■F BS^R rail .-' . ■ m 1 Hj £BS EM n» ■ 1% HH ■ I I *ui ny** m H&g-i ■■■■■■■■ i in — 1 HHl SHI Hi ■ ■ $$5 l^i H &j9mLjB ■■■ ■ ."??; ^sy ■ i*^air^ ■ fcj>,53!£3 mHKWS pSflffl £cm*»< vSBE |{&E ■■J BC 135 M129 '■ I OF CALIFORNIA SAN DIEGO 3 1822 01063 9508 '>^\ l , Kfi t;»vi ; t ■ iB fflBHSBfi a*fe : ABBS hH Pi LIBRARY UNIVERSITY OF CALIFORNIA SAN DIEGO j THE UNIVERSITY LIBRARY DIVERSITY OF CALIFORNIA, SAN DIEGO LA JOLLA, CALIFORNIA SYMBOLIC LOGIC AND ITS APPLICATIONS SYMBOLIC LOGIC AND ITS APPLICATIONS HUGH MacCOLL B.A. (London) LONGMANS, GREEN, AND CO. 39 PATERNOSTER ROW, LONDON NEW YORK AND BOMBAY 1906 All rights reserved PREFACE . This little volume may be regarded as the final con- centrated outcome of a series of researches begun in 1872 and continued (though with some long breaks) until to-day, My article entitled " Probability Notation No. 2," which appeared in 1872 in the Educational Times, and was republished in the mathematical " Reprint," con- tains the germs of the more developed method which I afterwards explained in the Proceedings of the London Mathematical Society and in Mind. But the most impor- tant developments from the logical point of view will be found in the articles which I contributed within the last eight or nine years to various magazines, English and French. Among these I may especially mention those in Mind and in the Athenceum, portions of which I have (with the kind permission of these magazines) copied into this brief epitome. Readers who only want to obtain a clear general view of symbolic logic and its applications need only attend to the following portions: §§ 1 to 18, §§ 22 to 24, §§ 46 to 53, §§ 7G to 80, §§ 112 to 120, §§ 144 to 150. Students who have to pass elementary examinations in ordinary logic may restrict their reading to §§ 1 to 18, §§ 46 to 59, §§ 62 to 0Q, §§ 76 to 109, § 112. Mathematicians will be principally interested in the last five chapters, from § 114 to § 156; but readers vi PREFACE who wish to obtain a complete mastery of my symbolic system and its applications should read the whole. They will find that, in the elastic adaptability of its notation, it bears very much the same relation to other systems (including the ordinary formal logic of our text-books) as algebra bears to arithmetic. It is mainly this nota- tional adaptability that enables it to solve with ease and simplicity many important problems, both in pure logic and in mathematics (see § 75 and § 15 7), which lie wholly beyond the reach of any other symbolic system within my knowledge. HUGH MacCOLL. August 17 th, 1905. CONTENTS INTRODUCTION SECS. PAGE 1-3. General principles — Origin of language ... 1 CHAPTER I 4-12. Definitions of symbols — Classification of propositions — Examples and formulae ...... 4 CHAPTER II 13-17. Logic of Functions — Application to grammar ... 9 CHAPTER III 18-24. Paradoxes — Propositions of the second, third, and higher degrees . . . . . . . . .12 CHAPTER IV 25-32. Formulae of operations with examples worked — Venn's problem ......... 20 CHAPTER V 33-38. Elimination — Solutions of implications and equations — Limits of statements 27 CHAPTER VI 39-43. Jevous's " Inverse Problem " ; its complete solution on the principle of limits, with examples ... 33 66 CONTENTS CHAPTER VII SECS. PAGE 44-53. Tests of Validity — Symbolic Universe, or Universe of Discourse — No syllogism valid as usually stated . 39 CHAPTER VIII 54-63. The nineteen traditional syllogisms all deducible from one simple formula — Criticism of the technical words ' distributed ' and ' undistributed ' — The usual syllogistic ' Canons ' unreliable ; other and simpler tests proposed 49 CHAPTER IX 64-66 (a). Enthymemes— Uiven one premise of a syllogism and the conclusion, to find the missing premise- Strongest conclusion from given premises CHAPTER X 67-75. To find the weakest data from which we can prove a given complex statement, and also the strongest conclusion deducible from the statement — Some contested problems — ' Existential Import of Pro- positions ' — Comparison of symbolic methods . 70 CHAPTER XI 76-80. The nature of inference — The words if, therefore, and because — Causation and discovery of causes . . 80 CHAPTER XII 81-89. Solutions of some questions set at recent examina- tions . 86 CHAPTER XIII 90-113. Definitions and explanations of technical terms often used in logic — Meaningless symbols and their uses ; mathematical examples — Induction: inductive reasoning not absolutely reliable ; a curious case in mathematics — ' Infinite ' and ' infinitesimal ' . .91 CONTENTS i x CALCULUS OF LIMITS CHAPTER XIV SECS. PAGE 114-131 Application to elementary algebra, with examples . 106 CHAPTER XV 132-140. Nearest limits — Table of Reference . . . .117 CHAPTER XVI 141-143. Limits of two variables — Geometrical illustrations . 123 CHAPTER XVII 144-150. Elementary probability — Meaning of 'dependent' and ' independent ' in probability, with geo- metrical illustrations . . . . . .128 CHAPTER XVIII 151-157. Notation for Multiple Integrals — Problems that re- quire the integral calculus . . . ' . . 132 ALPHABETICAL INDEX (The numbers indicate the sections, not the pages.) Alternative, 7, 41 Anipliative, 108 Antecedent, 28 Cause, 79 Complement, 46 Connotation, 93 Consequent, 28 Contraposition, 97 Contrary, 94 Conversion, 98 Couturat's notation, 132 (footnote) Dichotomy, 100 Dilemma, 101-103 Elimination, 33-38 Entliymeme, 64 Equivalence, 11, 19 Essential, 108 Excluded Middle, 92 Existential import of proposi- tions, 72, 73 Factor, 7, 28 Formal, 109 Functions, 13-17 Grammar, 17 Illicit process, 63 (footnote) Immediate inference, 91 Implication, 10, 18 114 from Induction, 112 Inference, nature of, 76-80 Infinite and infinitesimal, 113 Jevons's 'inverse problem,' 39-43 Limits of statements, 33 Limits of variable ratios, 143 Major, middle, minor, 54 Material, distinguished Formal, 109 Meaningless symbols, 110 Mediate inference, 91 Modality, 99 Multiple, 28 Particulars, 49 Ponendo ponens, &c, 104-107 Product, 7 Sorites, 90 Strong statements, 33, 34 Subalterns and subcontraries, 95, 96 Syllogisms, 54 Transposition, 56 Universals, 49 Universe of discourse, 46-50 Venn's problem, 32 Weak statements, 33, 34 SYMBOLIC LOGIC INTRODUCTION 1. In the following pages I have done my best to explain in clear and simple language the principles of a useful and widely applicable method of research. Symbolic logic is usually thought to be a hard and abstruse subject, and unquestionably the Boolian system and the more modern methods founded on it are hard and abstruse. They are, moreover, difficult of application and of no great utility. The symbolic system explained in this little volume is, on the contrary, so simple that an ordinary schoolboy of ten or twelve can in a very short time master its fundamental conceptions and learn to apply its rules and formulas to practical problems, especially in elementary mathematics (see §§ 114, 118). Nor is it less useful in the higher branches of mathematics, as my series of papers published in the Proceedings of the London Mathematical Society abundantly prove. There are two leading principles which separate my symbolic system from all others. The first is the principle that there is nothing sacred or eternal about symbols ; that all symbolic conventions may be altered when convenience requires it, in order to adapt them to new conditions, or to new classes of problems. The symbolist has a right, in such circumstances, to give a new meaning to any old symbol, or arrangement of symbols, provided the change- of sense be accompanied by a fresh definition, and provided the nature of the 2 SYMBOLIC LOGIC [§§ 1, 2 problem or investigation be such that we run no risk of confounding the new meaning with the old. The second principle which separates my symbolic system from others is the principle that the complete state- ment or proposition is the real unit of all reasoning. Provided the complete statement (alone or in connexion with the context) convey the meaning intended, the words chosen and their arrangement matter little. Every intelligible argument, however complex, is built up of individual statements ; and whenever a simple elementary symbol, such as a letter of the alphabet, is sufficient to indicate or represent any statement, it will be a great saving of time, space, and brain labour thus to repre- sent it. 2. The words statement and proposition are usually regarded as synonymous. In my symbolic system, however, I find it convenient to make a distinction, albeit the distinction may be regarded as somewhat arbitrary. I define a statement as any sound, sign, or symbol (or any arrangement of sounds, signs, or symbols) employed to give information ; and I define a proposition as a statement which, in regard to form, may be divided into two parts respectively called subject and predicate. Thus every proposition is a statement ; but we cannot affirm that every statement is a proposition. A nod, a shake of the head, the sound of a signal gun, the national flag of a passing ship, and the warning " Caw " of a sentinel rook, are, by this definition, statements but not propositions. The nod may mean " I see him " ; the shake of the head, " I do not see him " ; the warning " Caw " of the rook, " A man is coming with a gun," or " Danger approaches " ; and so on. These propositions express more specially and precisely what the simpler statements express more vaguely and generally. In thus taking statements as the ultimate constituents of sym- bolic reasoning I believe I am following closely the gradual evolution of human language from its primitive §§ 2, 3] INTRODUCTION 3 prehistoric forms to its complex developments in the languages, dead or living, of which we have knowledge now. There can be little doubt that the language or languages of primeval man, like those of the brutes around him, consisted of simple elementary statements, indivisible into subject and predicate, but differing from those of even the highest order of brutes in being uninherited — in being more or less conventional and therefore capable of indefinite development. From their grammatical structure, even more than from their com- munity of roots, some languages had evidently a common origin; others appear to have started independently; but all have sooner or later entered the propositional stage and thus crossed the boundary which separates all brute languages, like brute intelligence, from the human. 3. Let us suppose that amongst a certain prehistoric tribe, the sound, gesture, or symbol S was the understood representation of the general idea stag. This sound or symbol might also have been used, as single words are often used even now, to represent a complete statement or proposition, of which stag was the central and leading idea. The symbol S, or the word stag, might have vaguely and varyingly done duty for "It is a stag," or " I see a stag," or " A stag is coming," &c. Similarly, in the customary language of the tribe, the sound or symbol B might have conveyed the general notion of bigness, and have varyingly stood for the statement " It is big" or " I see a big thing coming," &c. By degrees primitive men would learn to combine two such sounds or signs into a compound statement, but of varying form or arrangement, according to the impulse of the moment, as SB, or BS, or S B , or S B , &c., any of which might mean "I see a big stag," or "The stag is big" or " A big stag is coming," &c. In like manner some varying arrangement, such as SK, or S K , &c, might mean " The stag has been killed," or " I have killed the stag" &c. 4 SYMBOLIC LOGIC [§§ 3, 4 Finally, and after many tentative or haphazard changes, would come the grand chemical combination of these linguistic atoms into the compound linguistic molecules which we call propositions. The arrangement S B (or some other) would eventually crystallize and permanently signify " The stag is big," and a similar form S K would permanently mean " The stag is killed" These are two complete propositions, each with distinct subject and predicate. On the other hand, S B and S K (or some other forms) would permanently represent " The big stag " and " The killed stag." These are not complete pro- positions ; they are merely qualified subjects waiting for their predicates. On these general ideas of linguistic development I have founded my symbolic system. CHAPTER I 4. The symbol A B denotes a proposition of which the individual A is the subject and B the predicate. Thus, if A represents my aunt, and B represents brown-haired, then A B represents the proposition " My aunt is brown- haired." Now the word aunt is a class term ; a person may have several aunts, and any one of them may be represented by the symbol A. To distinguish between them we may employ numerical suhixes, thus A 1} A 2 , A 3 , &c, Aunt No. 1, Aunt No. 2, &c. ; or we may distinguish between them by attaching to them different attributes, so that A B would mean my brown-haired aunt, A R my red-haired aunt, and so on. Thus, when A is a class term, A B denotes the individual (or an individual) of whom or of which the proposition A B is true. For example, let H mean " the horse " ; let w mean " it won the race " ; and let s mean " I sold it," or " it has been sold by me." Then H£, which is short for (H w ) s , represents the complex proposition " The horse which won the race has been sold by me," or " I have sold the horse which §§ 4-6] EXPLANATIONS OF SYMBOLS 5 won the race." Here we are supposed to have a series of horses, H r H 2 , H 3 , &c, of which H vv is one; and we are also supposed to have a series, S 1; S 2 , S 3 , &c, of things which, at some time or other, I sold ; and the proposition H* asserts that the individual H w , of the first series H, belongs also to the second series S. Thus the suffix w is adjectival; the exponent s predicative. If we inter- change suffix and exponent, we get the proposition H^, which asserts that "the horse which I have sold won the race." The symbol H w , without an adjectival suffix, merely asserts that a horse, or the horse, won the race without specifying which horse of the series H x , H 2 , &c. 5. A small minus before the predicate or exponent, or an acute accent affecting the whole statement, indicates denial. Thus if H° means " The horse has been caught " ; then H~° or (H c )' means " The horse has not been caught." In accordance with the principles of notation laid down, the symbol H_ c will, on this understanding, mean " The horse which has not been caught" or the " uncaught horse " ; so that a minus suffix, like a suffix without a minus, is adjectival. The symbol H c (" The caught horse ") assumes the statement H c , which asserts that " The horse has been caught." Similarly H_ c assumes the statement H~°. 6. The symbol denotes non-existence, so that , 2 , 3 , &c, denote a series of names or symbols which correspond to nothing in our universe of admitted realities. Hence, if we give H and C the same meanings as before, the symbol H° will assert that " The horse caught does not exist," which is equivalent to the statement that "No horse has been caught." The symbol H~ , which denies the statement H°, may therefore be read as " The horse caught does exist," or " Some horse has been caught." Following the same principle of notation, the symbol H° c may be read "An uncaught horse does not exist," or " Every horse has been caught," The context would, of course, indicate the particular totality of horses 6 SYMBOLIC LOGIC [§§ 6-8 referred to. For example, H° c may mean " Every horse that escaped has been caught," the words in italics being understood. On the same principle H:° denies H° c , and may therefore be read " Some uncaught horse does exist" or " Some horse has not been caught." 7. The symbol A B x C D , or its usually more convenient synonym A B -C r> , or (without a point) A B C D , asserts two things — namely, that A belongs to the class B, and that C belongs to the class D ; or, as logicians more briefly express it, that " A is B " and that " C is D." The symbol A B + C D asserts an alternative — namely, that " Either A belongs to the class B, or else C to the class D " ; or, as it is more usually and briefly expressed, that " Either A is B, or C is D." The alternative A B + C D does not necessarily imply that the propositions A B and C D are mutually exclusive ; neither does it imply that they are not. For example, if A B means " Alfred is a barrister," and C D means "Charles is a doctor"; then A B C D asserts that " Alfred is a barrister, and Charles a doctor" while A B + C D asserts that "Either Alfred is a barrister, or Charles a doctor," a statement which (apart from context) does not necessarily exclude the possibility of A B C D , that both A B and C D are true. # Similar conventions hold good for A B C D E F and A B + C p + E r , &c. From these con- ventions we get various self-evident formulae, such as (1) (A B C D )' = A- B + C- D ; (2) ( A B + C p )' = A- B C- D ; (3) (A B C- D )' = A" B + C B ; (4) ( A B + C^)' = A" B C D . 8. In pure or abstract logic statements are represented by single letters, and we classify them according to attributes as true, false, certain, impossible, variable, respec- tively denoted by the five Greek letters t, i, e, >/, 9. Thus the symbol A T B l C e D r 'E 9 asserts that A is true, that B is false, that C is certain, that D is impossible, that * To preserve mathematical analogy, A B and O may be called factors of the product A B C D , and terms of the sum A B +C D ; though, of course, these words have quite different meanings in logic from those they bear in mathematics. §§ 8-10] EXPLANATIONS OF SYMBOLS 7 E is variable (possible but uncertain). The symbol A T only asserts that A is true in a particular case or instance. The symbol A e asserts more than this: it asserts that A is certain, that A is always true (or true in every case) within the limits of our data and defini- tions, that its probability is 1. The symbol A' only asserts that A is false in a particular case or instance ; it says nothing as to the truth or falsehood of A in other instances. The symbol A 71 asserts more than this ; it asserts that A contradicts some datum or defini- tion, that its probability is 0. Thus A T and A 1 are simply assertive; each refers only to one case, and raises no general question as to data or probability. The symbol A e (A is a variable) is equivalent to A -7, A~' ; it asserts that A is neither impossible nor certain, that is, that A is p>ossible but uncertain. In other words, A 6 asserts that the probability of A is neither nor 1, but some proper fraction between the two. 9. The symbol A BC means (A B ) C ; it asserts* that the statement A B belongs to the class C, in which C may denote true, or false, or possible, &c. Similarly A BCD means (A BC ) D , and so on. From this definition it is evident that A VL is not necessarily or generally equivalent to A" 1 , nor A" equivalent to A' £ . 10. The symbol A B : C D is called an implication, and means (A B C" D )^, or its synonym (A" B + C D ) € . It may be read in various ways, as (1) A B implies C D ; (2) If A belongs to the class B, then C belongs to the class D ; (3) It is impossible that A can belong to the class B without C belonging to the class D ; (4) It is certain that either A does not belong to the class B or else C belongs to the class D. Some logicians consider these four propositions equivalent, while others do not ; but all ambiguity may be avoided by the convention, adopted * The symbol A BC must not be confounded with the symbol A BC , which I sometimes use as a convenient abbreviation for A B A C ; nor with the symbol A" r , which I use as short for A B + A c . 8 SYMBOLIC LOGIC [§§ 10, 11 here, that they are synonyms, and that each, like the symbol A B : C D , means (A B C" D )' 7 , or its synonym (A" B + C D ) e . Each therefore usually asserts more than (A B C- D )' and than (A- B + C D ) T , because A" and A e (for any statement A) asserts more than A' and A T respec- tively (see § 8). 11. Let the proposition A B be denoted by a single letter a ; then a will denote its denial A~ B or (A B ) . When each letter denotes a statement, the symbol A : B : C is short for (A : B)(B : C). It asserts that A implies B and that B implies C. The symbol (A = B) means (A : B)(B : A). The symbol A ! B (which may be called an inverse implication) asserts that A is implied in B ; it is therefore equivalent to B : A. The symbol A ! B ! C is short for (A ! B)(B ! C) ; it is therefore equivalent to C : B : A. When we thus use single letters to denote statements, we get numberless self-evident or easily proved formulae, of which I subjoin a few. To avoid an inconvenient multiplicity of brackets in these and in other formulae I lay down the convention that the sign of equivalence ( = ) is of longer reach than the sign of implication ( : ), and that the sign of implication ( : ) is of longer reach than the sign of disjunction or alternat ion( + ). Thus the equivalence a = ft : y means a = (ft: y), not (a = ft):y, and A + B : x means (A + B) : x, not A + (B : x). (I) x(a + ft)=xa + xP; (2) (aft)' = a' + ft' ; (3) (a + ft)' = a f ft f - (4) a:ft = ft':a'; (5) (x:a)(x:ft) = x:aft; (6) « + ft : x = (« : x)(ft :x) ; (7) (A:B:C):(A:C); (8) (A ! B ! C) : (A ! C) ; (9) (A!C)!(A!B!C); (10) (A: C) !(A:B: C) ; (II) (A + A r ) e ; (12) (A T + A') f ; (13) (AA r )\ (14) (A f + A» + A") e ; (15) A f :A T ; (16) A": A 1 ; (17) A e = (A'y; (18) A" = (A / ) f ; (19) A e = (A') 9 ; (20) e : A = A e ; (21) A : r, = A" ; (22) Ae = A; (23) A*i = *i. §| 11_U] LOGIC OF FUNCTIONS 9 These formulae, like all valid formulae in .symbolic logic, hold good whether the individual letters represent certainties, impossibilities, or variables. 12. The following examples will illustrate the working of this symbolic calculus in simple cases. ( 1 ) ( A + B'C)' = A'(B'C)' = A'(B + C) - A'B + A'C. (2) ( A e + B f C e )' = A^B-'C 6 / = A-(B* + C~ 6 ) = (A" + A e XB e + C + C). (3) (A" + A 9 B e ) f = A (A e B e Y = A e (A" e + B' 9 ) = A 9 A" 9 + A e B- = A e (B< + B") ; for A e A- e = r] (an impossibility), and B e = B e + B". CHAPTER II 13. Symbols of the forms F(x), f(x), (p(x), &c, are called Functions of x. A function of x means an expression containing the symbol x. When a symbol {x) is called valid when it is true for all admissible values (or meanings), ar x , x- 2 , x 3 , &c , of x. 10 SYMBOLIC LOGIC [§§ 14-16 becomes when a is put for x and /3 for y. Hence, / : x and x : e, which hold good whether the statement x be true or false. The former asserts that if an impossibility be true any statement x is true, or that an impossibility implies any statement. The latter asserts that the statement x (whether true or false) implies any certainty e, or (in other words) that if x is true e is true. The paradox will appear still more curious when we change x into e in the first formula, or x into rj in the second. We then get the formula r\ : e, which asserts that any im- possibility implies any certainty. The reason why the last formula appears paradoxical to some persons is probably this, that they erroneously understand >/ : e to mean Q^ : Q e , and to assert that if any statement Q is impossible it is also certain, which would be absurd. But }} : e does not mean this (see § 74) ; by definition it simply means (>/e / )'', which asserts that the statement t]e is an impossibility, as it evidently is. Similarly, r\ : x means {qx'J*, and asserts that nx is an impossibility, which is true, since the statement r\x' contains the im- possible factor n. We prove x : e as follows : x\e — (xe'y = (x>i) ri = if — e. For e =■>], since the denial of any certainty is some impossibility (see § 20). That, on the other hand, the implication Q 1 : Q e is not a valid formula is evident ; for it clearly fails in the case Q?. Taking Q = »/, we get Q" : Q e = rp : >f = * : '/ = («/)" = («0" = >/. 19. Other paradoxes arise from the ambiguity of the sign of equivalence ( = ). In this book the statement 14 SYMBOLIC LOGIC [§| 1 9, 20 (a = /3) does not necessarily assert that a and /3 are synonymous, that they have the same meaning, but only that they are equivalent in the sense that each implies the other, using the word 'implies' as denned in § 10. In this sense any two certainties, e x and e 2 , are equivalent, however different in meaning ; and so are any two im- possibilities, n l and n 2 \ but not necessarily two different variables, B x and 6 2 . We prove this as follows. By definition, we have (e l = e 2 ) = (e l :e 2 )(e 2 :e 1 ) = (e/ 2 ne/ i y =(vi) , (Vt) , = , »w=V4= e «; for the denial of any certainty e x is some impossibility r\ y . Again we have, by definition, = *K = Vi= But we cannot assert that any two variables, 6 X and # 2 , are necessarily equivalent. For example, 6 2 might be the denial of 6 V in which case we should get ( e l = e 2 ) = (0 X = e\) = (0 l : e\)(6\ ■. ej = ( 0A) Wi)" The symbol used to assert that any two statements, a and /3, are not only equivalent (in the sense of each implying the other) but also synonymous, is (a = /3); but this being an awkward symbol to employ, the symbol (a = /3), though it asserts less, is generally used instead. 20. Let the symbol it temporarily denote the word possible, let p denote probable, let q denote improbable, and let u denote uncertain, while the symbols e, r\, 6, t, i have their usual significations. We shall then, by definition, have (A 7r = A" ) ) and (A u = A' e ), while A p and A 5 will respectively assert that the chance of A is greater than \, that it is less than \. These conventions give us the nine-factor formula (*V)W(^WV)W § 20] PARADOXES AND AMBIGUITIES 15 which asserts (1, 2) that the denial of a truth is an untruth, and conversely; (3, 4) that the denial of a probability is an improbability, and conversely; (5, G) that the denial of a certainty * is an impossibility, and con- versely ; (7 ) that the denial of a variable is a variable ; (8, 9) that the denial of a possibility is an uncertainty, and conversely. The first four factors are pretty evident ; the other five are less so. Some persons might reason, for example, that instead of (tt') w we should have (-n-'y ; that the denial of a possibility * is not merely an uncer- tainty but an impossibility. A single concrete example will show that the reasoning is not correct. The state- ment " It will rain to-morrow " may be considered a possibility ; but its denial " It will not rain to-morrow," though an uncertainty is not an impossibility. The formula {tt') u may be proved as follows : Let Q denote any state- ment taken at random out of a collection of statements containing certainties, impossibilities, and variables. To prove {ir') u is equivalent to proving Q 77 : (Q') M . Thus we get (Tr'y = Q- : (Q'f = Q e + Q e : (Q'r + (Q7 = Q e + Q e :Q e + Q e = e; for (Q l , y = Q e , and (Q / f = Q, e , whatever be the statement Q. To prove that (^y, on the other hand, is not valid, we have only to instance a single case of failure. Giving Q the same meaning as before, a case of failure is Q 8 ; for we then get, putting Q = 6 V = e 1 ;r 1i = (e/ i y = (e 1 € 2 y = , l2 * By the " denial of a certainty " is not meant (A e )', or its synonym A-*, which denies that a particular statement A is certain, but (A e )' or its synonym A' e , the denial of the admittedly certain statement A e . This state- ment Ae (since a suffix or subscriptum is adjectival and not predicative) assumes A to be certain ; for both A x and its denial A'x assume the truth of A* (see §§ 4, 5). Similarly, "the denial of a possibility" does not mean A-"' but AV, or its synonym (Att)', the denial of the admittedly possible statement An-. 16 SYMBOLIC LOGIC [§21 21. It may seem paradoxical to say that the pro- position A is not quite synonymous with A T , nor A' with A 1 ; yet such is the fact. Let A = It rains. Then A' = It does not rain ; A T = it is true that it rains ; and A' = it is false that it rains. The two propositions A and A T are equivalent in the sense that each implies the other ; but they are not synonymous, for we cannot always substitute the one for the other. In other words, the equivalence (A = A T ) does not necessarily imply the equivalence (p(A) = (p(A T ). For example, let (p(A) denote A e ; then (A_) = A< = e;=(e T y = r , for a variable is never a certainty, though it may turn out true in a particular case. Again, we get ^(A T ) = (A T ) e = (^) e = e« = e; for 6 T T means (0 T ) T , which is a formal certainty. In this case, therefore, though we have A = A T , yet (p(A) is not equivalent to (p(A T ). Next, suppose A denotes t , a variable that happens to be false in the case considered, though it is not false always. We get 0(A') = (A') e = A* = 0? = »7; for no variable (though it may turn out false in a parti- cular case) can be an impossibility. On the other hand, we get (A l ). It is a remarkable fact that nearly all civilised languages, in the course of their evolution, as if impelled by some unconscious instinct, have drawn §§ 21, 22] DEGREES OF STATEMENTS 17 this distinction between a simple affirmation A and the statement A T , that A is true ; and also between a simple denial A' and the statement A 1 , that A is false. It is the first step in the classification of statements, and marks a faculty which man alone of all terrestrial animals appears to possess (see §§ 22, 99). 22. As already remarked, my system of logic takes account not only of statements of the second degree, such as A" 13 , but of statements of higher degrees, such as A a/3y , A afiyS , &c. But, it may be asked, what is meant by state- ments of the second, third, &c, degrees, when the primary subject is itself a statement ? The statement A a/iy , or its synonym (A a/3 ) 7 , is a statement of the first degree as re- gards its immediate subject A a/3 ; but as it is synonymous with (A a ) Py , it is a statement of the second degree as regards A tt , and a statement of the third degree as regards A, the root statement of the series. Viewed from another standpoint, A a may be called^a revision of the judgment A, which (though here it is the root statement, or root judgment, of the series) may itself laave been a revision of some previous judgment here unexpressed. Similarly, (A")* 3 may be called a revision of the judgment A a , and so on. To take the most general case, let A denote any complex statement (or judgment) of the n tb degree. If it be neither a formal certainty (see § 109), like (a/3 : a) e , nor a formal impossibility, like (a/3 : af, it may be a material certainty, impossibility, or variable, according to the special data on which it is founded. If it follows necessarily from these data, it is a certainty, and we write A* ; if it is incompatible with these data, it is an impossi- bility, and we write A'' ; if it neither follows from nor is incompatible with our data, it is a variable, and we write A". But whether this new or revised judgment be A e or A^ or A", it must necessarily be a judgment (or state- ment) of the (w+l) th degree, since, by hypothesis, the statement A is of the w th degree. Suppose, for ex- ample, A denotes a functional statement 2, nor a formal impossibility, like 3<2, so that when we have no data out the mere arrangement of words, " Mrs. Brown is not at home," we are justified in calling this proposition, that is to say, this intelligible arrangement of words, a variable, and in asserting A 6 . If at the moment the servant tells me that " Mrs. Brown is not at home " I happen to see Mrs. Brown walking away in the distance, then / have fresh data and form the judgment A e , which, of course, implies A T . In this case I say that " A is certain" because its denial A' (" Mrs. Brown is at home ") would contradict my data, the evidence of my eyes. But if, instead of seeing Mrs. Brown walking away in the distance, I see her face peeping cautiously behind a curtain through a corner of a window, I obtain fresh data of an opposite kind, and form the judgment A v , which implies A'. In this case I say that " A is im- possible," because the statement represented by A, namely, " Mrs. Brown is not at home," this time contra- dicts my data, which, as before, I obtain through the medium of my two eyes. To say that the proposition A is a different 'proposition when it is false from what it is when it is true, is like saying that Mrs. Brown is a different person when she is in from what she is when she is out. 20 SYMBOLIC LOGIC [§25 CHAPTER IV 25. The following three rules are often useful: — (1) A'(A) = A'tf>(e). (2) A"4>(A) = A*0(>;). (3) A e i); for, as proved in § 19, we have always (e x = e y ), and (t] x = r] y ), whatever be the certainties e x and e and whatever the impossibilities rj x ana " %• Suppose, for example, that \j/ denotes A'B'C^C : AB + CA). = A £ B T, C fl e = A e B"C 9 ; so that the fourth or bracket factor of \j/- may be omitted without altering the value or meaning of \f/. In this operation we assumed the formulas (1) (ariz=r]); (2) (ae = a); (3) (*i + a==a). Other formulae frequently required are (4) (AB)' = A' + B'; (5) (A + B)' = A'B'; (6) e + A = e; (7) AA' = >7; (8) A + A' = e; (9) / = *,; (10) >/ = *; (11) A + AB = A; (12) (A + B)(A + C) = A + BC. We get §§ 26, 27] FORMULAE OF OPERATION 21 26. For the rest of this chapter we shall exclude the consideration of variables, so that A, A T , A* will be con- sidered mutually equivalent, as will also A', A', A''. On this understanding we get the formulas (1) A<£(A) = A(A', B) = AB'^)(»;, n)\ (7) AB'<£(A', B') = AB'<£(>/, e), and so on; like signs, as in A(p(A) or A / ^)(A / ), in the same letter, producing (p(e) ; and unlike signs, as in B'(p(B) or B^>(B / ), producing (>/)- The following ex- amples will show the working of these formulas : — Let /)=AB'C. A'B0'(A, B) = A / B(AB / C + A / BC / / = A / B( w C + eeC / y = A'B(C')' = A'BC. Next, let cp(B, D) = (CD' + CD + B / C / ) / . Then, B'D'0(B, D) = B'D'(CD' + CD + B'C')' = B / D , (Ce + C / >; + eC'/ = B'D'(C + C)' = B f D'e , = B'D'>i = > ] . The application of Formulas (4), (5), (11) of § 25 would, of course, have obtained the same result, but in a more troublesome manner. 27. If in any product ABC any statement-factor is implied in any other factor, or combination of factors, the implied factor may be omitted. If in any sum (i.e., 2% SYMBOLIC LOGIC [§§ 27, 28 alternative) A + B + C, any term implies any other, or the sum of any others, the implying term may be omitted. These rules are expressed symbolically by the two formulae — (1) (A:B):(AB = A); (2) (A:B):(A + B = B). By virtue of the formula (x : a)(x : /3) = x : a/3, these two formulae may be combined into the single formula — (3) (A:B):(AB = A)(A + B = B). As the converse of each of these three formula? also holds good, we get (4) A:B = (AB = A) = (A + B = B). Hence, we get A + AB = A, omitting the term AB, because it implies the term A; and we also get A(A + B) = A, omitting the factor A + B, because it is implied by the factor A. 28. Since A : B is equivalent to (AB = A), and B is a factor of AB, it follows that the consequent B may be called a factor of the antecedent A, in any implication A : B, and that, for the same reason, the antecedent A may be called a multiple of the consequent B. The equivalence of A : B and (A = AB) may be proved as follows : — (A = AB) = (A : AB)(AB : A) = ( A : AB)e = A:AB = (A:A)(A:B) = e(A:B) = A:B. The equivalence of A : B and (A + B = B) may be proved as follows : — (A + B = B) = (A + B:B)(B:A + B) = (A + B:B)e = A + B:B = (A:B)(B:B) = A:B. The formula? assumed in these two proofs are (x : aft) = (x : a)(x : /3), and a + /3 : x = (a : x)(fi : x), both of which may be considered axiomatic. For to assert that " If x is true, then a and /3 are both true " is equivalent to asserting that " If x is true a is true, and §§ 28, 29] REDUNDANT TERMS 23 if x is true /5 is true." Also, to assert that " If either a or |8 is true x is true " is equivalent to asserting that " If a is true x is true, and if /3 is true x is true." 29. To discover the redundant terms of any logical sum, or alternative statement. These redundant terms are easily detected by mere inspection when they evidently imply (or are multiples of) single co-terms, as in the case of the terms underlined in the expression a fty + a'y + aft/ + ft/, which therefore reduces to a!y + fiy'. But when they do not imply single co-terms, but the sum of two or more co-terms, they cannot generally be thus detected by inspection. They can always, however, be discovered by the following rule, which includes all cases. Any term of a logical sum or alternative may be omitted as redundant when this term multiplied by the denial of the sum of all its co-terms gives an impossible product rj ; but if the product is not rj, the term must not be omitted. Take, for example, the alternative statement CD' + C'D + B'C' + B'D'. Beginning with the first term we get CD'(C'D + B'C + B'D')' = CD'(w + B'» 7 + B'e)' = CD / (B / )' = BCD / . Hence, the first term CD' must not be omitted. Taking next the second term CD, we get CTKCD' + B'C/ + B'D'/ = C'D( w + B'e + B',,)' = C / D(BY=BC / D. Hence, the second term CD must not be omitted. We next take the third term B'C, getting B^CD 7 -I- C/D + B'D'/ = B'C'(>iI)' + eD + eD'/ = B'C'(D + D , / = B / C'>/ = >/. This shows that the third term B'C can be omitted as 24 SYMBOLIC LOGIC [§§ 29-31 redundant. Omitting the third term, we try the last term B'D', thus B'D'(CD' + CD)' = B'D'(Ce + C>,)' = B'D'C. This shows that the fourth term B'D' cannot be omitted as redundant if we omit the third term. But if we retain the third term B'C, we may omit the fourth term B'D', for we then get B'D'(CD' + CD + B'C 7 )' = B'D'(Ce + C'n + eC')' = B / D , (C + C / ) , =B / D / J? = i7. Thus, we may omit either the third term B'C, or else the fourth term B'D', as redundant, but not both. 30. A complex alternative may be said to be in its simplest form* when it contains no redundant terms, and none of its terms (or of the terms left) contains any re- dundant factor. For example, a + ab + m + m'n is reduced to its simplest form when we omit the redundant term ab, and out of the last term strike out the unnecessary factor m' . For a + ab = a, and m + m'n = m + n, so that the simplest form of the expression is a + m + n. (See § 31.) 31. To reduce a complex alternative to its simplest form, apply the formula (a + /3)' = a'/3' to the denial of the alternative. Then apply the formula (a/3/ = a' + ft' to the negative compound factors of the result, and omit the redundant terms in this new result. Then develop the denial of this product by the same formulae, and go through the same process as before. The final result will be the simplest equivalent of the original alternative. Take, for example, the alternative given in § 30, and denote it by (p. We get cp = a + ab + m + m'n = a + m + m!n. (p' = (a + m + m'n)' = a'm'(m'n)' = a'm'(m + nf) = a' m'n'.

