) (^= o) (c = o); 
 (abc + ab'c = I) = (a= i) (b'=c); 
 (abc + ab' c = i) = (a = (5 = /); 
 {abc + at) c = i) = (^ = o) (a = ^); 
 {ab'c+ ab'c = i) = (b = 6) (c = i); 
 (a^c+ d i c ^ i) = (^ = o) (a = c); 
 {ad c^ d d c = \) = (a = o) (1^ = o); 
 
GEOMETRICAL DIAGRAMS OF VENN. 73 
 
 11. {abc + ab'c-\- a b' c = i) = (b = c) {c <ia); 
 
 12. {abc -^ ab'c -\-a'b'c' ^ i) = {be = o) (a = b + c); 
 
 13. {abc + a'b'c+ a b' c = i) = (ac = o) {a = b); 
 
 14. {ab'c-\-a'b'c+a'b'c' = 1) = (b = o) (a < <:). 
 
 The last two causes, as we know, are the equality (i) 
 itself and the absurdity (i = o). It is evident that the 
 cause independent of a is the S^^ (b ^ o) (c = i), and the 
 cause independent of c is the lo* (a = o) (b ^ o). There 
 is no cause, properly speaking, independent of b. The most 
 "natural" cause, the one which may be at once divined 
 simply by the exercise of common sense, is the i2'ii: 
 
 {be -= o) {a = b + c). 
 
 But other causes are just as possible; for instance the 9* 
 (b =0) (a = <r), the 7th {c =0) {a = b), or the 13A 
 {ac = 0) {a ^ b). 
 
 We see that this method furnishes the complete enumera- 
 tion of all possible cases. In particular, it comprises, among 
 the forms of an equality, the solutions deducible therefrom 
 with respect to such and such an "unknown quantity", and, 
 among the consequences of an equality, the resultants of the 
 elimination of such and such a term. 
 
 48. The Geometrical Diagrams of Venn. — Poretsky's 
 method may be looked upon as the perfection of the methods 
 of Stanley Jevons and Venn. 
 
 Conversely, it finds in them a geometrical and mechanical 
 illustration, for Venn's method is translated in geometrical 
 diagrams which represent all the constituents, so that, in 
 order to obtain the result, we need only strike out (by 
 shading) those which are made to vanish by the data of the 
 problem. For instance, the universe of three terms a, b, c, 
 represented by the unbounded plane, is divided by three 
 simple closed contours into eight regions which represent the 
 eight constituents (Fig. i). 
 
GEOMETRICAL DIAGRAMS OF VENN. 
 
 cb'b'c' 
 
 Fig. I. 
 
 To represent geometrically the data of Venn's problem we 
 must strike out the regions abc, ai c , a be and a be; there 
 will then remain the regions abc , abc, abc, and abc 
 which will constitute the universe relative to the problem, 
 being what Poretsky calls his logical whole (Fig. 2). Then 
 
 Fig. 
 
 every class will be contained in this universe, which will give 
 for each class the expression resulting from the data of the 
 problem. Thus, simply by inspecting the diagram, we see 
 that the region be does not exist (being struck out); that the 
 region b is reduced to abc (hence to ab); that all a is ^ 
 or c, and so on. 
 
LOGICAL MACHINE OF JEVONS. 75 
 
 This diagrammatic method has, however, serious incon- 
 veniences as a method for solving logical problems. It does 
 not show how the data are exhibited by canceling certain 
 constituents, nor does it show how to combine the remaining 
 constituents so as to obtain the consequences sought. In 
 short, it serves only to exhibit one single step in the argument, 
 namely the equation of the problem; it dispenses neither with 
 the previous steps, i. e., "throwing of the problem into an 
 equation" and the transformation of the premises, nor with 
 the subsequent steps, :. e., the combinations that lead to 
 the various consequences. Hence it is of very little use, 
 inasmuch as the constituents can be represented by algebraic 
 symbols quite as well as by plane regions, and are much 
 easier to deal with in this form. 
 
 49. The Logical Machine of Jevons. — In order to 
 make his diagrams more tractable, Venn proposed a me- 
 chanical device by which the plane regions to be struck out 
 could be lowered and caused to disappear. But Jevons 
 invented a more complete mechanism, a sort of logical piano. 
 The keyboard of this instrument was composed of keys in- 
 dicating the various simple terms {a, b, c, d), their negatives, 
 and the signs -h and ^. Another part of the instrument 
 consisted of a panel with movable tablets on which were 
 written all the combinations of simple terms and their neg- 
 atives; that is, all the constituents of the universe of dis- 
 course. Instead of writing out the equalities which represent 
 the premises, they are "played" on a keyboard like that of 
 a typewriter. The result is that the constituents which vanish 
 because of the premises disappear from the panel. When 
 all the premises have been "played", the panel shows only 
 those constituents whose sum is equal to i, that is, forms 
 the universe with respect to the problem, its logical whole. 
 This mechanical method has the advantage over Venn's geo- 
 metrical method of performing automatically the "'throwing 
 into an equation", although the premises must first be ex- 
 pressed in the form of equalities; but it throws no more light 
 than the geometrical method on the operations to be per- 
 
76 TABLE OF CONSEQUENCES. 
 
 formed in order to draw the consequences from the data 
 displayed on the panel. 
 
 50. Table of Consequences.— But Poretsky's method 
 can be illustrated, better than by geometrical and mechanical 
 devices, by the construction of a table which will exhibit 
 directly all the consequences and all the causes of a given 
 equality. (This table is relative to this equality and each 
 equality requires a different table). Each table comprises 
 the 2« classes that can be defined and distinguished in the 
 universe of discourse of n terms. We know that an equality 
 consists in the annulment of a certain number of these 
 classes, viz., of those which have for constituents some of 
 the constituents of its logical zero N. Let m be the number 
 of these latter constituents, then the number of the sub- 
 classes ofiV is 2"' which, therefore, is the number of classes 
 of the universe which vanish in consequence of the equality 
 considered. Arrange them in a column commencing with 
 o and ending with N (the two extremes). On the other 
 hand, given any class at all, any preceding class may be 
 added to it without altering its value, since by hypothesis 
 they are null (in the problem under consideration). Conse- 
 quently, by the data of the problem, each class is equal to 
 2™ classes (including itself). Thus, the assemblage of the 
 2« classes of discourse is divided into 2"-'" series of 2'" 
 classes, each series being constituted by the sums of a certain 
 class and of the 2'« classes of the first column (sub-classes 
 of N). Hence we can arrange these 2'" sums in the 
 following columns by making them correspond horizontally 
 to the classes of the first column which gave rise to them. 
 Let us take, for instance, the very simple equality a = b, 
 which is equivalent to 
 
 ab + a b = o. 
 
 The logical zero (JV) in this case is ab' -\- a b. It com- 
 prises two constituents and consequently four sub-classes: 
 o, a^ , a'b, and ab' + a'b. These will compose the first 
 column. The other classes of discourse are ab, a b' , ab + a b , 
 
TABLE OF CAUSES. TJ 
 
 and those obtained by adding to each of them the four 
 classes of the first column. In this way, the following table 
 is obtained: 
 
 o ab a b ab -V a b 
 
 ab a b a + b 
 
 a b b a a + b 
 
 ab + a b a + b a + b i 
 
 By construction, each class of this table is the sum of 
 those at the head of its row and of its column, and, by the 
 data of the problem, it is equal to each of those in the 
 same column. Thus we have 64 different consequences for 
 any equality in the universe of discourse of 2 letters. They 
 comprise 16 identities (obtained by equating each class to 
 itself) and 16 forms of the given equality, obtained by 
 equating the classes which correspond in each row to the 
 classes which are known to be equal to them, namely 
 
 o ^ ab + a b, ab = a + b, a b = a + b , ab + a b ^ 1 
 
 a ^= b, b = a , ab = a b, a + b ^ a + b. 
 
 Each of these 8 equahties counts for two, according as it 
 is considered as a determination of one or the other of its 
 members. 
 
 51. Table of Causes. — The same table may serve to 
 represent all the causes of the same equality in accordance 
 with the following theorem: 
 
 When the consequences of an equality N = o are ex- 
 pressed in the form of determinations of any class U, the 
 causes of this equality are deduced from the consequences 
 of the opposite equality, N —- 1 , put in the same form, 
 by changing i7 to 6'' in one of the two members. 
 
 For we know that the consequences of the equality j?V = o 
 have the form 
 
 U= {JV + X) U + NYU', 
 and that the causes of the same equality have the form 
 U = I/X U-^ {N+ Y) U'. 
 
78 TABLE OF CAUSES. 
 
 Now, if we change U into U' in one of the members o 
 this last formula, it becomes 
 
 and the accents of X and Y can be suppressed since these 
 letters represent indeterminate classes. But then we have 
 the formula of the consequences of the equality .A^^ = o or 
 N= I. 
 
 This theorem being established, let us construct, for in- 
 stance, the table of causes of the equality a = b. This will 
 be the table of the consequences of the opposite equality 
 a = i, for the first is equivalent to 
 
 ab' + a b = o, 
 and the second to 
 
 {ab + ab' = o) == (a^ + a'b = i). 
 o ab a b ab ■\- a b 
 
 ab a b a + b 
 
 ' 1' i' ' '11' 
 
 a b b a a + b 
 
 ab + ab' a + b' a' + b i 
 
 To derive the causes of the equality a = b from this table 
 instead of the consequences of the opposite equality a = b , 
 it is sufficient to equate the negative of each class to each 
 of the classes in the same column. Examples are: 
 
 a ^ ^ = o, a + b = a b , a+b = ab + ab, 
 
 a' + b = a, a' + b = b', a' + b — a + b' ; . . . . 
 
 Among the 64 causes of the equality under consideration 
 there are 16 absurdities (consisting in equating each class of 
 the table to its negative); and 16 forms of the equality (the 
 same, of course, as in the table of consequences, for two 
 equivalent equalities are at the same time both cause and 
 consequence of each other). 
 
 It will be noted that the table of causes differs from the 
 table of consequences only in the fact that it is sym- 
 metrical to the other table with respect to the principal diagonal 
 
NUMBER OF POSSIBLE ASSERTIONS. 79 
 
 (o, i); hence they can be made identical by substituting the 
 word "row" for the word "column" in the foregoing state- 
 ment And, indeed, since the rule of the consequences con- 
 cerns only classes of the same column, we are at liberty so to 
 arrange the classes in each column on the rows that the 
 rule of the causes will be verified by the classes in the 
 same row. 
 
 It will be noted, moreover, that, by the method of con- 
 struction adopted for this table, the classes which are the 
 negatives of each other occupy positions symmetrical with 
 respect to the center of the table. For this result, the sub- 
 classes of the class N" (the logical whole of the given 
 equality or the logical zero of the opposite equality) must 
 be placed in the first row in their natural order from o to N'; 
 then, in each division, must be placed the sum of the classes 
 at the head of its row and column. 
 
 With this precaution, we may sum up the two rules in the 
 following practical statement: 
 
 To obtain every consequence of the given equality (to 
 which the table relates) it is sufficient to equate each class 
 to every class in the same column; and, to obtain every 
 cause, it is sufficient to equate each class to every class in 
 the row occupied by its symmetrical class. 
 
 It is clear that the table relating to the equality iV= o 
 can also serve for the opposite equality iV = i, on condition 
 that the words "row" and "column" in the foregoing statement 
 be interchanged. 
 
 Of course the construction of the table relating to a given 
 equality is useful and profitable only when we wish to 
 enumerate all the consequences or the causes of this equal- 
 ity. If we desire only one particular consequence or cause 
 relating to this or that class of the discourse, we make use 
 of one of the formulas given above. 
 