= AB'C + AB(D + D') + A'B'(D' + D) = AB'C + ABe + A'B'e = AB'C' + AB + A'B'. <£ / =(AB / C / ) , (AB)'(A / B , ) / = (A' + B + C)(A' + B')(A + B) = (A' + B'C)( A + B) = A'B + AB'C. i p = «p')' = (A'B + AB'C)' = (A + B')(A' + B + C) = AB + AC' + A'B' + B'C. Applying the test of § 29 to discover redundant terms, we find that the second or fourth term (AC or B'C') may be omitted as redundant, but not both. We thus get (p = AB + A'B' + B'C = AB + AC + A'B', either of which may be taken as the simplest form of (p. 32. We will now apply the preceding principles to an interesting problem given by Dr. Venn in his " S} T m- bolic Logic" (see the edition of 1894, page 331). Suppose we were asked to discuss the following set of rules, in respect to their mutual consistency and brevity. a. The Financial Committee shall be chosen from amongst the General Committee. /3. No one shall be a member both of the General and Library Committees unless he be also on the Finan- cial Committee. y. No member of the Library Committee shall be on the Financial Committee. 26 SYMBOLIC LOGIC [§32 Solution. Speaking of a member taken at random, let the symbols F, G, L, respectively denote the statements " He will be on the Financial Committee," " He will be on the General Committee," " He will be on the Library Committee." Putting >;, as usual, for any statement that contradicts our data, we have a = (F:G); /3 = (GLF' : >,) ; 7 = (LF:#,); so that a/3 7 = (F:G)(GLF / :i?)(FL:i7) . = (FG':*i)(GLF':T,)(FL:r,) = FG' + GLF / + FL:>/. Putting (J> for the antecedent FG' + GLF' + FL, we get ^(F' + GXG' + L' + FXF' + I/) (See § 25, Formulae (4) and (5)) = (F' + GL')(G' + L' +■ F) = F'G' + F'L' + GL' + FGL' = F'G' + GL'; for the term FGL', being a multiple of the term GL', is redundant by inspection, and F / L / is also redundant, because, by § 29, F / L / (F , G / + GL')' = F'L'(eG' + G<)' = F'L'(G' + G)' = n . Hence, finally, we get (omitting the redundant term FL) <£ = (<£')' = (F'G' + GL')' = FG' + GL, and there I ore a/3y = (f>:ri= (FG' + GL : rf) = (F : G)(G : L')- That is to say, the three club rules, u, (3, 7 , may be replaced by the two simple rules F : G and G : L', which assert, firstly, that " If any member is on the Financial Committee, he must be also on the General Committee," which is rule a in other words ; and, secondly, that " If any member is on the General Committee, he is not to be on the Library Committee." §33] SOLUTIONS, ELIMINATIONS, LIMITS 27 CHAPTER V 33. From the formula (a : b)(c : d) = ah' + cd' : »/ the product of any number of implications can always be expressed in the form of a single implication, « + fi + 7 + &c : i], of which the antecedent is a logical sum (or alternative), and the consequent an impossibility. Suppose the im- plications forming the data of any problem that contains the statement x among its constituents to be thus re- duced to the form Ax + B,v' + C:tj, in which A is the coefficient or co-factor of x, B the co- efficient of x', and C the term, or sum of the terms, which contain neither x nor x'. It is easy to see that the above data may also be expressed in the form (B:asX«:A')(C!:9) J which is equivalent to the form (B^iA'XCm). When the data have been reduced to this form, the ^iven implication, or product of implications, is said to be solved with respect to x ; and the statements B and A' (which are generally more or less complex) are called the limits of x; the antecedent B being the strong * or superior limit ; and the consequent A', the weak or inferior limit. Since the * When from our data we can infer a: /3, but have no data for inferring |8 : u, we say that el is stronger than /3. For example, since we have AB:A:A + B, we say that AB is stronger than A, and A stronger than A+B. 28 SYMBOLIC LOGIC [§§ 33, 34 factor (B : x : A') implies (B : A'), and our data also imply the factor (C : >j), it follows that our data imply (B:A')(C:>,), which is equivalent to AB + C : *). Thus we get the formula of elimination (Ac + B,/ + C:>,):(AB + C:>;), which asserts that the strongest conclusion deducible from our data, and making no mention of x, is the im- plication AB + C : >/. As this conclusion is equivalent to the two-factor statement C^ABy, it asserts that the state- ment C and the combination of statements AB arc both impossible. 34. From this we deduce the solution of the follow- ing more general problem. Let the functional symbol ;) = (B : z : A')(B : A')(C : >/). § 34] SOLUTIONS, ELIMINATIONS, LIMITS 29 The expression represented by Az + Bz' + G is understood to have been reduced to its simplest form (see §§ 30, 31), before we collected the coefficients of z and z'. The limits of z are therefore B and A' ; and the result after the elimination of z is (B : A')(C : >/), which = AB + C : n . To find the limits of y from the implication AB + C : >/, we reduce AB + C to its simplest form (see §§ 30, 31), which we will suppose to be By + Ey' + F. We thus get, as in the previous expression in z, AB + C : r\ = By + E/ + F : r, = (E : y : D')(E : D')(F : >/)• The limits of y are therefore E and D', and the result after the successive elimination of z and y is (E : D')(F : >,), which = ED + F : >/. To find the limits of x from the implication ED + F : >/, we proceed exactly as before. We reduce ED + F to its simplest form, which we will suppose to be Gx + Hx + K, and get ED + F : n = Gx + Ha/ + K : r, = (H : x : G')(H : G')(K : >;). The limits of x are therefore H and G', and the result after the successive elimination of z, y, x is (H : G0(K : >/), which = HG + K : >,. The statements z, y, x having thus been successively eliminated, there remains the implication GH + K : }j, which indicates the relation (if any) connecting the remaining constituent statements a and b. Thus, we finally get (/) = (B : z : A')(E : // : D')(H : x : G')(GH + K : ,,). in which A and B do not contain z (that is, they make no mention of z) ; D and E contain neither z nor y ; G and H contain neither z nor y nor x ; and the expression K 30 SYMBOLIC LOGIC [§§ 34, 35 in the last factor will also be destitute of (i.e., will make no mention of) the constitutents x, y, z, though, like G and H, it may contain the constituent statements a and b. In the course of this process, since >) : a and a : e are certainties whatever the statement a may be (see § 18), we can supply >/ for any missing antecedent, and e for any missing consequent. 35. To give a concrete example of the general prob- lem and solution discussed in § 34, let (p denote the data e : xyza + xyb +xy z +y z a . We get, putting (p for these data,

= (y'+ ab)z + (]jy)z + {x'y + ax') : n putting A for y' + ab, B for by, and C for x'y + ax'. As in § 34, we get (B:s:A')(AB + C:>7), so that the limits of z are B and A', and the result after the elimination of z is AB + C : »/. Substituting their values for A, B, C, this last implication becomes {ab + ,c)y + ax' : ?/, which we will denote by *Dy + Ey' + F : n, putting J) for ab + x, E for n, and F for ax. Thus we get (f> = (B : z : A')(Dy + E/ + F : >/) = (B :z : A0(E : y : D')(ED + F : »;). Having thus found the limits {ix., the weakest ante- §§35,36] SOLUTIONS, ELIMINATIONS, LIMITS 31 cedents and strongest consequents) of z and y, we proceed to find the limits of x from the implication ED + F : n, which is the strongest implication that remains after the elimination of z and y. Substituting for D, E, F the values which they represent, we get DE + F : n = {ah + J)n + «J : n = Gx + BJ + K : n, in which G, H, K respectively denote >/, a, n- We thus get DE + F : >i = (H : x : G')(HG + K : tj) ; so that our final result is <$> = (B : z : A')(E : // : D')(H : a; : G0(HG + K : i,) = (by : z : a'y + b f y){n : y : rt ; ;« + b'x){a : £C : e)(>/ : »/) = (/>// :z:a'y + b'y)(y : a'x + b'x)(a : x). To obtain this result we first substituted for A, B, D, E, G, H, K the values we had assigned to them ; then we omitted the redundant antecedent >/ in the second factor, the redundant consequent e in the third factor, and the redundant certainty (»/ : »/), which constituted the fourth factor. The fact that the fourth factor (HG + K:>/) reduces to the form (n : rj), which is a formal certainty (see § 18), indicates that, in this particular problem, nothing can be implicationally affirmed in terms of a or b (without mentioning either x or y or z) except formal certainties such as (ab : a), (aa f : >;), ab(a + b') : >i, &c, which are true always and independently of our data (p. 36. If in the preceding problem we had not reduced the alternative represented by As + Bz' + C to its sim- plest form (see §§ 30, 31), we should have found for the inferior limit or consequent of z, not a'y + b'y, but x(a'y + b'y). From this it might be supposed that the strongest conclusion deducible from z (in conjunction with, or within the limits of, our data) was not A' but xk'. But though xh! is formally stronger than A', that 32 SYMBOLIC LOGIC [§§ 36-38 is to say, stronger than A' token we have no data but our definitions, here we have other data, namely,