 52. The Number of Possible Assertions. — If we regard 
 logical functions and equations as developed with respect to 
 all the letters, we can calculate the number of assertions or 
 difierent problems that may be formulated about n simple 
 
8o PARTICULAR PROPOSITIONS. 
 
 terms. For all the functions thus developed can contain only 
 those constituents which have the coefficient i or the coef- 
 ficient o (and in the latter case, they do not contain them). 
 Hence they are additive combinations of these constituents; 
 and, since the number of the constituents is 2", the number 
 of possible functions is 2^". From this must be deducted 
 the function in which all constituents are absent, which is 
 identically o, leaving 2^"— i possible equations (255 when 
 71 = 3). But these equations, in their turn, may be combined 
 by logical addition, i. e., by alternation; hence the number 
 of their combinations is 22""— i— i, excepting always the 
 null combination. This is the number of possible assertions 
 affecting n terms. When n ^ 2, this number is as high as 
 32767." We must observe that only universal premises are 
 admitted in this calculus, as will be explained in the follow- 
 ing section. 
 
 53. Particular Propositions. — Hitherto we have only 
 considered propositions with an affirmative copula (i. e., in- 
 clusions or equalities) corresponding to the -universal prop- 
 ositions of classical logic' It remains for us to study prop- 
 ositions with a negative copula (non inclusions or inequalities), 
 which translate particular propositions^; but the calculus of 
 
 1 G. Peano, Calcob geometrico (1888) p. x; Schr6der, Algebra der 
 Logik, Vol. II, p. 144 — 148. 
 
 2 The universal affirmative, "All a's are b's" , may be expressed by 
 the formulas 
 
 (a < *) == (a = ab) = (ab' = o) = (a -|- * = I), 
 
 and the universal negative, "No a's are ^'s", by the formulas 
 
 [a < U)= {a = ab'] = {ab = o) = (a* +3'= I). 
 
 3 For the particular affirmative, "Some a's are ^'s", being the negation 
 of the universal negative, is expressed by the formulas 
 
 {a 4; b') = (a 4= ab-) = {ab + o) = (a'-f *'4= 1), 
 
 'articular negative, "Some a's are not i's", being 1 
 versal affirmative, is expressed by the formulas 
 
 (a <f:; *) = (a 4= ab) = (ab' ^ o) = (a' + * =f= I), 
 
 and the particular negative, "Some a's are not i's", being the negation 
 of the universal affirmative, is expressed by the formulas 
 
SOLUTION OF AN INEQUATION WITH ONE UNKNOWN. Si 
 
 propositions having a negative copula results from laws al- 
 ready known, especially from the formulas of De Morgan 
 and the law of contraposition. We shall enumerate the chief 
 formulas derived from it 
 
 The principle of composition gives rise to the following 
 formulas: 
 
 (^a + b^c) = {a<^c) + {b-^c), 
 whence come the particular instances 
 
 (a+ (J+o) = (a + o) + (<r+o). 
 
 From these may be deduced the following important im- 
 plications: 
 
 (« + o)<(« + -5 + 0), 
 
 {a^iXiab^i). 
 
 From the principle of the syllogism, we deduce, by the 
 law of transposition, 
 
 {a<b) (a + o)<(/5 + o), 
 
 {a<b) (3+ !)<(«+ I). 
 
 The formulas for transforming inclusions and equalities 
 give corresponding formulas for the transformation of non- 
 inclusions and inequalities, 
 
 (a<^) = (a^'+o) =(a'+<5+i), 
 
 (a =)= <J) = (a^' -H a'b + o) = {ab + aV+ i). 
 
 54. Solution of an Inequation with One Unknown. — 
 
 If we consider the conditional inequahty {inequation) with 
 one unknown 
 
 ax + bx =^ o, 
 
 we know that its first member is contained in the sum of 
 its coefficients 
 
 ax ■\- bx <^a + b. 
 
82 SOLUTION OF AN INEQUATION WITH ONE UNKNOWN. 
 
 From this we conclude that, if this inequation is verified, 
 we have the inequality 
 
 a + i + o. 
 
 This is the necessary condition of the solvability of the 
 inequation, and the resultant of the elimination of the un- 
 known X. For, since we have the equivalence 
 
 Y\ {ax + ix == a) = (a + 6 = o), 
 
 we have also by contraposition the equivalence 
 2 (<^x + 6x^o)^(a+6=^ o). 
 
 X 
 
 Likewise, from the equivalence 
 
 2 {ax + bx ^ o) ^ (a^ = o), 
 
 we can deduce the equivalence 
 
 Y\ {ax + bx =)= o) = {ab =t= o), 
 
 which signifies that the necessary and sufficient condition for 
 the inequation to be always true is 
 
 {ab^o); 
 and, indeed, we know that in this case the equation 
 
 {ax + bx = o) 
 is impossible (never true). 
 
 Since, moreover, we have the equivalence 
 
 {ax + bx = o) = (a- = d X + bx), 
 we have also the equivalence 
 
 {ax Ar bx ^ 6) = {x ^ a X ^ bx). 
 Notice the significance of this solution: 
 {ax+ bx=^o) = {ax^o) + {bx^o) = {x^a) + (*<*). 
 
 "Either x is not contained in a , or it does not contain b". 
 This is the negative of the double inclusion 
 
 b <ix <ia . 
 
EQUATION AND AN INEQUATION. 83 
 
 Just as the product of several equalities is reduced to one 
 single equality, the sum (the alternative) of several inequalities 
 may be reduced to a single inequality. But neither several 
 alternative equalities nor several simultaneous inequalities can 
 be reduced to one. 
 
 55. System of an Equation and an Inequation. — We 
 
 shall limit our study to the case of a simultaneous equality 
 aad inequality. For instance, let the two premises be 
 
 (ax + bx = o) (ex + (/a: 4= o)- 
 
 To satisfy the former (the equation) its resultant ab = o 
 must be verified. The solution of this equation is 
 
 X = a X ■\- bx . 
 
 Substituting this expression (which is equivalent to the 
 equation) in the inequation, the latter becomes 
 
 (a c ■\- ad)x + {be + b d)x =j= o. 
 