= (z:y: b'x + xz)(z + a : x){z : a' + b'). 38. The preceding method of finding what I call the " limits " of logical statements is closely allied to, and was suggested by, my method (published in 1877, in the Proc. of the Lond. Math. Soc.) for successively finding the limits of integration for the variables in a multiple integral (see § 138). In the next chapter the method will be applied to the solution (so far as solution is pos- sible) of Professor Jevons's so-called " Inverse Problem," which has given rise to so much discussion, not only among logicians but also among mathematicians. §39] JEVONS'S "INVERSE PROBLEM" 33 CHAPTER VI 39. Briefly stated, the so-called "inverse problem" of Professor Jevons is this. Let tp denote any alternative, such as abc + a'bc + aV V '. It is required to find an im- plication, or product of implications,* that implies this alternative. Now, any implication whatever (or any product of implications) that is equivalent to f : rj), &c, must necessarily imply the given alternative cp, so that the number of possible solutions is really unlimited. But though the problem as enun- ciated by Professor Jevons is thus indeterminate, the number of possible solutions may be restricted, and the problem rendered far more interesting, as well as more useful and instructive, by stating it in a more modified form as follows : — Let cp denote any alternative involving any number of constituents, a, b, c, &c. It is required to resolve the implication e : cp into factors, so that it will take the form (M : a : N)(P : b : Q)(R : c : S), &c, in which the limits M and N (see § 33) may contain b, c, &c, but not a; the limits P and Q may contain c, d, &c, but neither a nor b ; the limits R and S may contain d, e, &c, but neither a nor b nor c ; and so on to the last constituent. When no nearer limits of a con- stituent can be found we give it the limits >; and e ; the former being its antecedent, and the latter its con- sequent (see §§ 18, 34). * Professor Jevons calls these implications "laws," because he arrives at them by a long tentative inductive process, like that by which scien- tific investigators have often discovered the so-called " laws of nature " (see§ 112). C 34 SYMBOLIC LOGIC [§39 As a simple example, suppose we have * (p = abc + a'bc + ab'c', the terms of which are mutually exclusive. Reducing

/. This alternative equivalent of cp' may be simplified (see § 31) by omitting either the first or the third term, but not both ; so that we get e : (p = b'c + a'c' + be' : rj = a'b' + b'c + be : 17. Taking the first equivalent of e : ; : c), and (c : e) are all formal certainties, they may be omitted as factors, so that we get e :

denotes an alternative. In §§ 34, 35 the symbol denotes a given implication, which may take either such a form ase:a + /3 + 7 + &c. , or as a + /3 + 7 + &c. : 7/. §§39,40] JEVONS'S "INVERSE PROBLEM" 35 We will now take the second equivalent of e : ] = (b' + c': a)(c = b) = {1/ : a)(c' : a)(c = b), in which either the factor (b' : a) or the factor {c : a) may be omitted as redundant, but not both. For though the factor (c = b) alone neither implies (b' : a) nor (/), or its equivalent (b' : a), in the result ; and the omission of the term a'c' in the alternative leads, in like manner, to the omission of the factor (a'c' : rf), or its equivalent (c' : a), in the result. 40. I take the following alternative from Jevons's "Studies in Deductive Logic" (edition of 1880, p. 254, No. XII.), slightly changing the notation, abed + abe'd + ab'cd' + a'bed' + a'b'c'd'. Let (p denote this alternative, and let it be required to 36 SYMBOLIC LOGIC [§40 find successively the limits of a, b y c, d. In other words, we are required to express e : ,, S = e, T = r,, U = 6. Omitting the last two factors R : c : S and T : d : U because they are formal certainties, we get e : (p = (d + be' + b'c : a : bd + b'c){d :b:c + d). A glance at the given alternative <£> will verify this result, which asserts ( 1 ) that whenever we have either d or be' or b'c, then we have a ; (2) that whenever we have a, then we have either bd or b'c ; (3) that whenever we have d, then we have b ; (4) that whenever we have b, then we have either c or d; and (5) that from the implication e -. (p we can infer no relation connecting c with c£ without making mention of a or b ; or, in other words, that c cannot be expressed in terms of d alone, since the factor >/ : c : e is a formal certainty and therefore true from our definitions alone apart from any special data. The final factor is only added for form's sake, for it must always have >/ for antecedent and e for consequent. In other words, when we have n constituents, if x be the n th or last in the order taken, the last factor must necessarily be >; : x : e, and therefore a formal certainty which may be left understood. Others of the factors may (as in the case of n : c : e here) turn out to be formal certainties also, but not necessarily. We have found the limits of the constituents a, b, c, d, taken successively in alphabetic order. If we take the reverse order d, c, b, a, our result will be e : (p = (ab + ac' + bd : d : ab)(ab' + a'b : c : a + b), §§ 40, 41] ALTERNATIVES 37 omitting the third and fourth factors >) : b : e and >; : a : e because they are formal certainties. There is one point in this result which deserves notice. Since every double implication a : x : (3 always implies a : /3, it follows that (in the first bracket) ab + ac' + he implies ab. Now, the latter is formally stronger than the former, since any statement x is formally stronger than the alternative x + y. But the formally stronger statement x, though it can never be weaker, either formally or materially, than x + y, may be materially equivalent to x + y; and it must be so whenever y materially (i.e., by the special data of the problem) implies x, but not otherwise. Let us see whether our special data, in the present case, justifies the inferred implication ab + ac + be : ab. Call this implica- tion \J/-. By virtue of the formula a + (3 + y : x = (a : x) (/3 : x)(y : x), we get (putting ab for a and for x, ac for (3, and be for y) \|z = (ab : al)){ac' : ab)(bc : ab) = e(ac : ab)(bc : ab) = (ac : a)(ac : b)(bc : a)(bc : b) = e(ac' : b)(bc' : a)e = (ac : b)(bc : a). This asserts that (within the limits of our data in this problem) whenever we have ac we have also b, and that whenever we have be we have also a. A glance at the given fully developed alternative

7 : a : b'c + ce')(>; : b : d' + c)(d : c : e)(e : d : e)(r) :e:e). This is the final result with every limit expressed. Omit- §§ 42-44] UNRESTRICTED FUNCTIONS 39 ting the superior limit >/ and the inferior limit e wherever they occur, and also the final factor >j : c : e because it is a formal certainty (see § 18), we get e : \Jr = (a : &'c + ce')(b : ri' + e)(d : c)(e : rf). Suppose next we arc required to find the limits in the order d, e, c, a. b. Our final result in this case will be e : y$r = (e : d : &'c + 7 : a : e)(>/ :b:e) = (e : d : b'c + ce)(e : a'c + b'c)(a : c). 43. When an alternative

/ : a : e ; that is to say, the order changes, and both, being certainties, may be omitted. It will be observed that when the order of limits is prescribed, the exact solution is prescribed also : no two persons can (without error) give different solu- tions, though they may sometimes appear different in form (see §§39, 40). CHAPTER VII 44. Let ~F u (x, y, z), or its abbreviated synonym F„, re- present the functional proposition F(x, y, z), when the values or meanings of its constituents x, y, z are unre- stricted ; while the symbol F r (x, y, z), or its abbreviated synonym F r , represents the functional proposition F( l i , ) y; z) when the values of x, y, z are restricted. For example, if x can have only four values. x y x, 2 x. A , x 4 ; y 40 SYMBOLIC LOGIC [§§ 44, 45 the four values y , y 2 , y z , y ; and z the three values z„ z v z. s ; then we write F r , and not F M . But if each of the three symbols x, y, z may have any value (or meaning) whatever out of the infinite series x v x 2 , x 3 , &c, y v y 2 , y 3 , &c., z v «„, z , &c. ; then we write F M , and not F r The suffix r is intended to suggest the adjective restricted, and the suffix u the adjective unrestricted. The symbols F e , F n , F e , as usual, assert respectively that F is certain, that F is impossible, that F is variable ; but here the word certain is understood to mean true fur all the admissible values of .<•, y, z in the functional statement F(x, y, z) ; impossible means false for every admissible value of x, y, z in the statement F(x, y, z); and variable means neither certain nor impossible. Thus F e asserts that Fix, y, z) is neither always true nor always false ; it is synonymous with F _e F~", which is synonymous with (F^F"/. 45. From these symbolic conventions we get the three formulae : (1)(F-F<); (2)(F? t :F?.); (3)(F?:F? f ); but the converse (or inverse) implications are not neces- sarily true, so that the three formulae would lose their validity if we substituted the sign of equivalence ( = ) for the sign of implication (:). The first two formulae need no proof; the third is less evident, so we will prove it as follows. Let

2 to be a certainty, so that we get the deductive sorites e: 2 :(F-F;:)(F£:F?) : (F; e : F-)(F7 : 1?) [for a : /3 = /3' : «'] : (F-F7 : FfFJ) [for (A : a)(B : b) : (AB : ah)] : (F*: F*) [for A-'A^ = A e , by definition]. This proves the third formula