 Its resultant (the condition of its solvability) is 
 
 {dc + ad + be + b'd+o) =-- [{a + b) e + {a + b')d+o], 
 
 which, taking into account the resultant of the equality, 
 
 {ab = o) = («' + (5 = a') = (a + b' = B' ) 
 
 may be reduced to 
 
 ae + b' d=^ o. 
 
 The same result may be reached by observing that the 
 equality is equivalent to the two inclusions 
 
 {x<a) {x'<b'), 
 
 and by multiplying both members of each by the same term 
 
 {ex<. dc) {dx < b'd) <i(ex + dx < a'e + b'd) 
 
 (ex + dx 4= oX i'^'^ + ^'^+ °)- 
 
 This resultant implies the resultant of the inequality taken 
 
 alone 
 
 e + d+o, 
 
 so that we do not need to take the latter into account. It 
 
 6* 
 
84 CALCULUS OF PROPOSITIONS. 
 
 is therefore sufficient to add to it the resultant of the equality 
 to have the complete resultant of the proposed system 
 
 {ab = o) {a c + ^^4= o). 
 
 The solution of the transformed inequality (which conse- 
 quently involves the solution of the equality) is 
 
 x=^{dc' ■\- act)x + {be + b'd)x'. 
 
 56. Formulas Peculiar to the Calculus of Propositions, 
 
 — All the formulas which we have hitherto noted are valid 
 alike for propositions and for concepts. We shall now 
 establish a series of formulas which are valid only for prop- 
 ositions, because all of them are derived from an axiom 
 peculiar to the calculus of propositions, which may be called 
 the principle of assertion. 
 
 This axiom is as follows: 
 (Ax. X.) (<z == i) = a. 
 
 P. I.: To say that a proposition a is true is to state the 
 proposition itself. In other words, to state a proposition is 
 to affirm the truth of that proposition."^ 
 
 Corollary: 
 
 a = {a = \) ^ {a = o). 
 
 P. I.: The negative of a proposition a is equivalent to the 
 affirmation that this proposition is false. 
 
 By Ax. IX (§ 20), we already have 
 
 (<7 ^ i) {a = o) = o, 
 
 "A proposition cannot be both true and false at the same 
 time", for 
 
 (Syll.) {a = I) (a = o)< (I = o) = o. 
 
 I We can see at once that this fonnnla is not susceptible of a con- 
 ceptual interpretation (C. I.) ; for, if a is a concept, (a ^ I) is a prop- 
 osition, and we would then have a logical equality (identity) between 
 a concept and h proposition, which is absurd. 
 
IMPLICATION AND ALTERNATIVE. 8$ 
 
 But now, according to Ax. X, we have 
 
 (a = i) + (fl = p) = a + a = i. 
 
 "A proposition is either true or false". From these two 
 formulas combined we deduce directly that the propositions 
 (a = i) and (a = o) are contradictory, i. e., 
 
 (a 4= i) = (a = o), (a + o) = (a = i). 
 
 From the point of view of calculation Ax. X makes it 
 possible to reduce to its first member every equality whose 
 second member is i, and to transform inequalities into 
 equalities. Of course these equalities and inequalities must 
 have propositions as their members. Nevertheless all the 
 formulas of this section are also valid for classes in the 
 particular case where the universe of discourse contains only 
 one element, for then there are no classes but o and i. In 
 short, the special calculus of propositions is equivalent to the 
 calculus of classes when the classes can possess only the 
 two values o and i. 
 
 57. Equivalence of an Implication and an Alternative. 
 
 — The fundamental equivalence 
 
 (a < ^) = (a' + ^ = i) 
 gives rise, by Ax. X, to the equivalence 
 
 which is no less fundamental in the calculus of propositions. 
 To say that a implies d is the same as affirming "not-a or 
 d", i. e., "either a is false or b is true." This equivalence 
 is often employed in every day conversation. 
 
 Corollary. — For any equality, we have the equivalence 
 
 {a = b) = ab + a b' . 
 
 Demonstration: 
 
 (a = b)='(a< b) {b<a)= (a + b) {b' + a)^ab + ab'. 
 
 "To affirm that two propositions are equal (equivalent) 
 is the same as stating that either both are true or both are 
 false". 
 
86 IMPLICATION AND ALTERNATIVE. 
 
 The fundamental equivalence established above has im- 
 portant consequences which we shall enumerate. 
 
 In the first place, it makes it possible to reduce secondary, 
 tertiary, etc., propositions to primary propositions, or even 
 to sums (alternatives) of elementary propositions. For it 
 makes it possible to suppress the copula of any proposition, 
 and consequently to lower its order of complexity. An im- 
 plication (A ■< B), in which A and £ represent propositions 
 more or less complex, is reduced to the sum A^ + B, in 
 which only copulas within A and B appear, that is, prop- 
 ositions of an inferior order. Likewise an equality (A = B) 
 is reduced to the sum (AB + A'B') which is of a lower 
 order. 
 
 We know that the principle of composition makes it 
 possible to combine several simultaneous inclusions or equal- 
 ities, but we cannot combine alternative inclusions or equal- 
 ities, or at least the result is not equivalent to their alter- 
 native but is only a consequence of it. In short, we have 
 only the implications 
 
 (<z<r) ^{b<^cX{ab<c\ 
 {c<a)^ {c<bX{c<a^b), 
 
 which, in the special cases where ^ = o and c = \, become 
 (a= 6) + {b = o)<i {ah = o), 
 (a= i) -i- (3= i)<(a + ^= i). 
 