/)', (z : >/)' (which are respectively synony- mous with x* 1 , y' 1 *, z" ) must always be considered to form part of our data, whether expressed or not ; and their denials, (x : »/), (y : n), (« : »?), must be considered impossible. With these conventions we get — (A) Every (or all) X is Y = S x : S Y = (x : y) = {xy'f (0) Some X is not Y = (S x : S Y / = (x : y)' = (xy'y (E) No X is Y = S x : S- Y = x : y = (xyY (1) Some X is Y = (S x : S" T )' = (x : y')' = {xyj*. § 50] GENERAL AND TRADITIONAL LOGIC 45 In this way we can express every syllogism of the traditional logic in terms of x, y, z, which represent three propositions having the same subject S, but different predicates X, Y, Z. Since none of the propositions x, y, z (as already shown) can in this case belong to the class r\ or e, the values (or meanings) of x, y, z are restricted. Hence, every traditional syllogism expressed in terms of x, y, z must belong to the class of restricted functional statements F r (x, ?/, z), or its abbreviated synonym F r) and not to the class of unrestricted functional statements FJx, y, z), or its abbreviated synonym F w , as this last statement assumes that the values (or meanings) of the propositions x, y, z are wholly unrestricted (see § 44). The proposition F w (x, y, z) assumes not only that each constituent statement x, y, z may belong to the class >/ or e, as well as to the class 9, but also that the three statements x, y, z need not even have the same subject. For example, let F (x, y, z), or its abbreviation F, denote the formula (x : y)(y : z) : (x : z). This formula asserts that " If x implies y, and y implies z, then x implies z." The formula holds good whatever be the statements x, y, z ; whether or not they have (as in the traditional logic) the same subject S ; and whether or not they are certainties, impossibilities, or variables. Hence, with reference to the above formula, it is always correct to assert F 6 whether F denotes F M or F r . When x, y, z have a common subject S, then F e will mean F^. and will denote the syllogism of the traditional logic called Barbara ;* whereas when x, y, z are wholly unrestricted, F e will mean F^ and will therefore be a more general formula, of which the traditional Barbara will be a particular case. * Barbara asserts that " If every X is Y, and every Y is Z, then every X is Z," which is equivalent to (S x : S v ) (S v : S z ) : (S x : S z ). 46 SYMBOLIC LOGIC [§§50,51 But now let F, or Y(x, y, z), denote the implication (y : z)(y : x) : (x : z')'. It' we suppose the propositions x, y, z to be limited by the conventions of §§46, 50, the traditional syllogism called Darapti will be represented by F r and not by F M . Now, by the first formula of § 45, we have F,' ( : F, 6 ., and, consequently, F; 6 : F~ e , but not necessarily F~ e : F; e . Thus, if F u be valid, the traditional Darapti must be valid also. We find that F w is not valid, for the above implication represented by F fails in the case f(xzy, as it then becomes (>1 : z){ri : x) : (xz)~ v , which is equivalent to ee : if, and consequently to e : »/, which = {er/f = (ee) 7 ' = rj. But since (as just shown) F; 6 does not necessarily imply F; 6 , this discovery docs not justify us in concluding that the traditional Darapti is not valid. The only case in which F fails is y\xz) n , and this case cannot occur in the limited formula F r (which here repre- sents the traditional Darapti), because in F r the pro- positions x, y, z are always variable and therefore possible. In the general and non-traditional implication F M , the case x yi y v z r ', since it implies [piiczf, is also a case of failure; but it is not a case of failure in the traditional logic. 51. The traditional Darapti, namely, "If every Y is Z, and every Y is also X, then some X is Z," is thought by some logicians (I formerly thought so myself) to fail when the class Y is non-existent, while the classes X and Z are real but mutually exclusive. But this is a mistake, as the following concrete example will show. Suppose we have Y = (0 1( 2 , ;i ), Z = (e v e 2 , e 3 ), X = (« 4> e a , e 6 ). Let P denote the first premise of the given syllogism, Q the second, and R the conclusion. We get P = Every Y is Z = > h ; Q = Every Y is X = >; 2 ; and R = Some X is Z = >/ 3 ; three statements, >/ r »/ 2 , »/ 3 , §§51,52] TRADITIONAL SYLLOGISMS 17 each of which contradicts our data, since, by our data in this case, the three classes X, Y, Z arc mutually exclusive. Hence in this case we have PQ : R = ( V / 2 : >i,) = (>i, : *1 3 ) = {%n^ = e 1 ; so that, when presented in the form of an implication, Darapti does not fail in the case supposed. (But see § 52.) 52. Startling as it may sound, however, it is a demonstrable fact that not one syllogism of the traditional logic — neither Darapti, nor Barbara, nor any other — is valid in the form in which it is usually presented in our text-books, and in which, I believe, it has been always presented ever since the time of Aristotle. In this form, every syllogism makes four positive assertions : it asserts the first premise ; it asserts the second ; it asserts the conclusion ; and, by the word ' therefore,' it asserts that the conclusion follows necessarily from the premises, i.e. that if the premises be true, the conclusion must be true also. Of these four assertions the first three may be, and often are, false ; the fourth, and the fourth alone, is a formal certainty. Take the standard syllogism Barbara. Barbara (in the usual text-book form) says this : " Every A is B ; every B is C ; therefore every A is C." Let \f/(A, B, C) denote this syllogism. If valid it must be true whatever values (or meanings) we give to A, B, C. Let A— ass, let B = bear, and let C = camel. If \J/(A, B, C) be valid, the following syllogism must therefore be true : " Every ass is a bear ; every bear is a camel; therefore, every ass is a camel." Is this concrete syllogism really true ? Clearly not ; it contains three false statements. Hence, in the above form, Barbara (here denoted by \|/) is not valid ; for have we not just adduced a case of failure ? And if we give random values to A, B, C out of a large number of classes taken haphazard (lings, queens, sailors, doctors, stones, cities, horses, French, Europeans, white things, black things, &c, &c), we shall find that the cases in which this syllogism will 48 SYMBOLIC LOGIC [§§ 52, 53 turn out false enormously outnumber the cases in which it will turn out true. But it is always true in the following form, whatever values we give to A, B, C : — " If every A is B, and every B is C, then every A is C." Suppose as before that A = ass, that B = bear, and that C = camel. Let P denote the combined premises, " Every ass is a bear, and every bear is a camel," and let Q denote the conclusion, " Every ass is a camel." Also, let the symbol .'. , as is customary , denote the word therefore. The first or therefore -form asserts P .". Q, which is equivalent* to the two-factor statement P(P:Q); the second or if-form asserts only the second factor P : Q. The therefore-form vouches for the truth of P and Q, which are both false ; the if-form vouches only for the truth of the implication P : Q, which, by definition, means (PQ'y. and is a formal certainty. (See § 10.) 53. Logicians may say (as some have said), in answer to the preceding criticism, that my objection to the usual form of presenting a syllogism is purely verbal ; that the premises are always understood to be merely hypothetical, and that therefore the syllogism, in its general form, is not supposed to guarantee either the truth of the premises or the truth of the conclusion. This is virtually an admission that though (P •'• Q) is asserted, the weaker statement (P : Q) is the one really meant — that though logicians assert " P therefore Q," they only mean " If P then Q." But why depart from the ordinary common- sense linguistic convention ? In ordinary speech, when we say " P is true, therefore Q is true," we vouch for the truth of P ; but when we say " If P is true, then Q is true," we do not. As I said in the Athenmum, No. 3989 : — " Why should the linguistic convention be different in logic ? . . . Where is the necessity ? Where is the advantage 1 Suppose a general, whose mind, during his past university days, had been over-imbued with the traditional logic, were in war time to say, in speaking of an untried and possibly innocent prisoner, ' He is a spy ; therefore he * I pointed out this equivalence in Mind, January 1880. §§ 53, 54] TRADITIONAL SYLLOGISMS 49 must be shot,' and that this order were carried out to the letter. Could he afterwards exculpate himself by saying that it was all an un- fortunate mistake, due to the deplorable ignorance of his subordinates ; that if these had, like him, received the inestimable advantages of a logical education, they would have known at once that what he really meant was ' If he is a spy, he must be shot'? The argument in defence of the traditional wording of the syllogism is exactly parallel." It is no exaggeration to say that nearly all fallacies are due to neglect of the little conjunction, If. Mere hypotheses are accepted as if they were certainties. CHAPTER VIII 54. In the notation of § 50, the following are the nine- teen syllogisms of the traditional logic, in their usual order. As is customary, they are arranged into four divisions, called Figures, according to the position of the " middle term " (or middle constituent), here denoted by y. This constituent y always appears in both pre- mises, but not in the conclusion. The constituent z, in the traditional phraseology, is called the " major term," and the constituent x the " minor term." Similarly, the premise containing z is called the " major premise," and the premise containing x the " minor premise." Also, since the conclusion is always of the form " All X is Z," or " Some X is Z " or " No X is Z," or " Some X is not Z," it is usual to speak of X as the ' subject ' and of Z as the ' predicate.' As usual in text-books, the major premise precedes the minor. Figure 1 Barbara =(y Celarent = (y Darii = (y Ferio = (y z)(x :y):(x:z) z'){x : y) : (x : z) z)(x : y')' : (x : z 1 )' z')(, : y')' : (x : z) f D 50 SYMBOLIC LOGIC [§ 54 Figure 2 Cesare = (z : y'){x : y) : (x : z*) Camestres = (: : y\x : y') : (x : z) Festino = («:/)(« :/)':(*: z)' Baroko = (a : y)(x : y)' : (a: : z)' Figure 3 Darapti = (y : z)(y : x) : (x : z')' Disamis = (y : z , )\y : x) : (x : z')' Datisi = (y : z)(y : a/)' : (a; : z'f Felapton = (y : z')(y : «) : (x : z)' Bokardo = ( y : z)\y : x) : {x : z)' Ferison = (y : z'){y : x')' : (x : z)' Figure 4 Bramantip = (z : y)(y : x) : (x : z 1 )' Camenes = (z : y)(y : x') : (x : z') Dismaris = {z : y')\y : x) : (x : z)' Fesapo = (z : y')(y : x) : (x : z)' Fresison = (z : y')(y : x')' : (x : z)' Now, let the symbols (Barbara),,, (Celarent) M , &c. ; denote, in conformity with the convention of § 44, these nineteen functional statements respectively, when the values of their constituent statements x. y, z are unrestricted ; while the symbols (Barbara),., (Celarent),., &c, denote the same functional statements when the values of x, y, z are restricted as in § 50. The syllogisms (Barbara),., (Celarent),., &c, with the suffix r, indicating restriction of values, are the real syllogisms of the traditional logic ; and all these, without exception, are valid — within the limits of the understood restriction*. The nineteen syllogisms of general logic, that is to say, of the pure logic of statements, §§ 54-5 0] GENERAL LOGIC 51 namely, (Barbara),,, (Celarent),,, &c., in which x, y, z are a n restricted in values, are more general than and imply the traditional nineteen in which x, y, z are restricted as in § 5 ; and four of these unrestricted syllogisms, namely, (Darapti),,, (Felapton),,, (Bramantip),,, and (Fesapo),,, fail in certain cases. (Darapti) w fails in the case y 7 '(".:)\ (Felapton),, and (Fesapo) w fail in the case y%ez / ) TI , and (Bramantip ) u fails in the case &(x'yf. 55. It thus appears that there are two Barbaras, two Celarents, two Dai'ii, &c, of which, in each case, the one belongs to the traditional logic, with restricted values of its constituents x, y, z; while the other is a more general syllogism, of which the traditional syllogism is a particular case. Now, as shown in § 45, when a general law F w , with unrestricted values of its constituents, implies a general law F,., with restricted values of its constituents, if the former is true absolutely and never fails, the same may be said of the latter. This is expressed by the formula F„ : F*. But an exceptional case of failure in F„ does not necessarily imply a corresponding case of failure in F,. ; for though F, e , : F;. is a valid formula, the implication F M e : F; e (which is equivalent to the converse implica- tion F e r : F e ,) is not necessarily valid. For example, the general and non-traditional syllogism (Darapti),, implies the less general and traditional syllogism (Darapti),.. The former fails in the exceptional case y\xzj i ; but in the traditional syllogism this case cannot occur because of the restrictions which limit the statement y to the class 6 (see § 50). Hence, though this case of failure necessitates the conclusion (Darapti);;*, we cannot, from this conclusion, infer the further, but incorrect, conclusion (Darapti); 6 . Similar reasoning applies to the unrestricted non-traditional and restricted traditional forms of Felapton, Bramantip, and Fesapo. 56. All the preceding syllogisms, with many others not recognised in the traditional logic may. by means of the formulae of transposition a : j3 = /3 r : a! and a/3' \y' = ay:f$, 52 SYMBOLIC LOGIC [§§ 56, 57 be shown to be only particular cases of the formula (x'.y)(y:z):(x:z), which expresses Barbara. Two or three examples will make this clear. Lut j) : n = (y : xz){xz : >/) : (y : »/) ; for, in the traditional logic, (y:rf) = »/, since, by the con- vention of S 5 0, y must always be a variable, and, there- lore, always possible. Thus, finally (Darapti),. = (f)(y, xz, n). Lastly, take (Bramantip),.. We get (Bramantip), = (z : y)(y : x) : (x : z")' = (z : y)(y : x ){x : z) : i] = (z : y){z : x')(y : x) : n = (z : yx)(y : x) : >i = (z : yx')(yx' : >i) :i] = (z: yx')(yx : r,) : (z : //) ; for, in the traditional logic, (z:r]) = r), since z must be a variable and therefore possible. Hence, finally, we get (Bramantip),. = /). 57. By similar reasoning the student can verify the following list (see §§ 54-56): (p(x, y, z) = Barbara ; (p(x, y, z') = Celarent = Cesare : ip(y y z , x') = Darii = Datisi ; (p{x, z, y') = Ferio = Festino = Ferison = Fresison ; (p(z, y, x) = Camestres = Camenes ; §§57-59] TESTS OF SYLLOGISTIC VALIDITY 53 (y, xz, »/) = (Darapti),. ; (p(y, xz, »/) = (Felapton) r = (Fesapo) r ; ?. When thus trans- formed the validity of AB' : C, that is, of AC : B, may be tested in the same way as the validity of AB : C. The test is easy. Suppose the conclusion C to be x : z, in which z may be affirmative or negative. If, for example, z — He is a soldier; then z' = He is not a soldier. But it z—He is not a soldier; then z' — He is a soldier. The conclusion C being, by hypothesis, x:z, the syllogism AB : C, if valid, becomes (see § 11) either (x :y:z):(x: z), or else {x : y' : z) : (x : z), in which the statement y refers to the middle class (or " term ") Y, not mentioned in the conclusion x : z. If any supposed syllogism AB : C cannot be reduced to either of these two forms, it is not valid ; if it can be reduced to either form, it is valid. To take a concrete example, let it be required to test the validity of the following implicational syllogism : If no Liberal approves of Protection, though some Liberals approve of fiscal Retaliation, it follows that some person or persons who approve of fiscal Retaliation do not approve of Protection. Speaking of a person taken at random, let L = He is a Liberal; let P = He approves of Protection; and let R = He approves of fiscal Retaliation. Also, let Q denote the syllogism. We get Q=(L:P')(L:R'/:(R:P)'. To get rid of the non-implications, we transpose them (see § 56) and change their signs from negative to affirmative, thus transforming them into implications. This transposition gives us Q = (L:P , )(R:P):(L:R'). §§59, 00] TESTS OF SYLLOGISTIC VALIDITY 55 Since in this form of Q, the syllogistic propositions are all three implications (or " universale "), the combination of premises, (L : P')(R:P), must (if Q be valid) be equi- valent either to L : P : R' or else to L : P' : R' ; in which P is the letter left out in the new consequent or conclusion L : R'. Now, the factors L : P and P : R' of L : P : R' are not equivalent to the premises L : P' and R : P in the second or transposed form of the syllogism Q ; but the factors L : P' and P' : R' (which is equivalent to R : P) of L : P' : R' are equivalent to the premises in the second or transformed form of the syllogism Q. Hence Q is valid. As an instance of a non-valid syllogism of the form AB : C, we may give (x:y')(y:z'):(x:z'); for since the y's in the two premises have different signs, the one being negative and the other affirmative, the combined premises can neither take the form x:y:z nor the form x : y' : z' , which are respective abbreviations for (x>\y){y:z) and (x t y')(y' : /). The syllogism is there- fore not valid. 00. The preceding process for testing the validity of syllogisms of the forms AB : C and AB' : C apply to all syllogisms without exception, whether the values of their constituents x, y, z be restricted, as in the traditional logic, or unrestricted, as in my general logic of state- ments. But as regards syllogisms in general logic of the form AB : C (a form which includes Darapti, Felapton, Fesapo, and Bramantip in the traditional logic), with two implicational premises and a non-implicational con- clusion, they can only be true conditionally ; for in general logic (as distinguished from the traditional logic) no syllogism of this type is a formal certainty. It therefore becomes an interesting and important problem to deter- 56 SYMBOLIC LOGIC [§§ GO, 61 mine the conditions on which syllogisms of this type can be held valid. We have to determine two things, firstly, the iveakest premise (see § 33, footnote) which, when joined to the two premises given, would render the syllo- gism a formal certainty ; and, secondly, the weakest con- dition which, when assumed throughout, would render the syllogism a formal impossibility. As will be seen, the method we are going to explain is a general one, which may be applied to other formulae besides those of the syllogism. The given implication AB : C is equivalent to the implication ABC : y, in which A, B, C are three impli- cations (see § 59) involving three constituents x, y, z. Eliminate successively x, y, z as in § 34, not as in finding the successive limits of x, y, z, but taking each variable independently. Let a denote the strongest con- clusion deducible from ABC and containing no reference to the eliminated x. Similarly, let /3 and y respectively denote the strongest conclusions after the elimination of y alone (x being left), and after the elimination of z alone (x and y being left). Then, if we join the factor a or /3' or y' to the premises (ix. the antecedent) of the given implicational syllogism AB : C, the syllogism will become a formal certainty, and therefore valid. That is to say, ABa' : C will be a formal certainty ; and so will AB/3' : C and AB?' : C. Consequently, AB (a +fi'+ y) : C is a formal certainty ; so that, on the one hand, the weakest premise needed to be joined to AB to render the given syllogism AB : C valid {i.e. a formal certainty) is the alternative a' + fi' + y', and, on the other, the weakest datum needed to make the syllogism AB : C a formal impossibility is the denied of a + /?' + y , that is, a(3y. 61. Take as an example the syllogism Darapti. Here we have an implication AB : C in which A, B, C respectively denote the implications (y : x), (y : z), (x : z). By the method of § 34 we get ABC = yx + yz' + xz : >; = M* + N./ + P : r,, say, §61] CONDITIONS OF VALIDITY 57 in which M, N, P respectively denote the co-factor of x, the co-factor of %', and the term not containing x. The strongest consequent not involving x is MN + P : *), in which hero M = z, N = y, and P = yz' ; so that we have MN + P : n = zy + yz' : n = //( - + z') : 1 = ye : >/ = y : v\. Thus we get a = y: >/, so that the premise required when we eliminate x is (y : >;/ ; and therefore ( r .x)(y.z)(y.ri) f -(x:z , ) t should be a formal certainty, which is a fact ; for, getting rid of the non-implications by transposition, this complex implication becomes (y : x)(y : z){x : z) : (y : 17), which = (y : xz)(xz : n) ■ (y ■ n) ; and this is a formal certainty, being a particular case of the standard formula (f)(x, y, z), which represents Barbara both in general and in the traditional logic (see § 55). Eliminating y alone in the same manner from AB : C, we find that (3 = xz : *i = x : z' ; so that the complex implication {y:x)(y:z)(x:zy:(x:z')' should be a formal certainty. That it is so is evident by inspection, on the principle that the implication PQ : Q, for all values of P and Q, is a formal certainty. Finally, we eliminate z, and find that y = y: n- This is the same result as we obtained by the elimination of x, as might have been foreseen, since x and z are evidently inter- changeable. Thus we obtain the information sought, namely, that « / + /3 / + 7 / , the weakest premise to be joined to the premises of Darapti to make this syllogism a formal certainty in general logic is (y : >/) / + (xz : >/)' + (// : •?)', which = y*> + (xz)- 1 " ; 58 SYMBOLIC LOGIC [§§ 61, 62 and that a/3y, the Aveakest presupposed condition that would render the syllogism Darapti a logical impossi- bility, is therefore / ,p + (,,.,)--; j ' t w hich = y\ocz)\ Hence, the Darapti of general logic, with unrestricted values of its constituents x, y, z, fails in the case y\xzy ; but in the traditional logic, as shown in § 50, this case cannot arise. The preceding