 In the calculus of classes, the converse implications are 
 not valid, for, from the statement that the class ab \s null, 
 we cannot conclude that one of the classes a or ^ is null 
 (they can be not-null and still not have any element in 
 common); .and from the statement that the sum (a -I- b) is 
 equal to i we cannot conclude that either a or ^ is equal 
 to I (these classes can together comprise all the elements of 
 the universe without any of them alone comprising all). But 
 these converse implications are true in the calculus of prop- 
 ositions 
 
 {ab<cX{,a<,c)+ {b<c), 
 {c<a + b)<{c<a) + {c<b); 
 
IMPLICATION AND ALTERNATIVE. 87 
 
 for they are deduced from the equivalence established above, 
 or rather we may deduce from it the corresponding equal- 
 ities which imply them, 
 
 (i) {.ab < f) = \a < ^) + (^ < c), 
 
 (2) (^<fl + ^) = (<:<a) + (<r<^). 
 
 Demonstration : 
 
 (i) (a3<<r) = a'+ 3'+ ^, 
 
 (a < ^) + (^< f) = (a + <r) + (d' + c) = a + b' + c; 
 
 (2) (f<a + ^) = / + a + ,J, 
 
 (<;<«) + (<r< 3) = (f ' + a) + (^' + ^) = / + a + iJ. 
 
 In the special cases where r = o and c = \ respectively, 
 we find 
 
 (3) {<ib = o) = (a = o) + (-5 = o), 
 
 (4) (a + ^ = I) = (a = 1) + (3 = i). 
 
 P. I: (i) To say that two propositions united imply a 
 third is to say that one of them implies this third proposition. 
 
 (2) To say that a proposition implies the alternative of 
 two others is to say that it implies one of them. 
 
 (3) To say that two propositions combined are false is to 
 say that one of them is false. 
 
 (4) To say that the alternative of two propositions is true 
 is to say that one of them is true. 
 
 The paradoxical character of the first three of these state- 
 ments will be noted in contrast to the self evident character 
 of the fourth. These paradoxes are explained, on the one 
 hand, by the special axiom which states that a proposition 
 is either true or false; and, on the other hand, by the fact 
 that the false implies the true and that only the false is not 
 implied by the true. For instance, if both premises in the 
 first statement are true, each of them implies the conse- 
 quence, and if one of them is false, it implies the conse- 
 quence (true or false). In the second, if the alternative is 
 true, one of its terms must be true, and consequently will, 
 like the alternative, be implied by the premise (true or false). 
 
88 LAW OF IMPORTATION AND EXPORTATION. 
 
 Finally, in the third, the product of two propositions cannot 
 be false unless one of them is false, for, if both were true, 
 their product would be true (equal to i). 
 
 58. Law of Importation and Exportation. — The funda- 
 mental equivalence (a<^ 6) '= a + d has many other inter- 
 esting consequences. One of the most important of these 
 is i^e law of importation and exportation, which is expressed 
 by the following formula: 
 
 \a<{b<c)\ = {ab<c) 
 
 "To say that if a is true b implies c, is to say that a 
 and b imply c". 
 
 This equality involves two converse implications: If we 
 infer the second member from the first, we import into the 
 implication {b<^c) the hypothesis or condition a; if we infer 
 the first member from the second, we, on the contrary, 
 export from the implication (ab<^c) the hypothesis a. 
 
 Demonstration : 
 
 [a<{b<c)] = a + {b<^c) = a'+ b' + c, 
 (ab<^c) = {ab)' + c = a + b' -\- c. 
 
 Cor. I. — Obviously we have the equivalence 
 [<7<(^<r)] = \b<^{a<^cj\, 
 
 since both members are equal to {ab<^c), by the commu- 
 tative law of multiplication. 
 
 Cor. 2.— We have also 
 
 [<Z < (a < b)] = (<7 < b\ 
 
 for, by the law of importation and exportation, 
 
 \a<{a<b)\ = (,aa<b) = {a<^b). 
 
 If we apply the law of importation to the two following 
 formulas, of which the first results from the principle of 
 identity and the second expresses the principle of contra- 
 position, 
 
LAW OF IMPORTATION AND EXPORTATION. 89 
 
 (a<3)<(a<3), (a<<J)<(^'<a), 
 
 we obtain the two formulas 
 
 {a<ib)a<^b, (a <: 6) d' <: a , 
 
 which are the two types of hypothdical reasoning: -'If a 
 implies b, and if a is true, b is true" {modus pmens) ; "If a 
 implies b, and if b is false, a is false" {modus tollens). 
 
 Remark. These two formulas could be directly deduced 
 by the principle of assertion, from the following 
 
 {a <b) {b==o)<{a = o), 
 
 which are not dependent on the law of importation and 
 which result from the principle of the syllogism. 
 
 From the same fundamental equivalence, we can deduce 
 several paradoxical formulas: 
 
 1. <z<(*<3), a'<{a<b). 
 
 "If a is true, a is implied by any proposition b; if a is 
 false, a implies any proposition b". This agrees with the 
 known properties of o and i. 
 
 2. a < [{a <b)< b], a'< [{b < a)< b'l 
 
 "If a is true, then 'a implies b' implies b; if a is false, 
 then '^ implies a' implies not-3.'' 
 
 These two formulas are other forms of hypothetical reason- 
 ing {modus ponens and modus tollens). 
 
 3. [(a<^)<<7] = a^ i{b<a)<a\ = a, 
 
 "To say that, if a implies ^, a is true, is the same as 
 affirming a; to say that, if b implies a, a is false, is the 
 same as denying a". 
 
 Demonstration : 
 
 l{a<:ib)<ia\ = {a -V b<^a) = ai -^ a = a, 
 \{b <^ a) <^a \ = {b + a<^a)^a^ + a=a. 
 
 I This formula is Bertrand Russell's "principle of reduction'" See 
 The Prmdples oj iiathfma'lcs. Vol. I, p. 17 (Cambridge, 1903). 
 
90 INEQUALITIES REDUCED TO EQUALITIES. 
 
 In formulas (i) and (3), in which b is any term at all, 
 we might introduce the sign ]^ with respect to b. In the 
 following formula, it becomes necessary to make use of this 
 
 sign. 
 
 4. n i[«<(^<*)]<^| = «^- 
 
 X 
 
 Demonstration : 
 
 {[<7<(^<*j]<*j = |[<^'+ (^<^)]<^j 
 
 = \(a + b' ■\- x)<ix\^ abx ■\- x=^ab -\r X. 
 We must now form the product ]^ {at + x), where x 
 
 X 
 
 can assume every value, including o and i. Now, it is 
 clear that the part common to all the terms of the form 
 (ab + x) can only be ah. For, (i) ab is contained in each 
 of the sums (,ab -V x) and therefore in the part common to 
 all; (2) the part common to all the sums {ab -^ x) must be 
 contained in (ab -\- o), that is, in ab. Hence this common 
 part is equal to ab^., which proves the theorem. 
 
 59. Reduction of Inequalities to Equalities. — As we 
 
 have said, the principle of assertion enables us to reduce 
 inequalities to equalities by means of the following formulas: 
 
 (a + o) = (a: = i), (fl 4= i) = (a = o), 
 
 (^ + ^) = (« = b'). 
 For, 
 
 {a^b) = {ab + ab + o) = {ab' + ab' = i) = (a = b'). 
 
 Consequently, we have the paradoxical formula 
 
 (a 4= Z.) = (a = b'). 
 
 ' This argument is general and from it we can deduce the fonnala 
 f] (a + ;r) = a, 
 
 X 
 
 whence may be derived the correlative formula 
 
INEQUALITIES REDUCED TO EQUALITIES. 9 1 
 
 This is easily understood, for, whatever the proposition b, 
 either it is true and its negative is false, or it is false and 
 its negative is true. Now, whatever the proposition a may 
 be, it is true or false; hence it is necessarily equal either to 
 ^ or to ^ . Thus to deny an equality (between propositions) 
 is to affirm the opposite equality. 
 
 Thence it results that, in the calculus of propositions, we 
 do not need to take inequalities into consideration — a fact 
 which greatly simplifies both theory and practice. More- 
 over, just as we can combine alternative equalities, we can 
 also combine simultaneous inequalities, since they are redu- 
 cible to equalities. 
 
 For, from the formulas previously established (§ 57) 
 
 iab = o) = (a == Q>) -\- {b = o), 
 (a -f ^ = i) = (<7 = i) -t- (3 = ij, 
 we deduce by contraposition 
 
 (<x + o) (^ + o) = (a^ + o), 
 (a4=i) (^4=i) = (a-t-^4=i). 
 
 These two formulas, moreover, according to what we have 
 just said, are equivalent to the known formulas 
 
 {a =1) (*=!) = {ab = I), 
 (a = o) {b== o) = {a -^ b= o). 
 
 Therefore, in the calculus of propositions, we can solve 
 all simultaneous systems of equalities or inequalities and all 
 alternative systems of equalities or inequalities, which is not 
 possible in the calculus of classes. To this end, it is necessary 
 only to apply the following rule: 
 
 First reduce the inclusions to equahties and the non- 
 inclusions to inequalities; then reduce the equalities so that their 
 second members will be i, and the inequalities so that their 
 second members will be o , and transform the latter into equal- 
 ities having i for a second member; finally, suppress the 
 second members i and the signs of equality, i. e., form the 
 product of the first members of the simultaneous equalities and 
 the sum of the first members of the alternative equalities, 
 retaining the parentheses. 
 
92 CONCLUSION. 
 
 60. Conclusion. — The foregoing exposition is far from 
 being exhaustive; it does not pretend to be a complete 
 treatise on the algebra of logic, but only undertakes to make 
 known the elementary principles and theories of that science. 
 The algebra of logic is an algorithm with laws peculiar to 
 itself. In some phases it is very analogous to ordinary al- 
 gebra, and in others it is very widely different. For in- 
 stance, it does not recognize the distinction of degrees; the 
 laws of tautology and absorption introduce into it great 
 simplifications by excluding from it numerical coefficients. 
 It is a formal calculus which can give rise to all sorts of 
 theories and problems, and is susceptible of an almost in- 
 finite development 
 
 But at the same time it is a restricted system, and it is 
 important to bear in mind that it is far from embracing all 
 of logic. Properly speaking, it is only the algebra of 
 classical logic. Like this logic, it remains confined to the 
 domain circumscribed by Aristotle, namely, the domain of 
 the relations of inclusion between concepts and the relations 
 of implication between propositions. It is true that classical 
 logic (even when shorn of its errors and superfluities) was 
 much more narrow than the algebra of logic. It is almost 
 entirely contained within the bounds of the theory of the 
 syllogism whose limits to-day appear very restricted and 
 artificial. Nevertheless, the algebra of logic simply treats, 
 with much more breadth and universality, problems of the 
 same order; it is at bottom nothing else than the theory 
 of classes or aggregates considered in their relations of in- 
 clusion or identity. Now logic ought to study many other 
 kinds of concepts than generic concepts (concepts of classes) 
 and many other relations than the relation of inclusion (of 
 subsumption) between such concepts. It ought, in short, to 
 develop into a logic of relations, which Leibniz foresaw, 
 which Peirce and Schroder founded, and which Peano and 
 Russell seem to have established on definite foundations. 
 
 While classical logic and the algebra of logic are of 
 hardly any use to mathematics, mathematics, on the other 
 hand, finds in the logic of relations its concepts and fun- 
 
CONCLUSION. 93 
 
 damental principles; the true logic of mathematics is the logic 
 of relations. The algebra of logic itself arises out of pure 
 logic considered as a particular mathematical theory, for it 
 rests on principles which have been implicitly postulated and 
 which are not susceptible of algebraic or symbolic expression 
 because they are the foundation of all symbolism and of all 
 the logical calculus.' Accordingly, we can say that the al- 
 gebra of logic is a mathematical logic by its form and by 
 its method, but it must not be mistaken for the logic of 
 mathematics. 
 
 J The principle of deduction and the principle of substitution. See 
 the author's Manuel de Logistique, Chapter I, SS 2 and 3 [not published], 
 and Les Principes des Mathematiques, Chapter I, A. 
 
INDEX. 
 
 Absorption, Law of, 13, 92. ! 
 