reasoning may be applied to the syllogisms Felapton and Fesapo by simply chang- ing z into z! . Next, take the syllogism Bramantip. Here we get ABC = yx' + zy' + xz : >i, and giving u, /3, y the same meanings as before, we get a = z r >, /3 = z\ y = (x'y)\ Hence, a^y — z\xyf, and a' + ft' + y' = z~ n + (£c'y)~ r '. Thus, in general logic, Bra- mantip is a formal certainty when we assume z~ v + {x'yY*, and a formal impossibility when we assume &{x'yf ; but in the traditional logic the latter assumption is inadmis- sible, since z v is inadmissible by § 50, while the former is obligatory, since it is implied in the necessary assump- tion 2f. 62. The validity tests of the traditional logic turn mainly upon the question whether or not a syllogistic ' term ' or class is ' distributed ' or ' undistributed.' In ordinary language these words rarely, if ever, lead to any ambiguity or confusion of thought ; but logicians have somehow managed to work them into a perplexing tangle. In the proposition " All X is Y," the class X is said to be ' distributed,' and the class Y ' undistributed.' In the proposition " No X is Y," the class X and the class Y are said to be both ' distributed.' In the pro- position " Some X is Y," the class X and the class Y are said to be both 'undistributed.' Finally, in the pro- position " Some X is not Y," the class X is said to be ' undistributed,' and the class Y ' distributed.' § 6 2] < DISTRIBUTED — < UNDISTRIBUTED , 59 Let us examine the consequences of this tangle of technicalities. Take the leading syllogism Barbara, the validity of which no one will question, provided it bo expressed in its conditional form, namely, " If all Y is Z, and all X is Y, then all X is Z." Being, in this form (see § 52), admittedly valid, this syllogism must hold good whatever values (or meanings) we give to its con- stituents X, Y, Z. It must therefore hold good when X, Y, and Z are synonyms, and, therefore, all denote the same class. In this case also the two premises and the conclusion will be three truisms which no one would dream of denying. Consider now one of these truisms, say " All X is Y." Here, by the usual logical convention, the class X is said to be ' distributed,' and the class Y 1 undistributed.' But when X and Y are synonyms they denote the same class, so that the same class may, at the same time and in the same proposition, be both ' dis- tributed' and 'undistributed.' Does not this sound like a contradiction ? Speaking of a certain concrete collec- tion of apples in a certain concrete basket, can we con- sistently and in the same breath assert that " All the apples are already distributed " and that " All the apples are 'still undistributed " ? Do we get out of the dilemma and secure consistency if on every apple in the basket we stick a ticket X and also a ticket Y ? Can we then con- sistently assert that all the X apples are distributed, but that all the Y apples are undistributed ? Clearly not ; for every X apple is also a Y apple, and every Y apple an X apple. In ordinary language the classes which we can respectively qualify as distributed and undistributed are mutually exclusive ; in the logic of our text-books this is evidently not the case. Students of the traditional logic should therefore disabuse their minds of the idea that the words ' distributed ' and ' undistributed ' neces- sarily refer to classes mutually exclusive, as they do in everyday speech ; or that there is anything but a forced and fanciful connexion between the ' distributed ' and 60 SYMBOLIC LOGIC [§ 02 ' undistributed ' of current English and the technical ' distributed ' and ' undisturbed ' of logicians. Now, how came the words ' distributed ' and ' undis- tributed ' to be employed by logicians in a sense which plainly does not coincide with that usually given them ? Since the statement " No X is Y " is equivalent to the statement "All X is "Y," in which (see §§ 46-50) the class Y (or non-Y) contains all the individuals of the Symbolic Universe excluded from the class Y, and since " Some X is not Y " is equivalent to " Some X is *Y," the definitions of ' distributed ' and ' undistributed ' in text- books virtually amount to this : that a class X is dis- tributed with regard to a class Y (or *Y) when every individual of the former is synonymous or identical with some individual or other of the latter ; and that when this is not the case, then the class X is undistributed with regard to the class Y (or'Y). Hence, when in the state- ment " All X is Y " we are told that X is distributed with regard to Y, but that Y is undistribided with regard to X, this ought to imply that X and Y cannot denote exactly the same class. In other words, the proposition that " All X is Y " ought to imply that " Some Y is not X." But as no logician would accept this implication, it is clear that the technical use of the words ' distributed ' and ' undistributed ' to be found in logical treatises is lacking in linguistic consistency. In answer to this criticism, logicians introduce psychological considerations and say that the proposition " All X is Y " gives us infor- mation about every individual, X 1; X 2 , &c, of the class X, but not about every individual, Y v Y 2 , &c, of the class Y ; and that this is the reason why the term X is said to be 'distributed' and the term Y 'undistributed.' To this explanation it may be objected, firstly, that formal logic should not be mixed up with psychology — that its for- mulae are independent of the varying mental attitude of individuals ; and, secondly, that if we accept this ' infor- mation-giving ' or ' non-giving ' definition, then we should §62] 'DISTRIBUTED'— < UNDISTRIBUTED 1 fil say, not that X is distributed, and Y undistributed, but that X is known or inferred to be distributed, while Y is not known to be distributed — that the inference requires further data. To throw symbolic light upon the question we may proceed as follows. With the conventions of 8 50 we have (1) All X is Y = x:y; (2) No X is Y = x : // (3) Some X is Y = (x : //)'; (4) Some X is not Y = (x : //)'. The positive class (or ' term ') X is usually spoken of by logicians as the subject'; and the positive class Y as the ' predicate.' It will be noticed that, in the above examples, the non-implications in (3) and (4) are the respective denials of the implications in (2) and (1). The definitions of ' distributed ' and ' undistributed ' are as follows. (a) The class (or ' term ') referred to by the ante- cedent of an implication is, in text-book language, said to be ' distributed ' ; and the class referred to by the conse- quent is said to be ' undistributed.' (/$) The class referred to by the antecedent of a non- implication is said to be ' undistributed ' ; and the class referred to by the consequent is said to be ' distributed.' Definition (a) applies to (1) and (2); definition (/3) applies to (3) and (4). Let the symbol X d assert that X is ' distributed' and let X u assert that X is ' un- distributed.' The class 'X being the complement of the class X, and vice versa (see 8 46), we get (*X)* = X M , and (X)" = X d . From the definitions (a) and (/3), since (Y) d = Y", and ( y Y) u = Y d , we therefore draw the following four conclusions : — In (1) X d Y u ; in (2) X d Y d ; in (3) X U Y U : in (4) X u Y d . For in (2) the definition (a) gives us X d ( r Yf , and CY) u = Y d . Similarly, in (3) the definition (/3) gives us X u CY) d , and ( , Y) d = Y M . If we change y into x in proposition (1) above, we 62 SYMBOLIC LOGIC [§§ 62, 63 get " All X is X "=x:x. Here, by definition (a), we have X d X" ; which shows that there is no necessary antagonism between X rf and X" ; that, in the text-book sense, the same class may be both ' distributed ' and ' undistri- buted ' at the same time. 63. The six canons of syllogistic validity, as usually given in text-books, are : — (1) Every syllogism has three and only three terms, namely, the ' major term,' the ' minor term,' and the ' middle term ' (see § 5 4). (2) Every syllogism consists of three and only three propositions, namely, the ' major premise,' the ' minor premise,' and the 'conclusion' (see § 54). (3) The middle term must be distributed at least once in the premises ; and it must not be ambiguous. (4) No term must be distributed in the conclusion, unless it is also distributed in one of the premises.* (5) We can infer nothing from two negative pre- mises. (6) If one premise be negative, the conclusion must be so also ; and, vice versa, a negative conclusion requires one negative premise. Let us examine these traditional canons. Suppose \//('', y, z) to denote any valid syllogism. The syllogism being valid, it must hold good whatever be the classes to which the statements x, y, z refer. It is therefore valid when we change y into x, and also z into x ; that is to say, \|/(.'", ,/', :>-,) is valid (§ 13, footnote). Yet this is a case which Canon (1) appears arbitrarily and need- lessly to exclude. Canon (2) is simply a definition, and requires no comment. The second part of Canon (3) applies to all arguments alike, whether syllogistic or not. * Violation of Canon (4) is called "Illicit Process." When the term illegitimately distributed in the conclusion is the major term, the fallacy is called " Illicit Process of the Major " ; when the term illegitimately dis- tributed in the conclusion is the minor term, the fallacy is called " Illicit Process of the Minor " (see § 54). §63] 'CANONS 1 OF TRADITIONAL LOGIC 63 It is evident that if we want to avoid fallacies, we must also avoid ambiguities. The first part of Canon (3) cannot be accepted without reservation. The rule about the necessity of middle-term distribution does not apply to the following perfectly valid syllogism, " If every X is Y, and every Z is also Y, then something that is not X is not Z." Symbolically., this syllogism may be expressed in either of the two forms (x-.y){z:y):{x :z)' (1) {xy'nzyj'.ix'z'r (2) Conservative logicians who still cling to the old logic ; finding it impossible to contest the validity of this syllo- gism, refuse to recognise it as a syllogism at all, on the ground that it has four (instead of the regulation three) terms, namely, X, Y, Z, % the last being the class con- taining all the individuals excluded from the class X. Yet a mere change of the three constituents, x, y, z, of the syllogism Darapti (which they count as valid) into their denials x', //, z' makes Darapti equivalent to the above syllogism. For Darapti is _ {y:x\y:z):{x:zy (3); and by virtue of the formula a : (3 = /3' : a, the syllogism (l) in question becomes (/:*')(/ :*'):(*':*)' (4). Thus, if \^(f;, y, z) denote Darapti, then y\s(x', //', ;') will denote the contested syllogism (1) in its form (4); and, vice versa, if ^(x, y, z) denote the contested syllo- gism, namely, (1) or (4), then ^(a/, y ', z') will denote Darapti. To assert that any individual is not in the class X is equivalent to asserting that it is in the com- plementary class 'X. Hence, if we call the class 'X the non-X class, the syllogism in question, namely, (/:./)(/:/) :(,/:*)' (4), may be read, " If every non-Y is a non-X, and every non- 64 SYMBOLIC LOGIC [§ 03 Y is also a non-Z, then some non-X is a non-Z." For (x':z)' is equivalent to (./ z , )' r> , which asserts that it is possible for an individual to belong at the same time both to the class non-X and to the class non-Z. In other words, it asserts that some non-X is non-Z. Thus read, the contested syllogism becomes a case of Darapti, the classes X, Y, Z being replaced by their respective complementary classes 'X, 'Y, 'Z. It is evident that when we change any constituent x into x in any syllo- gism, the words ' distributed ' and ' undistributed ' inter- change places. Canon (4) of the traditional logic asserts that " No term' must be distributed in the conclusion, unless it is also distributed in one of the premises." This is another canon that cannot be accepted unreservedly. Take the syllogism Bramantip, namely, (z : y)(y : x) : (x : z')' , and denote it by \f/(V). Since the syllogism is valid within the restrictions of the traditional logic (see § 50), it should be valid when we change z into /, and consequently z into z. We should then get >},{/) = (*' :y)(y:x):(x:z)'. Here (see § 02) we get Z w in the first premise, and Z rf in the conclusion, which is a flat contradiction to the canon. Upholders of the traditional logic, unable to deny the validity of this syllogism, seek to bring it within the application of Bramantip by having recourse to distortion of language, thus : — " If every non-Z is Y, and every Y is X, then some X is non-Z." Thus treated, the syllogism, instead of having Z" in the first premise and Z d in the conclusion, which would contradict the canon, would have ( V Z)'' in the first premise and ( y Z) u in the conclusion, which, though it means exactly the same thing, serves to "save the face" of the canon and to hide its real failure and inutility. § G3] TESTS OF SYLLOGISTIC VALIDITY 65 Canon (5) asserts that " We can infer nothing from two negative premises." A single instance will show the unreliability of the canon. The example is (2, :0(^*') :(*':*)', Avhich is obtained from Darapti by simply changing z into z', and x into x . It may be read, " If no Y is X, and no Y is Z, then something that is not X is not Z." Of course, logicians may " save the face " of this canon also by throwing it into the Daraptic form, thus : " If all Y is non-X, and all Y is also non-Z, then some non-X is non-Z." But in this way we might rid logic of all negatives, and the canon about negative premises would then have no raison d'etre. Lastly, comes Canon (6), which asserts, firstly, that " if one premise be negative, the conclusion must be negative ; and, secondly, that a negative conclusion requires one negative premise." The objections to the preceding canons apply to this canon also. In order to give an appearance of validity to these venerable syllogistic tests, logicians are obliged to have recourse to distortion of language, and by this device they manage to make their negatives look like affirmatives. But when logic has thus converted all real negatives into seeming affirmatives the canons about negatives must disappear through want of negative matter to which they can refer. The following three simple formulae are more easily applicable and will supersede all the traditional canons : — (1) (a: y :z):(x:z) Barbara. (2) (z : y : x) : (x : z)' Bramantip. (3) (y:x)(y:z):(x:z')' .... Darapti. The first of these is valid both in general logic and in the traditional logic ; the second and third are only valid in the traditional logic. Apart from this limitation, they all three hold good whether any constituent be affirma- E 66 SYMBOLIC LOGIC [§§ 03, 64 tive or negative, and in whatever order we take the letters. Any syllogism that cannot, directly or by the formulae of transposition, a : /3 = /3' : a and a/3' : y' = ay : fi, be brought to one or other of these forms is invalid. CHAPTER IX Given one Premise and the Conclusion, to find the missing Complementary Premise.* 64. When in a valid syllogism we are given one premise and the conclusion, we can always find the weakest complementary premise which, with the one given, will imply the conclusion. AVhen the given conclusion is an implication (or " universal ") such as x : z or x : z\ the complementary premise required is found readily by mere inspection. For example, suppose we have the conclusion x:z f and the given major premise z : y (see § 5 4). The syllogism required must be either {x:y :z'): (x : z') or (x : y r : z') : (x : z'), the middle term being either y or y'. The major pre- mise of the first syllogism is y : z' ', which is not equivalent to the given major premise z : y. Hence, the first syllo- gism is not the one wanted. The major premise of the second syllogism is y' : z', and this, by transposition and change of signs, is equivalent to z : y, which is the given major premise. Hence, the second syllogism is the one wanted, and the required minor premise is x : y' . When the conclusion, but not the given premise, is a non-implication (or " particular "), we proceed as follows. Let P be the given implicational (or " universal ") pre- mise, and C the given non-implicational (or "particular") conclusion. Let W be the required weakest premise which, * A syllogism with one premise thus left understood is called an enthymeme. §§ G4, 05] TO FIND A MISSING PREMISE 67 joined to P, will imply C. We shall then have PW : C, which, by transposition, becomes PC : W. Let S be the strongest conclusion dcducible from PC. We shall then have both PC : S and PC : W'. These two implications having the same antecedent PC, we suppose their con- sequents S and W' to be equivalent. We thus get S = W', and therefore W = S'. The weakest 'premise required is therefore the denial of the strongest conclusion dedueible from PC {the given premise and the denial of the given conclusion). For example, let the given premise be y : x, and the given conclusion (x : z r )' . We are to have (y:x)W:(x:z'y. Transposing and changing signs, this becomes \{y:x){x:z')'.W. But, by our fundamental syllogistic formula, we have also (see § 5G) (y:x)(x:z'):(y:z'). We therefore assume W = y:z' ) and, consequently, W = (y : z f ) f . The weakest premise required * is therefore (y : //, and the required syllogism is (// : %)(y ■ *')' ■ (« : «')'■ 65. The only formulae needed in finding the weakest complementary premise are (1) a:(3 = (3':a'. (2) (a:/3)(/3: 7 ):(a: 7 ). (3) (a:/3)(a: 7 ):(/3 7 r\ The first two are true universally, whatever be the state- ments a, (3, y ; the third is true on the condition a*, that a is possible — a condition which exists in the * The implication y : «, since in the traditional logic it implies (y : s')', would also answer as a premise ; but it would not be the weakest (see § 33, footnote, and § 73). 68 SYMBOLIC LOGIC [§§ 65, 66 traditional logic, as here any of the statements a, (3, y may represent any of the three statements x, y, z, or any of their denials x , y', z , every one of which six state- ments is possible, since they respectively refer to the six classes X, Y, Z, %Y Z, every one of which is under- stood to exist in our Universe of Discourse. Suppose we have the major premise z:y with the conclusion (x : z')' ', and that we want to find the weakest complementary minor premise W. We are to have (z:y)W:(x:z'y, which, by transposition and change of signs, becomes (z:y)(x:z'):W. This, by the formula a : /3 = ft' : a , becomes (z:y)(z:x'):W. But by Formula (3) we have also (z:y)(z:x'):(yx'y. We therefore assume W' = (yz')' 71 , and consequently W = (yx'y = y:x. The weakest minor premise required is therefore y : x ; and the required syllogism is : y)(V ■ .') : (■'• : -')'- which is the syllogism Bramantip. As the weakest premise required turns out in this case to be an implica- tion, and not a non-implication, it is not only the weakest complementary premise required, but no other comple- mentary premise is possible. (See § 64, second footnote.) 66. When the conclusion and given premise are both non-implications (or " particulars "), we proceed as follows. Let P' be the given non-implicational premise, and C the non-implicational conclusion, while W denotes the required weakest complementary premise. We shall then have P'W : C or its equivalent WC : P, which we obtain by transposition. The consequent P of the second §§66, 66 (a)] THE STRONGEST CONCLUSION 69 implication being an implication (or " universal ") we have only to proceed as in § 64 to find W. For example, let the given non-implioational premise be (// : z)' \ and the given non-implicational conclusion {x : z)'. We are to have (yri/W :(*:«)'. By transposition this becomes W(x:z):(y:z). The letter missing in the consequent y : z is x. The syllogism WC : P must therefore be either (y : x : z) : (y : z) or else (y:x':z):(y:z); one or other of which must contain the implication C, of which the given non-implicational conclusion C, re- presenting (x : z)', is the denial. The syllogism WC : P must therefore denote the first of these two syllogisms, and not the second ; for it is the first and not the second that contains the implication C, or its synonym x : z. Hence W = y : x. Now, WC : P is equivalent, b} r trans- position, to WP' : C, which is the syllogism required. Substituting for W, P', C, we find the syllogism sought to be (// : '<■)(>/ ■ *)' ■ (? : *)', and the required missing minor premise to be y : x. 66 (a). By a similar process we find the strongest conclusion derivable from two given premises. One example will suffice. Suppose we have the combination of premises (z : y)(x : y)' '. Let S denote the strongest conclusion required. We get (z : y){x : //)' : S, which, by transposition, is (z : //)S / : (x : y). The letter missing in the implicational consequent of the second syllogism is z, so that its antecedent (z : y)S / must be either x : z : y or else x : z' : >/. 70 SYMBOLIC LOGIC [§§ 6G (a), G7 The first antecedent is the one that contains the factor z : y, so that its other factor x : z must be the one denoted by S'. Hence, we get S'=x:z, and S = (#:«)'. The strongest * conclusion required is therefore (x : z)' '. CHAPTER X 6 7. We will now introduce three new symbols, Wcp, Yep, Sep, which we define as follows. Let A v A 2 , A 3 , . . . A m be m statements which are all possible, but of which one only is true. Out of these m statements let it be understood that A r A 2 , A 3 , . . . A r imply (each sepa- rately) a conclusion cp ; that A r+1 , A r+2 , A.,. +3 , . . . A s imply cp' ; and that the remaining statements, A s+1 , A s+2 , . . . A m neither imply cp nor cp'. On this understanding we lay down the following definitions : — (1) W(/) = A 1 + A 2 + A 3 + . . . +A r . (2) W^) , = A r+1 + A r+2 + ... +A S . (3) V4> = V<£' = A s+1 + A g+2 + ... +A m . (4) S^ = W^ + V^ = W)', the denial of W. (7) S' means (S<£)', the denial of Sep. The symbol Wcp denotes the weakest statement that implies cp ; while Sep denotes the strongest statement that

' = W'<£; (3) V°<£ = (\\ r (^ = S<£ = (£). The last of these three formulas asserts that to deny the existence of Y(p in our arbitrary uni- verse of admissible statements, A , A 2 , &c, is equivalent to affirming that W<^>, Sep, and (p are all three equivalent, each implying the others. The statement Y° ; the former asserts that Y

/ ; whereas every statement of the class tj implies both

', since (as proved in § 18) the implication >/ : a is always true, whatever be the statement represented by a. The state- ment Y^cp also contradicts the convention laid down that all the statements A , A 2 , . . . A w are possible. Similarly, we may have W°<£ or W ^/. 68. The following examples will illustrate the mean- ings of the three symbols Wcp, Y(p, Sc£. Suppose our total (or " universe ") of possible hypotheses to consist of the nine terms resulting from the multiplication of the two certainties A' + A^ + A 9 and B« + B" + B fl . The product is A e B e + A^ + A e B" + A^B' + A"B" + A"B* + A*B e + A e B" + A 9 B 9 . 72 SYMBOLIC LOGIC [§ 68 Let (p denote (AB) e . We get (1) W(AB)* = A € B« + A fl B e . (2) S(AB) e = A*B 9 + A*B f + A e B* = A""B e + A e B"". (3) W(AB) e = S , (AB) 9 = A" + B T ' + A e B f . (See § 69.) (4) S( AB)- 9 = W'(AB) 9 = A" + B" + A'B' + A e B fl . (See § 69.) The first of the above formulae asserts that the weakest data from which we can conclude that AB is a variable is the alternative A e B 9 + A e B € , which affirms that either A is certain and B variable, or else A variable and B certain. The second formula asserts that the strongest conclusion we can draw from the statement that AB is a variable is the alternative A _T? B 9 + A^^, which asserts that either A is possible and B variable, or else A variable and B possible. Other formulae which can easily be proved, when not evident by inspection, are the following : — (5 (6 (8 (9 (10 (11 (12 (13 (14 (15 (16 (1< (18 W<£ : (p : S(f>. (W(j> = Sep) = (Wdj = ). W(AB)« = A e B e = S(ABy. W(A + B) e = A £ + B e . S(A + B) e = A 6 + B e + A e B e . W(A + B)" = A"B'' = S(A + B)" = (A + By. W(A + B) e = A"B 9 + A e B". S(A + B) e = A- £ B 9 + A e B^. W(AB)" = A" + B". S(AB) I) = A" + B' ! + A e B e . W(A : B) = W(AB')" = A" + B\ S(A : B) = S( AB')" = A" + B 6 + A e B". W(A : B/ = S'(A : B) = A £ B e + A"B". S(A : BY = W'(A : B) = A-"B" f . The formulae (15) and (16) may evidently be deduced from (13) and (14) by changing B into B'. Formula (17) asserts that the weakest data from which we can §§ 68, G9] APPLICATIONS OF SYMBOLS 73 conclude that A does not imply B is the alternative that either A is certain and B uncertain, or else A possible and B impossible. The formula may be proved as follows : W(A : B)' = S'(A : B) = (A" + B e + A fl BV = (A") , (B') / (A e B e ) / = A-^B-^A" 9 + B' e ) = A*B e + A-"B" ; for, evidently, A^A^M and B e B e = B". 69. All the formulae of § 68 may be proved from first principles, though some may be deduced more readily from others. Take, for example, (1), (2), (3). We are required to find W(AB) fl , S(AB) fl , W(AB)" 9 . We first write down the nine terms which constitute the product of the two certainties A e + A" + A fl and B' + B" + B fl , as in § 68. This done, we underdot every term that implies (AB) 9 , which asserts that AB is a variable ; we underline every term that implies (AB)" 5 , which asserts that AB is not a variable; and we enclose in brackets every term that neither implies (AB) 9 nor (AB)- . We thus get A e B e + A'B' 1 + A € B 9 + A"B e + A»B*» + A"B 9 + A 9 B e + A 9 B" + (A 9 B 9 ). By our definitions in § 67 we thus have W(AB) 9 = A £ B 9 + A 9 B e (1) By definition also Ave have V(AB) 9 = A 9 B 9 , and therefore S(AB) 9 = W(AB) 9 + V(AB) 9 = A f B 9 + A°B e + A 9 B 9 = A e B e + A 9 B e + A 9 B 9 + A 9 B 9 , for a = a + a = ( A e + A 9 )B 9 + A*(B' + B 9 ) = A"B 9 + A fl B-" (2). We may similarly deduce (3) and (4) from first principles, but they may be deduced more easily from the two formulae W((£ + ^) = W(£ + Wxfr .... (a) S(+x|O = S0 + S^ (£)> 74 SYMBOLIC LOGIC [§§ 69, 70 as follows : W(AB)-" = W{(AB) f + ( AB)" ( = W(AB)« + W(AB/> = A C B € + A" + B", from § 08, Formulae 7, 13. S( AB)- 9 = S { ( AB) £ + ( AB)" } = S( AB) e + S(AB)» = A e B« + A" + B" + A e B 9 , from § 08, Formulae 7, 14. 70. The following is an example of inductive, or rather inverse, implicational reasoning (see §§ 11, 112). The formula (A : x) + (B : x) : (AB : x) is always true ; when (if ever) is the converse, implication (AB : x) : (A : x) + (B : x), false ? Let

' e I (a/3)Xa" + /3")' ! (a/3)"a-"/3~" ! (a/SjV/S* ! (Ax' . Bx')\kxy(Bxy ! (ABxy(Ax') (Bxy Thus, the converse implication (p c fails in the case (a{$) r, a r, fir 7, i which represents the statement (ABa/yCAa/r^V • • • • ( 1 ); and it therefore also fails in the case (afiy>a fi 9 , which represents the statement (ABa/)"(A#')"(Ba/) 6 .... (2) ; for the second statement implies the first. The failure of

> T 2 o ; so that they are all variables. It is also clear that the respective chances of the three statements AB./, Axe', Bx', are 0, i 2 G , ^ ; so that we also have (ABx'y(Axy(Bx') 9 , which, by pure symbolic reason- insr, we found to be a case of failure. We may also show this by direct appeal to the diagram, as follows. The implication AB : x asserts that the point P cannot be W in both the circles A and B without being also in the ellipse x, a state- ment which is a material certainty, as it follows necessarily from the special data of our diagram (see § 109). The implication A : x asserts that P cannot be in A without being in x, a statement which is a material impossibility, as it is inconsistent with the data of our diagram ; and B : x is impossible for the same reason. Thus we have AB : x = e, A : x = v\, B : x = »/, so that we get ip = (A : x) + (B : x) : (AB : x) = >i + v : * = e cf) c = ( AB : x) : (A : x) + (B : x) = e : n + n = > h The Boolian logicians consider /). Every proposition is with them either certain or impossible, the propositions which I call variables (6) being treated as non-existent. The preced- ing illustration makes it clear that this is a serious and fundamental error. 71. The diagram above will also illustrate two other propositions which by most logicians are considered equivalent, but which, according to my interpretation of the word if, arc not equivalent. They are the complex 76 SYMBOLIC LOGIC [§§ 71, 72 conditional, " If A is true, then if B is true x is true" and the simple conditional, " If A and B are both true, then x is true!' Expressed in my notation, and with my inter- pretation of the conjunction if (see § 10), these con- ditionals are respectively A : (B : x) and AB : x. Giving to the propositions A, B, x, AB the same meanings as in § 70 (all having reference to the same subject, the random point P), it is evident that B : x, which asserts that the random point P cannot be in the circle B without being also in the ellipse x, contradicts our data, and is therefore impossible. The statement A, on the other hand, does not contradict our data ; neither does its denial A', for both, in the given conditions, are possible though uncertain. Hence, A is a variable, and B : x being impossible, the complex conditional A : (B : x) becomes 6 : 1}, which is equivalent to 0", and therefore an im- possibility. But the simple conditional AB : x, instead of being impossible, is, in the given conditions, a certainty, for it is clear from the figure that P cannot be in both A and B without being also in x. Hence, though A : (B : x) always implies AB : x, the latter does not always imply the former, so that the two are not, in all cases, equivalent. 72. A question much discussed amongst logicians is the " Existential Import of Propositions." When we make an affirmation A B , or a denial A" B , do we, at the same time, implicitly affirm the existence of A ? Do we affirm the existence of B ? Do the four technical propo- sitions of the traditional logic, namely, " All A is B," " No A is B," " Some A is B," " Some A is not B," taking each separately, necessarily imply the existence of the class A ? Do they necessarily imply the existence of the class B ? My own views upon this question are fully explained in Mind (see vol. xiv., N.S., Nos. 53-55); here a brief exposition of them will suffice. The convention of a "Symbolic Universe" (see §§ 46-50) necessarily leads to the following conclusions : — §§72,73] EXISTENTIAL IMPORT 77 Firstly, when any symbol A denotes an individvM ; then, any intelligible statement (A) implies that the individual represented by A has a real existence depends upon the context. Secondly, when any symbol A denotes a class, then, any intelligible statement <£(A) containing the symbol A implies that the whole class A has a symbolic existence ; but whether the statement (p(A) implies that the class A is wholly real, or wholly unreal, or partly real and partly unreal, depends upon the context. As regards this question of " Existential Import," the one important point in which I appear to differ from other symbolists is the following. The null class 0, which they define as containing no members, and which I, for convenience of symbolic operations, define as con- sisting of the null or unreal members V 2 , 3 , &c, is understood by them to be contained in every class, real or unreal ; whereas I consider it to be excluded from every real class. Their convention of universal inclusion leads to awkward paradoxes, as, for example, that " Every round square is a triangle," because round squares form a null class, which (by them) is understood to be con- tained in every class. My convention leads, in this case, to the directly opposite conclusion, namely, that "No round square is a triangle," because I hold that every purely unreal class, such as the class of round squares, is necessarily excluded from every purely real class, such as the class of figures called triangles. 73. Another paradox which results from this conven- tion of universal inclusion as regards the null class 0, is their paradox that the two universals " All X is Y " and " No X is Y " are mutually compatible ; that it is possible for both to be true at the same time, and that this is necessarily the case when the class X is null or non-existent. My convention of a " Symbolic Universe " 78 SYMBOLIC LOGIC [§§ 73, 74 leads, on the contrary, to the common-sense conclusion of the traditional logic that the two propositions " All X is Y " and " No X is Y " are incompatible. This may be proved formally as follows. Let (p denote the pro- position to be proved. We have (t> = (x:y)(x:y / ):v = (xy / )\xyy:f ] = (V + xy : >]) : >/ = {,/•(/ + y) : >/} : n = (xe : tj) : t] = (x : tj) : tj — (6 : tj) : >/ In this proof the statement x is assumed to be a variable by the convention of § 46. See also § 5 0. It will be noticed that (p, the proposition just proved, is equiva- lent to {x : y) : (x : y')' ', which asserts that " All X is Y " implies " Some X is Y." 74. Most symbolic logicians use the symbol A~< B, or some other equivalent (such as Schroeder's A=£ B), to assert that the class A is wholly included in the class B ; and they imagine that this is virtually equivalent to my symbol A : B, which asserts that the statement A implies the statement B. That this is an error may be proved easily as follows. If the statement A : B be always equivalent to the statement A -< B, the equivalence must hold good when A denotes >;, and B denotes e. Now, the statement >/ : e, by definition, is synonymous with (ye'y, which only asserts the truism that the impossibility r\e is an impossibility. (For the compound statement yja, whatever a may be, is clearly an impossibility because it has an impossible factor tj.) But by their definition the statement n -< e asserts that the class >? is wholly included in the class e; that is to say, it asserts that every individual impossibility. tj v >/ 2 , >; 3 , &c, of the class >; is also an individual (either e r or e 2 , or e 3 , &c.) of the class of certainties e ; which is absurd. Thus, >j : e is a formal certainty, whereas y -< e is a formal impossibility. (See 8 18.) § 75] CLASS INCLUSION AND IMPLICATION 79 75. Some logicians (see § 74) have also endeavoured to drag my formula (A:B)(B:C):(A:C) (1) into their systems under some disguise, such as (A -< B)(B -< C) -< (A -< C) .... (2). The meaning of (1) is clear and unambiguous; but how can we, without having recourse to some distortion of language, extract any sense out of (2) ? The symbol A -< B (by virtue of their definition) asserts that every individual of the class A is also an individual of the class B. Consistency, therefore, requires that the com- plex statement (2) shall assert that every individual of the class (A -< B)(B -< C) is also an individual of the class (A -< C). But how can the double-factor compound statement (A -< B)(B ■< C) be intelligibly spoken of as a class contained in the single-factor statement (A-/) may, in my system, be a certainty, an impossibility, or a variable, according to the special data of our problem or investi- gation (see §§ 22, 109). But how could the proposition A 09 be expressed in other systems ? In these it could 80 SYMBOLIC LOGIC [§§ 75, 76 not be expressed at all, for its recognition would involve the abandonment of their erroneous and unworkable hypothesis (assumed always) that true is synonymous with certain, and false with impossible. If they ceased to consider their A (when it denotes a proposition) as equivalent to their (A= 1), and their A' (or their corre- sponding symbol for a denial) as equivalent to their (A = 0), and if they employed their symbol (A=l) in the sense of my symbol A e , and their symbol (A=0) in the sense of my symbol A v , they might then express my statement A ee in their notation ; but the expression would be extremely long and intricate. Using A^B (in accordance with usage) as the denial of (A = B), my statement A e would then be expressed by (A=/=0)(A=/ r l), and my A 80 by {(A^0)(A^l)^0}{(A^0)(A=/=l)=£l}. This example of the difference of notations speaks for itself. CHAPTER XI 76. Let A denote the premises, and B the conclusion, of any argument. Then A .\ B (" A is true, therefore B is true "), or its synonym B v A (" B is true because A is true "), each of which synonyms is equivalent to A(AB / ) r ', denotes the argument. That is to say, the argument asserts, firstly, that the statement (or collec- tion of statements) A is true, and, secondly, that the affirmation of A coupled with the denial of B constitutes an impossibility^ that is to say, a statement that is incom- patible with our data or definitions. When the person to whom the argument is addressed believes in the truth of the statements A and (AB / )' ) , he considers the argument valid ; if he disbelieves either, he considers the argument invalid. This does not necessarily imply that he dis- §§76,77] 'BECAUSE 1 AND < THEREFORE' 81 believes either the premises A or the conclusion B ; he may be firmly convinced of the truth of both without accepting the validity of the argument. For the truth of A coupled with the truth of B does not necessarily imply the truth of the proposition (AB') 17 , though it does that of (AB')'. The statement (AB') 1 is equivalent to (AB')' (see § 23) and therefore to A' + B. Hence we have A(AB') 1 = A(A' + B) = AB = A T B\ But A .-. B, like its synonym A(AB / ) > ', asserts more than A T B T . Like A(AB / ) 1 , it asserts that A is true, but, unlike A(AB')\ it asserts not only that AB' is false, but that it is impossible — that it is incompatible with our data or definitions. For example, let k = He turned yah, and let B = Ife is guilty. Both statements may happen to be true, and then we have A T B T , which, as just shown, is equivalent to A(AB') 1 ; yet the argument A .-. B (" He turned pale : therefore he is guilty ") is not valid, for though the weaker statement A(AB')' happens on this occasion to be true, the stronger statement A(AB')'' is not true, because of its false second factor (KB'f. I call this factor false, because it asserts not merely (AB') 1 , that it is false that he turned pale without being guilty, an assertion which may be true, but also (AB')'', that it is impossible he should turn pale without being guilty, an assertion which is not true. 77. The convention that A .-. B shall be considered equivalent to A(A:B), and to its synonym A(AB'y, obliges us however to accept the argument A ,\ B as valid, even when the only bond connecting A and B is the fact that they are both certainties. For example, let A denote the statement 13 + 5 = 18, and let B denote the statement 4 + = 10. It follows from our symbolic conventions that in this case A .\ B and B .-. A are both valid. Yet here it is not easy to discover any bond of connexion between the two statements A and B ; we know the truth of each statement independently of F 82 SYMBOLIC LOGIC [§§ 77, 78 all consideration of the other. We might, it is true, give the appearance of logical deduction somewhat as follows : — By our data, 13 + 5 = 18. From each of these equals take away 9. This gives us (subtracting the 9 from the 13) 4 + 5 = 9. To each of these equals add 1 (adding the 1 to the 5). We then, finally, get 4 + G = 10 ; quod end demonstrandum. Every one must feel the unreality (from a psycho- logical point of view) of the above argument ; yet much of our so-called ' rigorous ' mathematical demonstrations are on lines not very dissimilar. A striking instance is Euclid's demonstration of the proposition that any two sides of a triangle are together greater than the third — a proposition which the Epicureans derided as patent even to asses, who always took the shortest cut to any place they wished to reach. As marking the difference be- tween A .-. B and its implied factor A : B, it is to be noticed that though A : e and >/ : A are formal certainties (see § 18), neither of the two other and stronger state- ments, A .-. e and »/ .*. A, can be accepted as valid. The first evidently fails when A = j/, and the second is always false ; for i] .: x, like its synonym >?(>/ : x), is false, because, though its second factor >j : x is necessarily true, its first factor 7] is necessarily false by definition. 78. Though in purely formal or symbolic logic it is generally best to avoid, when possible, all psychological considerations, yet these cannot be wholly thrust aside when we come to the close discussion of first principles, and of the exact meanings of the terms we use. The words if and therefore are examples. In ordinary speech, when we say, " If A is true, then B is true," or " A is true, therefore B is true," we suggest, if we do not positively affirm, that the knowledge of B depends in some way or other upon previous knowledge of A. But in formal logic, as in mathematics, it is convenient, if not absolutely necessary, to work with symbolic statements §§ 78, 79] CAUSE AND EFFECT 83 whose truth or falsehood in no way depends upon the mental condition of the person supposed to make them. Let us take the extreme case of crediting him with absolute omniscience. On this hypothesis, the word therefore, or its symbolic equivalent .-. , would, from the subjective or 'psychological standpoint, be as meaningless, in no matter what argument, as we feel it to be in the argument (7x9 = G3) therefore (2 + 1 = 3); for, to an omniscient mind all true theorems would be equally self- evident or axiomatic, and proofs, arguments, and logic generally would have no raison d'etre. But when we lay aside psychological considerations, and define the word 'therefore,' or its synonym .*. , as in § 7G, it ceases to be meaningless, and the seemingly meaningless argu- ment, (7 x 9 = 63)/. (2 + 1 = 3), becomes at once clear, definite, and a formal certainty. 79. In order to make our symbolic formula? and operations as far as possible independent of our changing individual opinions, we will arbitrarily lay down the following definitions of the word ' cause ' and ' explana- tion.' Let A, as a statement, be understood to assert the existence of the circumstance A, or the occurrence of the event A, while V asserts the posterior or simul- taneous occurrence of the event V ; and let both the statement A and the implication A : V be true. In these circumstances A is called a cause of V ; V is called the effect of A ; and the symbol A(A : V), or its synonym A.*. V, is called an explanation of the event or circum- stance V. To possess an explanation of any event or phenomenon V, we must therefore be in possession of two pieces of knowledge : we must know the existence or occurrence of some cause A, and we must know the law or implication A : V. The product or combination of these two factors constitute the argument A/. V, which is an explanation of the event V. We do not call A the cause of V, nor do we call the argument A .•. V the explanation of V, because we may have also 84 SYMBOLIC LOGIC [§§ 79, 80 B .•. V, in which case B would be another sufficient cause of V, and the argument B .-. V another sufficient explanation of V. 80. Suppose we want to discover the cause of an event or phenomenon x. We first notice (by experiment or otherwise) that x is invariably found in each of a certain number of circumstances, say A, B, C. We therefore provisionally (till an exception turns up) regard each of the circumstances A, B, C as a sufficient cause of x, so that we write (A : x)(B : x)(C : x), or its equivalent A + B + C : x. We must examine the different circum- stances A, B, C to see whether they possess some cir- cumstance or factor in common which might alone account for the phenomena. Let us suppose that they do have a common factor /. We thus get (see § 28) • (A:/)(B:/)(C:/),wmch=A + B + C:/. We before possessed the knowledge A + B + C : x, so that we have now A + B + C:/,'. If / be not posterior to x, we may suspect it to be alone the real cause of x. Our next step should be to seek out some circumstance a which is consistent with /, but not with A or B or C ; that is to say, some circum- stance a which is sometimes found associated with /, but not with the co-factors of / in A or B or C. If we find that fa is invariably followed by x — that is to say, if we discover the implication fa : x — then our suspicion is con- firmed that the reason why A, B, C are each a sufficient cause of x is to be found in the fact that each contains the factor /, which may therefore be provisionally con- sidered as alone, and independently of its co-factors, a sufficient cause of x. If, moreover, we discover that while on the one hand fa implies x, on the other f'a implies x' ; that is to say, if we discover (fa : %){fa : x' ) : our suspicion that / alone is the cause of x is confirmed § 80] CAUSE AND EFFECT 85 still more strongly. To obtain still stronger confirmation we vary the circumstances, and try other factors, (3, y, S, consistent with /, but inconsistent with A, B, C and with each other. If we similarly find the same result for these as for a ; so that (fa : x)(f'a : x'), which =/a : x :/+ a (//3 : x)(fp : x'), which = /]8 : x :f + /3' (/? : x )(f'y '• x ')> which =fy : x :/+ y' (/ = AB' : CD' = (AB' : C')(AB' : D'), since x : yz = (x : y) (x : z). In order that