 Absurdity, Type of, 27. 
 
 Addition, and multiplication, \ 
 Logical, V, vi, 9, 20; and i 
 multiplication, Modulus of, 
 1 9 ; and multiplication. The- 
 orems on, 14; Logical, not 
 disjunctive, 11. 
 
 Affirmative propositions, 80 n. 
 
 Algebra, of logic an algorithm, 
 92; of logic compared to 
 mathematical algebra, 13; 
 of thought, V. 
 
 Algorithm, Algebra of logic 
 an, 92. 
 
 Alphabet of human thought, v. 
 
 Alternative, 1 2 ; affirmation, 
 II, 20, 24; Equivalence of 
 an implication and an, 85. 
 
 Antecedent, 7. 
 
 Aristotle, iii, 2 in., 92. 
 
 Assertion, Principle of, 84. 
 
 Assertions, Number of pos- 
 sible, 79. 
 
 Axioms, 8, 10, 16, 17, 22, 
 
 27, 84. 
 
 Baldwin, 23n. 
 
 Boole, iii— ix, xiii, 11 n., 18, 21, 
 
 28, 29, 63; Problem of, 
 59—61. 
 
 Bryan, William Jennings, ix. 
 
 Calculus, Infinitesimal, v ; Logic- 
 al, viii, 3; ratiocinator, 
 V— viii. 
 
 Cantor, Georg, ion. 
 
 Categorical syllogism, 8. 
 
 Cause, 7, II. 
 
 Causes, Forms of, 69; Law 
 of, 67—69; Sixteen, 67, 72; 
 Table of, 77—79. 
 
 Characters, v. 
 
 Classes, Calculus of, 4, 86, 91. 
 
 Classification of dichotomy, 
 31 n. 
 
 Commutativity, 24. 
 
 Composition, Principle of, 
 II— 12, 86. 
 
 Concepts, Calculus of, 4. 
 
 Condition, 7; Necessary and 
 sufficient, 7—8, 45, 49, 57, 
 82; Necessary but not suffi- 
 cient, 41; of impossibility 
 and indetermination, 57. 
 
 Connaissances, 63. 
 
 Consequence, 7, 11. 
 
 Consequences, Forms of, 69; 
 Law of, 63—66; of the 
 syllogism, 9; Sixteen, 65, 
 71; Table of, 76-77. 
 
 Consequent, 7. 
 
 Constituents, 28; Properties 
 of, 29. 
 
INDEX. 
 
 95 
 
 Contradiction , Principle of, 
 
 22n., 23—24. 
 Contradictory propositions, 24 ; 
 
 terms, 29. 
 Contraposition, Law of, 26, 
 
 81; Principle of, 88. 
 Council, Members of, 71. 
 Couturat, v., i8n., 93. 
 
 Dedekind, ion. 
 
 Deduction, 61 ; Principle of, 93. 
 
 Definition, Theory of, x. 
 
 De Morgan, iii, iv, vi, viii, ix; 
 Formulas of, 32—33, 81. 
 
 Descartes, iv. 
 
 Development, 28; Law of, 
 30 — 32 ; of logical functions, 
 79; of mathematics, iv; of 
 symbolic logic, viii. 
 
 Diagrams of Venn, Geometric- 
 al, 73-74- 
 
 Dichotomy, Classification of, 
 
 Sin- 
 Disjunctive, Logical addition 
 
 not, 11; sums, 34. 
 Distributive law, 16. 
 Double inclusion, 37; ex- 
 pressed by an indeterminate, 
 48; Negative of the, 82. 
 Double negation, 24. 
 Duality, Law of, 20. 
 
 Economy of mental effort, iii. 
 
 Elimination of known terms, 
 63, 64—67 ; of middle terms, 
 61, 63; of unknowns, 53, 
 57» 59> 61; Resultant of, 
 40, 41,57,72,73,82; Rule 
 for resultant of, 43, 55. 
 
 Equalities, Formulas for trans- 
 forming inclusions into, 15, 
 25 — 26; Reduction of in- 
 equalities to, 85, 91. 
 
 Equality a primitive idea, 15; 
 Definition of, 6—8; Notion 
 of, ix. 
 
 Equation, and an inequation, 
 83; Throwing into an, 75. 
 
 Equations, Solution of, 50—53, 
 57—59, 61, 73. 
 
 Excluded middle. Principle of, 
 23—24. 
 
 Exclusion, Principle of, 23 n. 
 
 Exclusive, Mutually, 29. 
 
 Existence, Postulate of, 21, 27. 
 
 Exhaustion, Principle of, 23 n. 
 
 Exhaustive, Collectively, 29. 
 
 Forms, Law of, 62, 70; of con- 
 sequences and causes, 69. 
 
 Frege, vii, viii, x; Symbolism 
 of, vii. 
 
 Functions, iv; Development of 
 logical, 79; Integral, 29 n; 
 Limits of, 37—38; Logical, 
 29 — 30; of variables, 56; 
 Properties of developed, 
 34~37j Prepositional, iv; 
 Sums and products of, 44; 
 Values of, 55. 
 
 Hopital, Marquis de 1', vi. 
 Huntington, E. V., xiv, 4n., 
 
 i5n., 2 1 XL 
 Hypothesis, 7. 
 Hypothetical arguments, 2 7 ; 
 
 reasoning, 89; syllogism, 8. 
 
96 
 
 INDEX. 
 
 Ideas, Simple and complex, v. 
 
 Identity, vi; Principle of, 8, 
 2 1, 88; Type of, 27. 
 
 Ideography, v, vii, viii. 
 
 Implication, 5; and an alter- 
 native. Equivalence of an, 
 85; Relations of, 92. 
 
 Importation and exportation. 
 Law of, 88. 
 
 Impossibility, Condition of, 57. 
 
 Inclusion, vi; a primitive idea, 
 ix, 5; Double, 37; expressed 
 by an indeterminate, 46, 48; 
 Negative of the double, 82; 
 Relation of, x, 4—6, 92. 
 