= (/ : w'){y : e') : (y : w') = (y : e f : w') : (y : w'). 88 SYMBOLIC LOGIC [§§ 83-85 The given implication is therefore valid (see §§ 11, 56, 59). 84. Examine the argument, " His reasoning was correct, but as I knew his conclusion to be false, I was at once led to see that his premises must be false also." Let P = " his premises were true," and let C = "his con- clusion was true." Then P : C = " his reasoning (or rather implication) was valid." Let (p denote the implication of the argument to be examined. We get (see | 105) <£ = (P:C)C':P' = the valid form of the Modus tollendo tollens. Thus interpreted (p is valid. But suppose the word " premises " means P and Q, and not a single compound statement P. We then get <£=(PQ:C)C:P'Q' ; an interpretation which fails in the case CP'Q 1 , and also in the case C^P^Q 6 . To prove its failure in the latter case, we substitute for C, P, Q their respective exponential values r\, t}, e, and thus get

/e : rfirf : i/e' = (rj : ?])e : et] = ee : >/ = rj. 85. Supply the missing premise in the argument: " Not all mistakes are culpable ; for mistakes are some- times quite unavoidable." Let m = "it is a mistake," let c = "it is culpable," let u = " it is unavoidable," and let

X 2 , &c, or (AB) r (AB) 2 , &c, possessing the properties A and B. As a rule the greater the number of properties, A, B, C, &c, ascribed to X, the fewer the individuals possessing them ; or, in other words, the greater the connotation (or inten- sion), the smaller the denotation (or extension). In A a the symbol a connotes as predicate, and in A a it denotes as adjective. 94. Contrary and Contradictory. The two propositions " All X is Y " (or x : y) and " No X is Y " (or x : y f ) are §§ 94-98] TECHNICAL WORDS EXPLAINED 93 called contraries, each being the contrary of the other. The propositions " All X is Y " and " Some X is not Y," respectively represented by the implication x : y and its denial (x : y)' are called Contradictories, each being the contradictory or denial of the other (see § 50). Similarly "No X is Y" and "Some X is Y," respectively repre- sented by the implication x : y' and its denial (x : y') f , are called Contradictories. 95. Subcoutraries. The propositions "Some X is Y" and " Some X is not Y," respectively represented by the non-implications (x : y') r and (x : y)' ', are called Sub- contraries. It is easily seen that both may be true, but that both cannot be false (see § 73). 96. Subalterns. The universal proposition "All X is Y," or x : y, implies the particular " Some X is Y," or (x : y') f ; and the universal " No X is Y," or x : y' ' , implies the particular " Some X is not Y," or (x : y) f . In each of these cases the implication, or universal, is called the Subalternant, and the non-implication, or particular, is called the Subalternate or Subaltern. That x : y implies {x:y')' is proved in § 73; and by changing y into y' and vice versa, this also proves that x : y r implies (x : y)' . 97. Contraposition. This is the name given by some logicians to the formula x : y = ?/ : x, which, with the conventions of §§ 46, 50, asserts that the proposition " All X is Y " is equivalent to the proposition " All non-Y is non-X." But other logicians define the word differently. 98. Conversion. Let (p(x, y) denote any proposition, A, E, I, or O, of the traditional logic (see § 50); and let \j/(y, x) denote any other proposition which the first implies, the letters x and y being interchanged. The im- plication ' + A- e B e . 105. Modus tollendo tollens. Of this also there are two forms ; the first valid, the second not, namely, (A : B)B' : A' and (A : B)A' : B'. The first is evident ; the second fails, as before, in the case A^B"*, and in the case A~ e B e . For, denoting the 96 SYMBOLIC LOGIC [§§ 105-108 second form by (p, Ave get Wc// = A^B" + A" 6 B £ . (See §§ 67-69.) 106. Modus tollendo ponens. This also has two forms; the first valid, the other not. They are (A + B)A / :B and (AB)'B':A. The first may be proved formally as follows : — (A + B)A' : B = A'B'( A + B) : r, = (,, + >,) : ,/ = >j :>] = e. The second is not valid, for (AB)'B' : A = A'B'(AB)' : n = A'B' : n = (A + B) e ; which fails both in the case (A + By and in the case (A + B)". To prove its failure in the last case, let (p denote the given implication. We get (p = ( AB)'B' : A = (A + B) e , as already proved. Therefore, putting A + B = 0, we get (p = e* = n . 107. Modus poncndo tollens. This also has a valid and an invalid form, namely, (AB)'A : B' and (A + B)B : A'. The first is valid, for (AB)'A : B' = AB(AB)' : n = 1 : 1 = e. The second is not valid, for (A + B)B:A' = AB(A + B):>/ = AB:>,, which fails both in the case (AB) € and in the case (AB) e . In the first case the given implication becomes e : >;, which = t] ; and in the second case it becomes 6 : >/, which also = >]. 108. Essential (or Explicative) and Ampliative. Let x be any word or symbol, and let / + e = e. Next, suppose A a certainty and x meaningless. We get A x + A- r = e° + t-° = >; + e = f . Lastly, suppose A and x both meaningless. We get A x + A"* = 0° + 0-° = e + >/ = e. Let A x denote any function of x, that is, any expression containing the symbol x ; and let /x = x in mathematics. This is understood to be true for all positive values of x; but the symbol ^/x, and conse- quently also the symbol Jxjx, become meaningless when x is negative, for (unless we lay down further con- ventions) the square roots of negative numbers or fractions are non-existent. Mathematicians, therefore, have arrived tacitly, and, as it were, unconsciously, at the understanding that when x is negative, then, Avhatever meaning may be given to the symbol Jx itself, the combination y/x^x, like its synonym {^/xf, shall be synonymous with x ; and, further, that whatever meaning it may in future be found convenient to give to */— 1, that meaning must not conflict with any previous formula 100 SYMBOLIC LOGIC [§§ 111, 112 or definition. Those remarks bear solely on the algebraic symbol *J — 1, which we have given merely as a concrete illustration of the wider general principles discussed previously. In geometry the symbol *J — 1 now conveys by itself a clear and intelligible meaning, and one which in no way conflicts with any algebraic formula of which it is a constituent. 112. Induction. — The reasoning by which we infer, or rather suspect, the existence of a general law by observa- tion of particular cases or instances is called Induction. Let us imagine a little boy, who has but little experience of ordinary natural phenomena, to be sitting close to a clear lake, picking up pebbles one after another, throwing them into the lake, and watching them sink. He might reason inductively as follows: "This is a stone" (a); "I throw it into the water" (/3) ; "It sinks" (7). These three propositions he repeats, or rather tacitly and as it were mechanically thinks, over and over again, until finally he discovers (as he imagines) the universal law a/3 : y, that a/3 implies y, that all stones thrown into ivatcr sink. He continues the process, and presently, to his astonishment, discovers that the inductive law a/3 : y is not universally true. An exception has occurred. One of the pebbles which he throws in happens to be a pumice-stone and does not sink. Should the lake happen to be in the crater of an extinct volcano, the pebbles might be all pumice-stones, and the little boy might then have arrived inductively at the general law, not that all stones sink, but that all stones float. So it is with every so- called " law of nature." The whole collective experience of mankind, even if it embraced millions of ages and extended all round in space beyond the farthest stars that can ever be discovered by the most powerful telescope, must necessarily occupy but an infinitesimal portion of infinite time, and must ever be restricted to a mere infinitesimal portion of infinite space. Laws founded upon data thus confined, as it were, within the limits of an § 1 1 2] " LAWS OF NATURE " 101 infinitesimal can never be regarded (like most formulae in logic and in mathematics) as absolutely certain ; they should not therefore be extended to the infinite universe of time and space beyond — a universe which must necessarily remain for ever beyond our ken. This is a truth which philosophers too often forget (see § 80). Many theorems in mathematics, like most of the laws of nature, were discovered inductively before their validity could be rigorously deduced from unquestionable premises. In some theorems thus discovered further researches have shown that their validity is restricted within narrower limits than was at first supposed. Taylor's Theorem in the Differential Calculus is a well-known example. Mathematicians used to speak of the " failure cases " of Taylor's Theorem, until Mr. Homersham Cox at last investigated and accurately determined the exact con- ditions of its validity. The following example of a theorem discovered inductively by successive experiments may not be very important ; but as it occurred in the course of my own researches rather more than thirty years ago, I venture to give it by way of illustration. Let C be the centre of a square. From C draw in a random direction a straight line CP, meeting a side of the square at P. What is the average area of the circle whose variable radius is CP ? The question is very easy for any one with an elementary knowledge of the integral calculus and its applications, and I found at once that the average area required is equal to that of the given square. I next took a rectangle instead of a square, and found that the average area required (i.e. that of the random circle) was equal to that of the rectangle. This led me to suspect that the same law would be found to hold good in regard to all symmetrical areas, and I tried the ellipse. The result was what I had expected : taking C as the centre of the ellipse, and CP in a random direction meeting the curve at P, I found that the average area of the variable 102 SYMBOLIC LOGIC [§112 circle whose radius is CP must be equal to that of the ellipse. Further trials with other symmetrical figures confirmed my opinion as to the universality of the law. Next came the questions : Need the given figure be symmetrical ? and might not the law hold good for any point C in any area, regular or irregular ? Further trials again confirmed my suspicions, and led me to the discovery of the general theorem, that if there be any given areas in the same plane, and we take any point C anywhere in the plane (whether in one of the given areas or not), and draw any random radius CP meeting the boundary of any given area at a variable point P, the average area of the circle whose radius is CP is always equal to the sum of the given areas, provided we con- sider the variable circle as positive when P is a point of exit from any area, negative when P is a point of entrance, and zero when P is non-existent, because the random radius meets none of the given boundaries. Next came the question : Might not the same general theorem be extended to any number of given volumes instead of areas, with an average sphere instead of circle ? Experiment again led to an affirmative answer — that is to say, to the discovery of the following theorem which (as No. 3486) I proposed in the Educational Times as follows : Some shapeless solids lie about — No matter where they be ; Within such solid, or without, Let's take a centre C. From centre C, in countless hosts, Let random radii run, And meet a surface each at P, Or, may be, meet with none. Those shapeless solids, far or near, Their total prove to be The average volume of the sphere Whose radius is CP. §§112, 113] FINITE, INFINITE, ETC. 103 The sphere, beware, is positive When out at P they fly ; But, changing sign, 'tis negative When entrance there you spy. One caution more, and I have done : The sphere is naught when P there's none. In proposing the question in verse instead of in plain prose, I merely imitated the example of more dis- tinguished contributors. Mathematicians, like other folk, have their moments of exuberance, when they burst forth into song just to relieve their feelings. The theorem thus discovered inductively was proved de- ductively by Mr. G. S. Carr. A fuller and therefore clearer proof was afterwards given by Mr. D. Biddle, who succeeded Mr. Miller as mathematical editor of the Educational Times. 113. Infinite and Infinitesimal. Much confusion of ideas is caused by the fact that each of those words is used in different senses, especially by mathematicians. Hence arise most of the strange and inadmissible para- doxes of the various non- Euclidean geometries. To avoid all ambiguities, I will define the words as follows. The symbol a denotes any positive quantity or ratio too large to he expressible in any recognised notation, and any such ratio is called a positive infinity. As we may, in the course of an investigation, have to speak of several such ratios, the symbol a denotes a class of ratios called infinities, the respective individuals of which may be designated by a a 2 , a g , &c. An immensely large number is not necessarily infinite. For example, let M denote a million. The symbol M M , which denotes the millionth power of a million, is a number so inconceivably large that the ratio which a million miles has to the millionth part of an inch would be negligible in comparison ; yet this ratio M M is too small to be reckoned among the infinities a , a a y &c, of the class a, because, though inconceivably 104 SYMBOLIC LOGIC [§113 large, its exact value is still expressible in our decimal nota- tion ; for we have only to substitute 10° or 1,000,000 for M, and we get the exact expression at once. The symbol /3, or its synonym — a, denotes any negative in- finity ; so that fi v j3 2 , /3 3 , &c, denote different negative ratios, each of which is numerically too large to be expressible in any recognised notation. Mathematicians often use the symbols oo and — co pretty much in the sense here given to a and /3 ; but unfortunately they also employ oo and — oo indifferently to denote expres- 1 3 sions such as -, -, &c. which are not ratios at all, but mire non-existences of the class (see § 6). Mathematicians consider oo and — oo equivalent when they are employed in this sense; but it is clear that a and — a are not equivalent. They speak of all parallel straight lines meeting at a point at infinity ; but this is only an abbreviated way of saying that all straight lines which meet at any infinite distance a v or a 2 , or a,, &c, or fi v or /8 or /3 3 , &c, can never be distinguished by any possible instrument from parallel straight lines ; and may, therefore, for all practical purposes, be considered parallel. The symbol h, called a positive infinitesimal, denotes any positive quantity or ratio too small numerically to be expressible in any recognised notation; and the symbol 7c, called a negative infinitesimal, denotes any negative quantity or ratio too small numerically to be expressible in any recognised notation. Let c temporarily denote any positive finite number or ratio — that is to say, a ratio neither too large nor too small to be expres- sible in our ordinary notation; and let symbols of the forms xy, x + y, x — y, &c„ have their customary mathe- matical meanings. From these conventions we get various self-evident formula?, such as § 113] FINITE, INFINITE, ETC. 105 (1) (cay, (c(3f; (2) (ch)\ (ckf ; (3) (« - c)\ ; (4) (,±/0 c ; (5) ((3 + cf; (6) (f)", (|) fl ; (7) Q\ (£)*; (8) («Y, (/S 2 )"; (9) (aflP; (10) of : afar* ; ( 1 1 ) « a + s^ : ar° ; (12) (M)*. The first formula asserts that the product of a positive finite and a positive infinite is a positive infinite ; the tenth formula asserts that if any ratio x is a positive finite, it is neither a positive nor a negative infinite. The third formula asserts that the difference between a positive infinite and a positive finite is a positive infinite. Note 1.— A fuller discussion of the finite, the infinite, and the infini- tesimal will be found in my eighth article on " Symbolic Reasoning" in Mind. The article will probably appear next April. Note 2.— The four " Modals " of the traditional logic are the four terms in the product of the two certainties A T + A' and A f + A' + A". This pro- duct is A^ + A^ + A^A^ + A'A"; it asserts that every statement A is either necessarily true (A € ), or necessarily false (A''), or true in the case considered but not always (A T A"), or false in the case considered but not always (A'A"). See § 99. CALCULUS OF LIMITS CHAPTER XIV 114. We will begin by applying this calculus to simple problems in elementary algebra. Let A denote any number, ratio, or fraction. The symbol A x asserts that A belongs to the class x, the symbol x denoting some such word as positive, or negative, or zero* or imaginary, &c. The symbols A*B», A^ + B 2 ', A* : B y , A~ x , &c, are to be understood in the same sense as in §§ 4- 10. For example, let Y= positive, let N = negative, and let = zero* ; while all numbers or ratios not included in one or other of these three classes are excluded from our Universe of Discourse — that is to say, left entirely out of consideration. Thus we get (6 — 4) p , (4 — 6) N , (3 - 3)°, (f), (3 x 0)°, (3PJi* (P^/, (W, (N^f , ,3, (P 1 + P 2 ), p (N 1 + N 2 ) N , and many other self-evident for- mulas, such as (1) (AB) P = A P B P + A N B N . (2) (AB) N = A N B P + A P B N . (3)(AB)° = A° + B°. (4) {Ax - B) p = Ux - B )Y = k{x - ?Y + A*(x - B I V A/J \ A) \ A * In this chapter and after, the symbol 0, representing zero, denotes not simple general non-existence, as in § G, but that particular non- existence through which a variable passes when it changes from a positive infinitesimal to a negative infinitesimal, or vice verm. (See § 113.) 106 §§114, 115] CALCULUS OF LIMITS 107 (5)(A ,-B,={4-B)^4-By + 4_By. (7) (ax = ah) = (ax - ab)° = { a(x -b)}° = a" + (x - b)°. 115. The words greater and less have a wider meaning in algebra than in ordinary speech. In algebra, when we have (.« — a) p , we say that " x is greater than a," whether a is positive or negative, and whether x is positive or negative. Also, without any regard to the sign of x or a, when we have (x — ctf, we say that " x is less than a!' Thus, in algebra, whether x be positive or negative, and whether a be positive or negative, we have (x — of = (x > a), and (x — «) N = (x < a). From this it follows, by changing the sign of a, that (x + af = (x > - a), and (x + af = (x < - a) ; the symbols > and < being used in their customary algebraic sense. For example, let a - 3. We get (,r-sy = (x>3), and (x - 3f = (x < 3). In other words, to assert that x — 3 is positive is equivalent to asserting that x is greater than 3 ; while to assert that x — 3 is negative is equivalent to asserting that x is less than 3. Next, let a = - 3. We get ( x - a y = (x + 3) p = (x > - 3 ) (x - af = (x + 3 ) N = (x < - 3 ). Let x = 6, we get (x > - 3) = (x + 3) p = (6 + 3 ) p = e (a certainty). Let x= 0, we get (x > - 3 ) = {x + 3 ) p = (0 + 3 ) p = e (a certainty). 108 SYMBOLIC LOGIC [§§ 115-117 Let x= — 1, we get (x > - 3) = (x + 3) p = ( - 1 + 3) p = e (a certainty). Let a? = — 4, we get (x > - 3) = (x + 3) p = ( - 4 + 3 )'' = >/ (an impossibility). It is evident that (,/; > — 3) is a certainty (e) for all positive values of x, and for all negative values of x between and — 3 ; but that x> — 3 is an impossibility (>?) for all negative values of x not comprised between and —3. With (x< —3) the case is reversed. The statement (x< — 3) is an impossibility (>?) for all positive values of x and for all negative values between and — 3 ; while (x < — 3 ) is a certainty (e) for all negative values of x not comprised between and — 3. Suppose, for example, that x= — 8 ; we get (x< - 3) = (x + 3) N = ( - 8 + 3) N = e (a certainty). Next, suppose x= — 1 ; we get (x< - 3) = ( - 1 + 3) N = >? (an impossibility). 116. From the conventions explained in § 115, we get the formulas (A>B) = (-A)<(-B), and (A( -B); for{(-A)<(-B)} = {(-A)-(-B)} N = (-A + Bf = (A-B) P = (A>B), and{(-A)>(-B)} = {(-A)-(-B)} p = (-A + B) p = (A-B) N = (A a), we say that a is an inferior limit oix\ and when we have (x — cif, or its synonym (x x + 2 3 Let A denote this given statement of inequality. We get \ 2 3 = i6 2« — — — - _ )\ = (tx—3y=(x — 3 . Hence, — is an inferior limit of x. In other words, the 7 given statement A is impossible for any positive value 3 of x lower than -, and also impossible for all negative values of x. 119. Given the statements A and B, in which A denotes Sx — — — < — , and B denotes — 3x < - 2 4 3 4' Find the limits of x. We have A = Ux-°^- 1 -j = (12x-l() + 2x-lf =(^-n>»4-liy=(*|l). Hence we get AB = ( — > x > — 1. 8 \14 40/ 110 SYMBOLIC LOGIC [§§ 119, 120 Thus x may have any value between the superior 11 .... 21 limit - and the inferior limit -- ; but any value of 14 40 J x not comprised within these limits would be incom- patible with our data. For example, suppose x = 1 . We get a -(s - *=i - 1Y Ya - 2 - 1 Y- /3 V • » (an im ~ \ 2 4/ '\ 4/ 'U/ ' >? possibility). B : ( 6 ^ - 3 - ij Y| - 30 ; e (a certainty). Thus the supposition (#=1) is incompatible with A though not with B. Next, suppose x=0. We get / ^ 1 N A : ( ] : e (a certainty). B : ( - — ) : n (an impossibility). Thus, the supposition (x=0) is incompatible with B though not with A. 120. Next, suppose our data to be AB, in which A denotes ox — - > 4a; + -. 4 3 B denotes Qx — - < 4« + -. 2 4 We get 3 . IV / 13\ p / 13 ^-4- 4;/: -3J = (^12 / ) = (^12 §§ 120, 121] CALCULUS OF LIMITS 111 Hence we get 5 13 AB = ->£> — = >i (an impossibility) , /5 13\ /5 13\ t01 \8 >aJ> T2J : (8 > l2J : ' / - In this case therefore our data AB are mutually incompatible. Each datum, A or B, is possible taken by itself; but the combination AB is impossible. 121. Find for what positions of x the ratio F is positive, and for what positions negative, when F 2x-l 28 denotes — — . x — 6 x 2x 2 -29a; + 84 2(x- 4)(x - 10£) 1 = x(x-3) x(x - 3) As in § 113, let a denote positive infinity, and let /3 denote 'negative infinity. Also let the symbol (to, n) assert as a statement that x lies between the superior limit m and the inferior limit n, so that the three symbols (to, n), (m>x>ri), and (m — x)\x — nf are synonyms. We have to consider six limits, namely, a, 10i, 4, 3, 0, (3, in descending order, and the five intervening spaces corresponding to the five statements (a, 10i), (10J, 4), (4, 3), (3, 0), (0, (3). Since x must lie in one or other of these five spaces, we have e = (a, 10£) + (10l, 4) + (4, 3) + (3, 0) + (0, (3). Taking these statements separately, Ave get (a, 1 0+) : (x - 1 0|) p : (x - 1 0|)> - 4)> - 3) V : F p ( 1 Oh 4) : (x - 1 0|-)> - 4) p : (z - 1 Offix - 4) p (;v - 3 ) V : F K (4, 3) : (x - ±)"(x - S) ¥ : (x - 10i) N (fl - 4) N (.v - 3)V : F p (3, 0) : (x - 3)V : (x - 10|)> - ±f(x - 3)V : F N ( , /3) : x" : x\x - 3 f{x - 4) N (sc - 1 0i) N : F p . Thus, these five statements respectively imply F p , F N , F p , 112 SYMBOLIC LOGIC [§§ 121, 122 F N , F p , the ratio or fraction F changing its sign four times as x passes downwards through the limits 1 Oi, 4, 3, 0. Hence we get F p = («, 10*)+(4, 3) + (O,0); F N = (10i 4) + (3, 0). That is to say, the statement that F is 'positive is equiva- lent to the statement that x is either between a and 10 \, or between 4 and 3, or between and ft ; and the state- ment that F is negative is equivalent to the statement that x is either between 10i and 4 or else between 3 and 0. 2«-l_28 122. Given that — , to find the value or x — 3 x values of x. It is evident by inspection that there are two values of x which do not satisfy this equation ; they are and 3. m n . 2a; -1 1 ... 28 28 . . When x=0, we get = - while — = — ; and evi- 6 x-3 3' x dently a real ratio - cannot be equal to a meaningless o 28 ratio or unreality — (see § 113). Again when x=3, we . 2re-l 5 ... 28 28 , ., fl 5 get - — = — , while — = — ; and evidently - cannot 6 x-S x 3 J 28 be equal to — . Excluding therefore the suppositions (x=0) and (x=o) from our universe of possibilities, let A denote our data, and let F = — — - — . We get x — 3 x A . F o . / 2a- 1 _ 28\°. f 2(x- 8 4 4 8 to find the limits of x. Let A denote the given statement. We have . /13a; 3 3x G - 7.A 1 ' , 1Q B B , . _ XP A = | = (13# - 6 - 6x + - 7«) p \ 8 4 4 8/ = l ' = v. If in the given statement we substitute the sign < for the sign >, we shall get A = N = >/. Thus, the state- ,, ,13a; 3 . , ,i 3x 6 — 7x . . ment that is greater than is nnpos- 8 4 4 8 l 'tQ,--, Q sible, and so is the statement that — is less than 8 4 3% Q — 7x TT 13a; 3 , , , J ox G — 7a? . Hence must be equal to , 4 8 8 4 * 4 8 ' whatever value we give to x. This is evident from the fact 2,x 6 7x that -- . when reduced to its simplest form, is 4 8 r \2x — 6 , which, for all values of x, is equivalent to 13x 3 If in the given statement we substitute the 8 4' b sign = for the sign > , we shall get /13a_3_3. G-7,y = ()0 = \ 8 4 4 8/ so that, in this case, A is a formal certainty, whatever be the value of x. 124. Let A denote the statement x} + 3>2>x\ to find the limits of x. We have A = (x 2 - 2x + 3) p = { (x 2 - 2x + 1 ) + 2 } p = {{x- l) 2 + 2}" = e. H 114 SYMBOLIC LOGIC [§§ 124-128 Here A is a formal certainty whatever be the value of x, so that there are no real finite limits of x (see § 113). If we put the sign = for the sign > we shall get A={(,e-l)° + 2}° = > h Here A is a formal impossibility, so that no real value of x satisfies the equation x 2 + 3 = 2x. It will be remem- bered that, by § 114, imaginary ratios are excluded from our universe of discourse. 125. Let it be required to find the value or values of x from the datum x — s /x= 2. We get (x -Jx=2) = (x - Jx - 2)° = (x v + x* + x°) ( ;>J _ J x _ 2 )° = x\x - Jx - 2)° = x p {(x h - 2)(xi + 1)}° = A ' P (^ - 2)° = (x = 4) ; for (x = 4) implies x v , and x° and « N are incompatible with the datum (x - Jx - 2)°. 126. Let it be required to find the limits of x from the datum (x— Jx>2). (x-Jx>2) = (x-Jx-2y = (c i '+x"+x°)(x-Jx-2y = x p (x-Jx-2y = cc p {(x i -2)(x i + 1)}^ = ,^- 2) p = (.> ; >4) ; for (v>4) implies x F , and x° and re N are incompatible with the datum (x — Jx — 2) 1 '. 127. Let it be required to find the limits of x from the datum (x— Jx<2). (x- Jx<2) = (x- Jx- 2) N = (.^+^-M')<>- Jx- 2) N = (x v + x°)(x - Jx - 2) N = of(x - Jx - 2) N + x° = x ¥ {^ - 2)(x$ + 1)} N + x°=x*(x i - 2f+x° = x\x* < 2) + x° = x\x < 4) + x° = (4>^>0) + O=0). Here, therefore, x may have any value between 4 and zero, including zero, but not including 4. 128. The symbol gm denotes any number or ratio §§ 128, 129] CALCULUS OF LIMITS 115 greater than m, while Im denotes any number or ratio less than m (see § 115). The symbols g x m } g 2 m, g 3 m, &c., denote a series of different numbers or ratios, each greater than vi, and collectively-forming the class gm. Similarly, the symbols l-{m, l 2 m, l 3 m, &c, denote a series of different numbers or ratios, each less than m, and collectively forming the class Im. The symbol x gm asserts that the number or ratio x belongs to the class gm, while x 1 '" asserts that x belongs to the class Im (see § 4). The symbol x gm ' gn is short for x gm z gn ; the symbol xP mln is short iorx gm x ln ; and so on (see § 9, footnote). These symbolic conventions give us the formulae (1) x^ m = (x>m) = (x-my. (2) x lm = (x m)(x < n) = (x — mY(x — iif = (n> x > m). 129. Let m and n be two different numbers or ratios. We get the formula ( 1 ) ,,:<"" • 9* = X^V 71 + X a V n = (x > m > n) + (x > n > m). To prove this we have (since m and n are different numbers) af m.gn _ ^m.gn^jn + ^m^ for ^jn + n gm _ g = x gm x* n m 9n + x gm x ffn n 9m = {x 9m m 9W )x <)n + {,:»P n n am )x° m = x gm m gn + x 9 n n gm = (x > m > n) + (x > u > m ), for in each term the outside factor may be omitted, because it is implied in the compound statement in the bracket, since x>m>n implies x>n, and x>n>m im- plies x>m. Similarly, we get and prove the formula (2) x lm ■ ln = J m m ln + aV = (x < m < n) + (x < n < m). This formula may be obtained from (1) by simply sub- 116 SYMBOLIC LOGIC [§§ 129-131 stituting I for g ; and the proof is obtained by the same substitution. 130. Let m, n, r be the three different numbers or ratios. We get the formulae ( 1 ) aT fi - gn ■ gr = ,iP n in Jr + ,/;? W'" + aWV*. ( 2 ) rf m ■ ln ■ lr — J m ni ln m lr + x ln n lm n lr + J r r lm r ln . These two formulae are almost self-evident ; but they may be formally proved in the same way as the two formulas of § 129 ; for since m, n, r are, by hypothesis, different numbers or ratios, we have m gn ■ or + n gm . gr + ^m . gn _ ^ m ln.lr + n lm.lr + r lm.ln = € ^ while jf n -9n.gr = x gm.gn.gr e ^ by fas formula a = ae, and x im.in.i r= . x im.in.ir e ^ ^ ^q same formula. When we have multiplied o:° m ■ 9n -' jr by the alternative e v and omitted implied factors, as in § 129, we get Formula (1). When we have multiplied x lm - ln - lr by the alternative e 2) and omitted implied factors, as in § 129, we get Formula (2). The same principle evidently applies to four ratios, m, n, r, s, and so on to any number. 131. If, in § 130, we suppose m, n, r to be inferior limits of x, the three terms of the alternative e v namely, m gn-ir i n gm -° r , r gm - an , respectively assert that , n asserts that x m is a si^erior limit, and x n an inferior limit, of x. The* symbol x m , n , rs asserts that x m and x n are superior limits of x, while x r and x s are inferior limits of x. Thus aW.n means (x - x m f{x - x n J or (x m >x>x n ), x m'.n'.r. ■ means (x - x m f(x - x n f{x - x r ) p (x - x s f, and so on. 133. The symbol os m . (with an acute accent on the numerical suffix m) always denotes a proposition, and is synonymous with (x — x m y, which is synonymous with (x n t rs . The student may employ whichever he finds the more con- venient. From long habit I find the notation of the text easier ; but the other occupies rather less space, and has certain other advantages in the process of finding the limits. When, however, the limits have been found and the multiple integrals have to be evaluated, the notation of the text is preferable, as the other might occasionally lead to ambiguity (see §§ 151, 150). 118 SYMBOLIC LOGIC [§§ 134, 135 134. The employment of the symbol x m sometimes to denote the proposition (x — x, m ) v , and sometimes to denote the simple number or ratio x m , never leads to any ambiguity ; for the context always makes the meaning perfectly evident. For example, when we write X —I — \x — x z ) — x 3 , it is clear that the x s inside the bracket denotes the fraction -, which is supposed to be marked in the table of reference as the third limit of x; whereas the x 3 , outside the bracket, is affirmed to be equivalent to the statement (x — x 3 Y, and is therefore a statement also. Similarly, when we write A = ( 2,,; 2 + 8 4 > 2 9x) = (x - 1 0|) p + (x - 4) N = {x — x^f + (x — x 2 Y = x l + Xg, we assert that the statement A is equivalent to the alter- native statement x l + x^, of which the first term x 1 asserts (as a statement) that the limit x 1 (denoting 10|) is an inferior limit of x, and the second term »_, asserts that the limit x 2 (denoting 4) is a superior limit of x. Thus, the alternative statement x -\-x 2 > asserts that "either x l is an inferior limit of x, or else x 2 is a superior limit x. 135. The operations of this calculus of limits are mainly founded on the following three formula? (see §§ 129-131): ( 1 / x m . n = ''m\ x m ~ x n) "" x n\ x n ~ x m) • (Z) x m , n > = x m \x m x n ) + x n \x n — x m ) . \° ) x m'.n ''- m' .n\' vi '' n) ' In the first of the above formulae, the symbol x m n means /„,./„, and asserts that x m and x n are both inferior limits §§ 135, 136] CALCULUS OF LIMITS 119 of x. The statement (x m - x n f asserts that Xm is greater than x n and therefore a nearer inferior limit of x ; while the statement (x n -x m Y asserts, on the contrary, that x n and not x m is the nearer inferior limit (see §§ 129, 131). In the second formula, the symbol x m > n . asserts that x m and x n are both superior limits of x. The state- ment (x m - xj" asserts that x m is less than x n and there- fore a nearer superior limit of x ; while the statement (x n — x m ) K asserts, on the contrary, that x n and not x m is the nearer superior limit. The third formula is equiva- lent to ■' m .n • \ x m x n) > and asserts that if x m is a superior limit, and x n an inferior limit, of x, then x m must be greater than x w 13G. When we have three inferior limits, Formula (1) of § 135 becomes % m .n.r = x m « + Xnfi + X r y, in which a asserts that x m is the nearest of the three inferior limits, ft asserts that x n is the nearest, and y asserts that x r is the nearest. In other words, a — \ x m ~ x n) \ x m ~ X r) p = {x n x m ) (x n — x r ) y=(x r -x m f(x r -x n y. When we have three superior limits, Formula (2) of § 135 becomes x m'. W. ? = x m' a + x n'ft + x r'7> in which, this time, a asserts that x m is the nearest of the three superior limits, ft asserts that x n is the nearest, and y asserts that x r is the nearest. In other words, a = (x m x n ) \X m x r ) ft=( x n- X mf( x n- x rY y = (x r — x m f(x r — x n ) s . Evidently the same principle may be extended to any number of inferior or superior limits. 120 SYMBOLIC LOGIC [§§ 137, 138 137. There are certain limits which present themselves so often that (to save the trouble of consulting the Table of Limits) it is convenient to represent them by special symbols. These are positive infinity, negative infinity, and zero (or rather an infinitesimal). Thus, when we have any variable x, in addition to the limits x v x 2 , x 3 , &c, registered in the table, we may have always understood the superior limit x a , which will denote positive infinity, the limit x Q , which will denote zero (or rather, in strict logic, a positive or negative infinitesimal), and the always understood inferior limit x p , which will denote negative infinity (see § 113). Similarly with regard to any other variable y, we may have the three understood limits y a , y , y p , in addition to the registered limits y v y 2 , y 3 , &c. Thus, when we are speaking of the limits of x and y, we have x a — y a = a ; x = y = (or dx or dy) ; x fi = y p = - a. On the other hand, the statement x a ,_ m asserts that x lies between positive infinity x a , and the limit x m registered in the table of reference; whereas x m , p asserts that x lies between the limit x m and the negative infinity x p . Simi- larly, x m , tQ asserts that x lies between the superior limit x m and the inferior limit ; while ;% n asserts that x lies between the superior limit and the inferior limit x n . Thus, the statement « m ,. implies that x is positive, and x ff n implies that x is negative. Also, the statement x Q , is synonymous with the statement X s ; and the statement x is synonymous with the statement x p . As shown in § 134, the employment of the symbol x Q sometimes to denote a limit, and sometimes to denote a statement, need not lead to any ambiguity. 138. Just as in finding the limits of statements in pure logic (see §§ 33-40) we may supply the superior limit n when no other superior limit is given, and the inferior limit e when no other inferior limit is given, so in find- ing the limits of variable ratios in mathematics, we may supply the positive infinity a (represented by x a or y a or z , &c, according to the variable in question) when no §§ 138, 139] CALCULUS OF LIMITS 121 other superior limit is given, and the negative infinity (3 (represented by .^ or y p or z p , &c.) when no other inferior limit is given. Thus, when x m denotes a statement, namely, the statement (x — x^f, it may be written x a , m ; and, in like manner, for the statement x n >, which denotes (x — x n y, we may write x n , tP (see § 137). 139. Though the formulae of § 135 may generally be dispensed with in easy problems with only one or two variables, we will nevertheless apply them first to such problems, in order to make their meaning and object clearer when we come to apply them afterwards to more complicated problems which cannot dispense with their aid. Given that 7a?— 53 is positive, and 67 — 9a; negative; required the limits of x. Let A denote the first datum, and B the second. We get TABLE A = (7x-5Sy = (x-~X=x 1 =x a ,, 1 B = (67-9*) N = (9,:-G7) p = (^-y ) Hence, we get AB = av. x x a ,, 2 = x a ._ j By Formula (1) of § 135, we get a5j _ 2 = Xjlfa — x 2 ) p + x 2 (x 2 — x^f 53_67 Y 9~ = t r 1 (477-469) p + t r 2 (469-477) p ,forQ p = (63Q) p = x 1 e + aw = x 1 (see § 11, Formula? 22, 23). Thus we get AB = ,i' a - 12 = .r .i- From tne aata AB there- fore we infer that x lies between x a and ,i\ ; that is. 53 between positive infinity and — . 53 ,_4 greater than — - or 7-. ,67 53V In other words, x is 122 SYMBOLIC LOGIC [§§ 139, 140 Now, here evidently the formula of § 135 was not wanted ; for it is evident by mere inspection that u\ is greater than ,r 2 , so that a\ being therefore the nearest inferior limit, the limit ,r 2 is superseded and may be left out of account. In fact A implies B, so that we get AB = A = ,r aU . 140. Given that 7x — 53 is negative and 07 — 9* positive ; required the limits of x. Let A denote the first datum, and B the second. We get— A = (7£-53) N = (x- 53 x - — ■ x, 53 x 1 - 7 67 x 2 - 9 2'./3- Hence, we get AxS = Xy^ £#?2'. p ~~ ^'l'. 2' . 0. By Formula 2 of § 1 3 5 we get — 53 67\ N , / 7 53 = %(477 - 4G9) N +,%(469 - 477) N = Xyij + x 2 ,e = x% (see §11, Formulae 22, 23). This shows that the nearer superior limit x 2 super- sedes the more distant superior limit x\ ; so that we get A-b — ,Vy 2 '. — ®%. 07 Thus x lies between the superior limit x 2 (or — ] and negative infinity. 141] CALCULUS OF LIMITS 123 CHAPTER XVI 141. We will now consider the limits of two variables, and first with only numerical constants (see § 156). Suppose we have given that the variables x and y are both positive, while the expressions 2y — 3# — 2 and 3^ + 2^ — 6 are both negative; and that from these data we are required to find the limits of y and x in the order y, x. Table op Limits. Let A denote our whole data. We have A = y r x p (2y - 3x - 2) N (3?/ + 2x -6) N . Beginning with the first bracket factor, we get* (2y - 3x -2Y = (y-^x-lJ = (y- ? A ) N = Vv Then, taking the second bracket factor, we get o 2/i = ^ + l _ 6 2 y 2 =2--X o 2 X 2 — o a? 3 ~ o (3?/ + 2x - 6) N = ( y + - x - 2 I = Also 2/V = 2/ a '. a- a '.o ( seG §§ 137 ' 139), so that A = y a '. o»a'. <#i#2' = Va.: v. 2'. o ;v v. o = Vi>. 2'. a' «'. o ; for the nearer superior limits y 1 and y 2 supersede the more distant limit y a . Applying Formula (2) of § 135 to the statement y v v , we get /13 Vv. * = vvivi - VzY + y-Ay-2 - ?a) n = yA ^ - 1 + y 2 ^x-lj=y r (x-^ + !h\ x ~ 13 * The limits are registered in the table, one after another, as they are found, so that the table grows as the process proceeds. 124 SYMBOLIC LOGIC [§141 Substituting this alternative for y v 2 . in the expression for A, we get A = (y r x v + y^\)y^. o = (yv. 0% + feiftK-.o = VV. V C a. V . + yY = yv. 0(2/1)* = yv. d -® + l = yv.o(3x + 2) 1 ' = y v Jx + ^ J = yi'. (* - x ^f — 2/1'. 0^2 ! y%. = 2/2'. 0(2/2 - ?7o) P = 2/2'. o(2/ 2 ) P = 2/2'. o( 2 - -x J = y 2 , (6 - 2^ = ^,0(3 -xf = y^ Q (x- 3) N = 2/2'.0<%- Substituting these equivalents of ?y r and ?/ 2 . in A, we get A = }Jx. cftv. 2.0 "J~ 2/2'. O^a'. 3'. 1 = 2/l'. 0^1'. 1 2/2'. O'^V. 1 > for evidently x is a nearer inferior limit than ,r 2 , and therefore supersedes ,v 2 ; while x 3 is a nearer superior limit than x a (which denotes positive infinity), and there- fore supersedes x a . We have now done with the ?/-state- ments, and it only remains to apply Formula (3) of § 135 to the ^'-statements x vo and x si . It is evident, however, by mere inspection of the table, that this is needless, as it would introduce no new factor, nor discover any incon- sistency, since x x is evidently greater than x , that is, than zero, and x 3 is evidently greater than x x . The process therefore here terminates, and the limits are fully deter- §141] CALCULUS OF LIMITS 1 25 mined. We have found that either x varies between x x and zero, and y between y 1 and zero ; or else x varies between x B and x v and y between y 2 and zero. The figure below will illustrate the preceding process and table of reference. The symbol x denotes the distance of any point P (taken at random out of those in the shaded figure) from the line x , and the symbol y denotes the distance of the point P from the line y . The first equivalent of the data A is the statement x z x o x r llv 2- o x o> which asserts that y 1 and y 2 are superior limits of y, that y (or zero) is an inferior limit of y, and that x (or zero) is an inferior limit of x. It is evident that this compound statement A is true for every point P in the shaded portion of the figure, and that it is not true for any point outside the shaded portion. The final equivalent of the data A is the alternative y v% x r _ + Vv. o tl V. i> the first term of which is true for every point P in the quadrilateral contained by the lines y v y , x v x Q ; and the second term of which is true for the triangle contained by the lines y 2 , y 0) x v 126 SYMBOLIC LOGIC [§142 Table of Limits. 142. Given tliat y 2 — 4./.' is negative and y + 2x — 4 positive ; required the limits of y and x. Let A denote our data. We get A = (v/-4 t r)-\y + 2,,;-4) p = (7/ 2 -4 tt -) N (y-yi)"; tf - ± x y = {(y-2 JxXy + 2 Jx)Y = {y-2 s /xr(y+2 s fxy for (y — 2 s/^YiV + 2 x/^) N * s impossible. We therefore get a = 2/2'. 3(2/ - 2/i) p = 2/2'. s2/i = y-2.3. i By Formula (1) of § 135 we get 2/ 3 . i = 2/3(2/3 - Vif + y/yi - 7hY = y 3 (2tf - 2 ^ - 4) p + Vl {2x -2jx- 4)* = y 3 (# - x/« - 2 ) p + y^a? - s/x - 2 ) N (see §§ 126, 127) slx-l ^ X ~~l) ~i* Y -"((•"-D-lM^-i)-!}' = ? / 3 (.j-4) p +2/i(*-4) n = y 3 (# ~ «i) P + ^ - ^'i) N = 2/3^i + 2/r*r- Therefore A = 2/2'.3^1+//2'.l^l'- We now apply Formula (3) of § 135, thus !h. 3 = 2/2'. 3(2/2 - VsY = Ik. s( 2 */« + 2 xA')'' = y*. 3 e 2/2'. 1 = 2/2'. 1(2/2 - 2/i) r = 2/2'. i(2# + 2 V'/' - 4) P = yr. i(* + «/* - -)" = V*. i{( V* + 2J - (2) } = 2/2'. i(« - 1 ) r = h: M' ~ x -zY = 2/2'. i#2- §§ 142, 143] CALCULUS OF LIMITS 127 Thus the application of Formula (3) of § 135 to y 2 , 3 introduces no new factor, but its application to the other compound statement y 2 , 1 introduces the new statement x 2 , and at the same time the new limit x 2 . Hence we finally get (since Form 3 of § 135 applied to x a . a and Xy 2 makes no change) A^y.,.3^+^1%.2 (see §§137, 138). This result informs us that " either x lies between x a (positive infinity) and x ., and y between the superior oc jc z limit y 2 and the inferior limit y 3 ; or else x lies be- tween £&, and x 2 , and y between y 2 and y v The above figure will show the position of the limits. With this geometrical interpretation of the symbols x, y, &c., all the points marked will satisfy the conditions expressed by the statement A, and so will all other points bounded by the upper and lower branches of the para- bole, with the exception of the blank area cut off by the line y v 143. Given that y 2 — ±x is negative, and y + 2x — 4 also negative ; required the limits of y and x. Here the required limits (though they may be found 128 SYMBOLIC LOGIC [§§ 143-145 independently as before) may be obtained at once from the diagram in § 142. The only difference between this problem and that of § 142 is that in the present case y + 2x — 4 is negative, instead of being, as before, positive. Since y 2 — 4a; is, as before, negative, y. 2 will be, as before, a superior limit, and y 3 an inferior limit of y ; so that, as before, all the points will be restricted within the two branches of the parabola. But since y + 2x — 4 has now changed sign, all the admissible points, while still keeping between the two branches of the parabola, will cross the line y v The result will be that the only admissible points will now be restricted to the blank portion of the parabola cut off by the line y v instead of being, as before, restricted to the shaded portion within the two branches and extending indefinitely in the positive direction towards positive infinity. A glance at the diagram of § 142 will show that the required result now is 1J-2'. 3'%. ' V\'. 3^1'. 2> with, of course, the same table of limits. CHAPTER XVII A 144. The symbol — , when the numerator and denomi- nator denote statements, expresses the chance that A is true on the assumption that B is true; B being some state- ment compatible with the data of our problem, but not necessarily implied by the data. A 145. The symbol denotes the chance that A is true e when nothing is assumed but the data of our 'problem. This is what is usually meant when we simply speak of the " chance of A." § 146, 147] CALCULUS OF LIMITS 129 146. The symbol^—, or its synonym S(A, B), denotes B A A — — — ; and this is called the dependence* of the statement A upon the statement B. It indicates the increase, or (when negative) the decrease, undergone by the absolute chance — when the supposition B is added to our data. € The symbol <5° D , or its synonym S°(A, B), asserts that the B dependence of A upon B is zero. In this case the state- E E E Fig. 1. Fig. 2. Fig. 3. ment A is said to be independent oj the statement B ; which implies, as will be seen further on (see S 149), that B is independent of A. 147. The symbols a, b, c, &c. (small italics) respectively ABC represent the chances—, -, — , &c. (see S 145); and the € € € symbols a! ', I/, c ', &c, respectively denote the chances — , — , — , &c, so that we get e e e 1 = n + a' = b + b' = c + c' = &c. * Obscure ideas about ' dependence ' and ' independence ' in pro- bability have led some writers (including Boole) into serious errors. The definitions here proposed are, I believe, original. 130 SYMBOLIC LOGIC [§148 148. The diagrams on p. 129 will illustrate the pre- ceding conventions and definitions. Let the symbols A, B assert respectively as propositions that a point P, taken at random out of the total number of points in the circle E, will be in the circle A, that it will be in the circle B. Then AB will assert that P will be in both circles A and B ; AB' will assert that P will be in the circle A, but not in the circle B ; and similarly for the statements A'B and A'B'. In Fig. 1 we have A_ _ 3 . A'_ ,_10 7 _ft ~T3' 7~~ a ~1S AB_ 1 e "l3 : AB' 2 ' ~T~13 In Fig. 2 we have A_ _ 3 . A'_ ,_ £-*-?■£(•> *-}& The second of the above eight formulae shows that if any statement A is independent of another statement B, then B is independent of A ; for, by Formula (2), it is clear that :(x + y = a + b); (4) S°(A, B) = (AB) f '»; (5) (AB)" = (A + B) a+& ; 132 SYMBOLIC LOGIC [§§150,151 < 6 >(s4)=(s=f)=*( A - B >; „ /A B\ /A \ (7) [B = A) : \B = ! + {a = b):(AB)V + (a = h) - It is easy to prove all these formulae, of which the last may be proved as follows : A_B\ /A_Z> A\ /K_b A\° (A/ 6\)° B~Ay' ; \B~a'B/ : \B a'B/ : \ B\ X ~ a/ J \ A V /A \ : jjj(a-&)| :( B = 0J + («-^)°:(ABr+(a = &). The following chapter requires some knowledge of the integral calculus. CHAPTER XVIII 151. In applying the Calculus of Limits to multiple integrals, it will be convenient to use the following notation, which I employed for the first time rather more than twenty years ago in a paper on the " Limits of Multiple Integrals " in the Proc. of the Math. Society. The symbols ^>{x)x m!n and x m - n (p(x), which differ in the relative positions of . n , differ also in meaning. The symbol (%)% m >. n is short for the integra- tion (p(x)dx, taken between the superior limit x m and the inferior limit x n \ an integration which would be ex commonly expressed either in the form ' m dx(p(x) or fX ' ™(x m )— % . o = { (hA -cy-d- (hvl ~ cy 2 ) } x v , = { (I.* 2 - «b) - (W - cb) } x v . = (h^ 2 - ex - \tf + bc)x r . = #i\ o(^^ 3 — ikr 2 — iH> 2 # + &«') = £a 3 — lea 2 — \b % ci + bca. 152. The following formulae of integration are self- evident : — ( 1 ) *W . n = - %n' . m ; (2) #«>*V . „ = ~ <£OX'. m J (3) *W. „<£(■») = -X n '.rn{z)\ (4) ^' m <. n + X n , mt = X ni . r J ( 5 ) #(«)(#»' . n + *»» . r) = 0(^>m' . r I ( 6 ) fe . n + #„' . r )<£<>) = a? m . . ,#*') ; \ ' / //?»' . n' ' r' . s i/n' . rnfir' . s 2/m' . n^s' . >• — 2/«' . mP^s' . r 5 '. / '' m' . 71 ~r" "■ V . s "m' . s • " J r > . n ' ( 9 ) (x m . . „ + x r , . s )(p(x) = (x m . . , + ^ , n )(p(x) ; ( 1 0) . n + *V . s) = 0(#)(#m< . . + SYMBOLIC LOGIC [§§ 154, 155 x x =l "l =1 z=A •' 2 =1-2/ y + z — l y 8 =l -a ! e = A: = aj i'.o^i'.o 2! i'. D 154. The symbol IntA(x, y, z) denotes the integral Idxldyjdz, subject to the restrictions of the statement A, the order of variation being x, y, z. The symbol hit A, or sometimes simply A, may be used as an abbreviation for Int A(x, y, z) when the context leaves no doubt as to the meaning of the abbreviation. 155. Each of the Table op Limits. variables x, y, z is taken at random be- tween 1 and ; what is the chance that the . . z( 1 — x — y) traction — 1-y-yz will also be between 1 and ? Let the symbol Q, as a proposition, assert that the value of the fraction in question will lie between 1 and ; and let A denote our data .1',,?/,'^%. We have to find -, Avhich here =- 1 • (W 1 1 -0 A (see § 145). Also, let N denote the numerator z(l — x — y), and D the denominator 1 — y — yz of the fraction in ques- tion ; while, this time, to avoid ambiguity, the letter n will denote negative, and p positive (small italics instead of, as before, capitals). We get Q = N^D p (N - J)) n + N n D'\N - Bf. Taking the order of variation x, y, z, as in the table, we get, since z is given positive, W={l-x-yY = {x-{l-y)Y=Xt N" = ( 1 - x - yf ={.v-(l-y)Y = ,/- 2 E> p = (l -y-yzf= \y{\ +z)-l \-» = y 2 , D ,l = (l-?/-F) n =r = (z-z,v + y-iy = (z,r- // -:+iy> y + z- 1 = ■':, (N - Vy = (z - zx + y- l) p = (z,c -y-z+l) n = x 3 § 155] CALCULUS OF LIMITS 135 Substituting these results in our expression for Q, we shall have Multiplying by the given certainty x v -0 (see table), we get X V. oH == lV i\ 1'. 3. 0^2' + -'V. r. 2.02/-2- Applying Formulae (1) and (2) of § 135, we get (see § 137) #3. = X l X Z - '<0 )" + *oK - ^ = ^3 + •% t% _ r = x s (3C 3 - xj* + ar^ - x 3 ) n = X^e + x v n = x s , X 2. = ff«te f - ^ + • ?, o('' - V = X 2* + ^ = «* Substituting these results in our expression for x v Q,, we get #r. oQ = %(%2/3 + A W3')y2' + •'3' . 2^/2 == X 2' . 31/2' . 3 "r ^2' . 0^3' . 2' "•" #3 '. 2^2' We now apply Formula (3) of § 135 to the statements "*2\ 3' ^2'.o' ''3'. 2' **nUS '2' . 3 = ' V -l' . 3(^2 X i) = < r -2' . $%' *2' .0 == ^2' . Q\ X 2 ^0) = ''V . e "^3' . 2 = ■% . 2V^3 — ,?, 2 ) = X S . iVl- This shows that the application of § 135, Form 3, intro- duces no new statement in y ; so that we have finished with the limits of x, and must now apply the formulas of §135 to find the limits of y. Multiplying the expres- sion found for tr ro Q by the datum y v Q , we get #i'.o 7 /i' y. 2 = y 20 and substituting these results in the right-hand side of the last equivalence, we get <'V.2yi\oQ = ''V.3y2'.3 + <>2\oy3\o + <'V.2/'r.2 136 SYMBOLIC LOGIC [§§ 155, 156 The application of § 135, Form 3, to the y-statements will introduce no fresh statements in z, nor destroy any term by showing that it contains an impossible factor tj. We have therefore found the nearest limits of y ; and it only remains to find the limits of z. Multiplying the last expression by the datum z ro we get QA = Q,?v _ Q y v . oZ r . = Ov . 3y 2 . . 3 + dfe . o2/ 3 - . o + % . s#r . 2>r . o- The application of § 135, Form 3, to the factor z r _ will effect no change, since {z x — z ) p is a certainty. The pro- cess of finding the limits is therefore over ; and it only remains to evaluate the integrals. We get A Int A ^ = Int(/e.r. $2 . 3 + x* . y s . o + %? . Hl\- . 2K' . for Int A = Int x v ,& v . f fs l .. =l. The integrations are easy, and the result is log 2 (Naperian base), which is -± 5 a little above -. 9 156. Given that a is positive, that n is a positive whole number, and that the variables x and y are each taken at random between a and — a, what is the chance that {(x + y'T - a} is negative and {{x + y) n+1 - «} positive ? Let A denote our data y Y _ 2 x r#2 (see Table); let Q de- note the proposition {(x+y) n -a} s , and let R denote the proposition {(.?• + y) n+1 - a} p , in which the exponent N denotes negative, and the exponent P positive. „ , , . QR , . , Int QRA We have to find the chance — ^-, which = . In this problem we have only to find the limits of integration (or variation) for the numerator from the compound statement QRA, the limits of integration for the denominator being already known, since A = ?/ 1 .2^1.2. 156] CALCULUS OF LIMITS Vtf Table of Limits y 1 = a y 2 = ~ a i y 3 = a n — X 1 y i = — a n — x i 7i =a n + 1 —x •J 5 1 y, = - « n+l - x x x = a ,r 2 = -a i x 3 = a n — a i x. = a — a n 4 1 x =a + a n + 1 i x e = a + a n 6 1 x 7 = a n+1 — a i x 8 = — a — a n + 1 i a? g = « — a"+i n+l " 3 = (±) B A = e = 3/r.2 a a'.2 We take the order of integration y, #. The limits being registered in the table, one after another, as they are found, the table grows as the process goes on. For convenience of reference the table should be on a separate slip of paper. We will first suppose n to be even. Then Q = j {pc + y) - a n \ \ (as + y) + a" I = y - » — a?)| K | S I !/ + 's + yv^3')(y^5 + y& x h) = y$. ^ + y^.^s . a + yr. s% ; for by application of the formulae of § 135, ,r 4 5 = ,r 5 138 SYMBOLIC LOGIC [§156 #5 . 3 = «'■ s ; <'V. 5 = 1 ( an impossibili ty) ; and Xg . 5 - = .r 3 , For when a> 1 we have ( 2a — a n ] , and when ct< 1 we have ( a"+i — a Tl ] ; so that x 6 — a; 3 is always positive. We must now apply § 135, Form 3, to the statements in y. We get y s . 2 = y&. &#\ yz>. 5 z =yv.5 a i> Vv.^ — yv.^n- Substituting these results, we get Q%1'. 2 = VS. 2 == . ? /3'.5' ?, l'.3 a l "I" VM.ffiv.'fl r \ — (Vb'.S^V. 3 + yv.sfls'. 7/ (6 l J for ai.2 = «i = a 2.i3 an( i %.3 = > ? ( an impossibility). This is the final step in the process of finding the limits, and the result informs us that, when n is even, QRA is only possible when a x ( which =1) is an inferior limit of a. In other words, when n is even and a is not greater than 1, the chance of QR is zero. To find the chance when n is even and a is greater than 1, we have § L56] CALCULUS OF LIMITS 139 only to evaluate the integrals, employing the abbreviated notation of § 151. Thus Integral A = Int y v 2 ,?; r 2 = (y x — y 2 )^v. 2 — ( 2a)# r> 2 = x v , 2 {2ax) = 2cuc 1 — 2ax 2 = 4a 2 Integral QRA = y w 5 # r< 3 + y Vm 6 # 3 ,. 7 = (y-s - y^ v v. 3 + (Vi - y 5 )'%.7 = ( a Tl — a 1l + l W,_ 3 + ( a — ««+i+ ,i ] W 7 = ®V. 3 a?l ~ aH + 1 F + ' r 3'-7 *" ~ an+1 V + i^ = a n - a** 1 (^j - ^ 3 ) + ( a - a w +* )(a? 8 - x 7 ) + £(#?, - ^) = '( a n — a n + l Y 2a - £ a " - | a ™+ 1 QR = Int QRA _ Int QRA A 7w« A 4a 2 = — ( a" _ a ^+i Y 4a - a Tl - a"* 1 We have now to find the chance when % is odd. By the same process as before we get QR A = (y 3 , 5 ,/' r- 3 + y v . 5( %. 7 K + y & . &&. 2 «3- Here we have £wo inferior limits of a, namely, a x and a 3 , so that the process is not yet over. To separate the different possible cases, we must multiply the result obtained by the certainty (a 1 +a r )(a 3 + a$), which here reduces to a x +a v 3 + %, since a x is greater than a y For shortness sake let M x denote the bracket co- efficient (or co-factor) of a x in the result already obtained for QRA; and let M 3 denote y c «. 2 <% . 2. tne coefficient of a 3 . We get QRA = (M A + M 3 a 3 )(a 1 + a v . 3 + %) = (M 1 + M 3 )a 1 + M 3 a r>3 ; 140 SYMBOLIC LOGIC [§§ 156, 157 for a 13 = a r and a 9tl = rj (an impossibility). Hence, there are only two possible cases when n is an odd number, the case a 1 (that is to say, a>a v which here means a>l) and the case a r 3 . For the latter, a r -3 , we get OR Lit M, 1 /_ 4, N jL_ = ?= — J 2a — a n+1 A ira* A 8a' 2 \ For the first case, namely, the case a> 1, we get QR_ 7^(M 1 + M 8 ) [ A ~ 7w* A When the integrals in this case are worked out, the result will be found to be 9? = _L( o» - a«+i Y 2a - a" ) + — ( a^ 1 - a^+i A 4a 2 \ A / 8a\ 1 \2 + _ ( 2a — a»+ 1 The expression for the chance— —in the case a>l and the expression for it in the case a < 1 evidently ought to give the same result when we suppose a=l. This is easily seen to be the fact; for when we put a=l, each expression gives - as the value of the chance — — . 8 A 157. The great advantage of this " Calculus of Limits " is that it is independent of all diagrams, and can therefore be applied not only to expressions of two or three vari- ables, but also to expressions of four or several variables. Graphic methods are often more expeditious when they only require straight lines or easily traced and well- known curves ; but graphic methods of finding the limits of integration are, in general, difficult when there are three variables, because this involves the perspective representation of the intersections of curved surfaces. §157] CALCULUS OF LIMITS 141 When there are four or more variables, graphic methods cannot be employed at all. For other examples in pro- bability I may refer the student to my sixth paper in the Proceedings of the London Mathematical Society (June 10th, 1897), and to recent volumes of Mathematical Questions and Solutions from the Educational Times. It may interest some readers to learn that as regards the problems worked in §§ 155, 150, I submitted my re- sults to the test of actual experiment, making 100 trials in each case, and in the latter case taking a = 1 and 7i = 3. The theoretical chances (to two figures) are re- spectively -56 and -43, while the experiments gave the close approximations of *53 and - 41 respectively. THE END Printed by Ballantyne, Hanson & Co. Edinburgh &* London 1-IZ-61 lut rufc'-jjj Wf. gSB ljSC 95 ^^H ■ m "r /: ^ ■*■- ;v •■.•■:.::;:■• -^ ■■ .- '■' . ' ' ' I I '.:-''-' ' ::; '■ : ' "-'■■ "■•• " ■• ■ ■ M •.■■'■'-'..■•■-■• .'■<'■■'■■■:<■■■■-■■■■ '■'-•■" ' : ■■..'■' ^" ■'.■• : 3J •■'.-. | ;'..•■-■ ■ ■ . ■• ■■'■'■ '' ' ;: ' ^ -\ V. v ' ■ Pi :■"■-- -■■'■■ ' * ■■ ■,■■.'...'