 Inclusions into equalities. For- 
 mulas for transforming, 1 5, 
 25—26. 
 
 Indeterminate, 5 1 ; Inclusion 
 expressed by an, 46, 48. 
 
 Indetermlnation, 43; Condition 
 
 of, 57- 
 
 Inequalities, to equalities, Re- 
 duction of, 85, 91; Trans- 
 formation of non-inclusions 
 and, 81. 
 
 Inequation, Equation and an, 
 83; Solution of an, 81, 84. 
 
 Infinitesimal calculus, v. 
 
 Integral function, 29n. 
 
 Interpretations of the cal- 
 culus, 3f. 
 
 Jevons, viii, ix, xiii, iin., 73; 
 
 Logical piano of, 75. 
 Johnson, W. E., xiv, 59. 
 
 Known terms (connaissances), 
 63-64, 67. 
 
 Ladd-Franklin, Mrs., viii, xiii, 
 
 23 n., 42. 
 Lambert, iii, vi. 
 Leibniz, iii, ivflf., 4, 92. 
 Limits of a function, 37—38. 
 
 MacCoU, vi, ix, 2 1 n., 30. 
 
 MacFarlane, Alexander, xiii 
 
 Mathematical function, ix; 
 logic, iii, iv, 93. 
 
 Mathematics, Philosophy a 
 universal, iv. 
 
 Maxima of discourse, 29. 
 
 Middle, Principle of excluded, 
 23—24; terms. Elimination 
 of, 61, 63. 
 
 Minima of discourse, 28. 
 
 Mitchell, O., xiii, 42. 
 
 Modulus of addition and mul- 
 tiplication, 19. 
 
 Modus ponens, 89. 
 
 Modus tollens, 27, 89. 
 
 MiiUer, Eugen, ix, xiv, 46 n. 
 
 Multiplication. See s. v. "Ad- 
 dition." 
 
 Negation, v, vi, 9; defined, 
 21-23; Double, 24; Dual- 
 ity not derived from, 20, 22. 
 
 Negative, 21, 23; of the double 
 inclusion, 82; propositions, 
 80 n. 
 
 Non-inclusions and inequal- 
 ities. Transformation of, 81. 
 
 Notation, v, 2 in, 44. 
 
 Null-class, vi, 18, 20. 
 
 Number of possible assertions, 
 79- 
 
INDEX. 
 
 97 
 
 One, Definition of, ix, 17—20. 
 
 Particular propositions, 80. 
 
 Peano, iii, viii, x, Son, 92. 
 
 Peirce, C. S., viii, ix, xiii. 
 
 Philosophy a universal math- 
 ematics, iv. 
 
 Piano of Jevons, Logical, 75. 
 
 Poretsky, xiv, 28, 73, S2n, 
 53; Formula of, 38-39, 
 40; Method of, 62 — 70. 
 
 Predicate, 7. 
 
 Premise, 7. 
 
 Primary proposition, 6, 21. 
 
 Primitive idea. Equality a, 1 5 ; 
 Inclusion a, ix, 5. 
 
 Product, Logical, 10. 
 
 Propositions, ix; Calculus of, 
 4, 86, 91; Contradictory, 
 24; Formulas peculiar to 
 the calculus of, 84; Implica- 
 tion between, 92; reduced 
 to lower orders, 86; Un- 
 iversal and particular, 44, 80. 
 
 Reciprocal, 7, 21. 
 Reduclio ad absurdum, 27. 
 Reduction, Principle of, Sgn. 
 Relations, Logic of, 92. 
 Relatives, Logic of, ix. 
 Resultant of elimination, 40, 
 
 41, 57. 72, 73, 82; Rule 
 
 for, 43. 55- 
 Russell, B., vii, viii, 89n, 92. 
 
 Schroder, vi, viiif, xiii, 5, 2 in, 
 29j 41, 59> 61—62, Son, 
 92; Theorem of, 39. 
 
 Secondary proposition, 6, 21. 
 
 Simplification , Principle of, 
 II — 12, 21. 
 
 Simultaneous affirmation, 1 1 , 
 20, 24. 
 
 Solution of equations, 50—53, 
 57—59, 61, 73; of in- 
 equations, 81, 84. 
 
 Subject, 7. 
 
 Substitution, Principle of, 93. 
 
 Subsumption, 5. 
 
 Summand, 4. 
 
 Sums, and products of func- 
 tions, 44; Disjunctive, 34; 
 Logical, 10. 
 
 Syllogism, Principle of the, 8, 
 ! 15, 62n; Theory of the, 92. 
 : Symbolic logic, iii, v; Devel- 
 I opment of, viii. 
 ! Symbolism in mathematics, iv. 
 
 Symbols, Origin of, iv. 
 
 Symmetry, 7, 20, 24. 
 
 I 
 1 
 1 Tautology, Law of, 13, 92. 
 
 ' Term, 4. 
 
 Theorem, 7. 
 ' Thesis, 7. 
 
 Thought, Algebra of, v; Alpha- 
 \ bet of human, v; Economy 
 of, iii. 
 Transformation of inclusions 
 into equalities, 15, 25—26; 
 of inequalities into equal- 
 ities, 85, 91; of non-in- 
 clusions and inequalities, 81. 
 
 Universal characteristic of 
 Leibniz, v— viii; propositions, 
 80 n. 
 
98 
 
 INDEX. 
 
 Universe of discourse, i8, 
 
 2311, 27. 
 Unknowns, Elimination of, 53, 
 
 57, 59, 61. 
 
 Variables, Functions of, 56. 
 
 Venn, John, iii, viii, ix; Geo- 
 metrical diagrams of, 73-74; 
 Mechanical device of, 75; 
 Problem of, 71—73. 
 
 Viete, iv. 
 Voigt, 42. 
 
 Whitehead, A. N., viii, xiii, 
 
 56n., 59, 61 n. 
 Whole, Logical, 62. 
 
 Zero, Definition of, ix, 17—20; 
 Logical, 62, 76. 
 
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