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A UTHOR : 
 
 ELLIS, ALEXANDER J. 
 
 TITLE: 
 
 LOGIC FOR CHILDREN 
 
 PLACE: 
 
 LONDON 
 
 DA TE : 
 
 1882 
 
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 COLUMBIA UNIVERSITY LIBRARIES 
 
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\1 
 
 LOGIC FOE CHILDREN, 
 
 DEDUCTIVE AND INDUCTIVE. 
 
 BEING THE SUBSTANCE OF 
 
 TWO ADDRESSES TO TEACHERS 
 
 DELIVERED BEEORE THE COLLEGE OF PRECEPTORS. 
 
 ENTITLED 
 
 I 
 
 J. ON A METHOD OF TEACHING DEDUCTIVE OR 
 FORMAL LOGIC TO CHILDREN BY MEANS OF 
 WORDS AND COUNTERS, 8 May, 1872. 
 
 2. ON SCHOOL INDUCTIONS, 12 Feb., 1873. 
 
 -/. 
 
 ,\ 
 
 BY 
 
 ALEXANDEE J. ELLIS, 
 
 P.R.S., F.S.A., F.C.P.S., F.C.P., 
 
 PRESIPKNT (for THE SECOND TIME) OF THE PHILOLOGICAL SOCIETY, MEMBER OF THE 
 
 MATHEMATICAL SOCIETY, FORMERLY A VICE-PRESIDENT OF THE COLLEGE 
 
 OF PRECEPTORS, AND FORMERLY SCHOLAR OF TRINITY 
 
 COLLEGE, CAMBRIDGE, B.A. 1837. 
 
 The Jirst lecture reprinted from the Educational Times for June, July, and 
 August^ 1872 ; the second from the same for March^ April, and May, 1873 ; 
 both with numerous additions. 
 
 Printed off in 1872 and 1873, and Published in 1882. 
 
 Ii 
 
 ' i 
 
 /( 
 
 LONDON: 
 
 C. F. HODGSON & SON, 1, GOUGH SQUARE, 
 
 ^FLEET STREET. 
 
 Price Two Shillings,* 
 
tit 
 
 LOGIC FOE CHILDTIEN, . 
 
 DEDUCTIVE AND INDUCTIVE. 
 
 BEINO THE SUBSTANCE OF 
 
 TWO ADDRESSES TO TEACHERS 
 DELIVERED BEFORE THE COLLEGE OF PRECEPTORS. 
 
 ENTITLED 
 
 I 
 
 1. ON A METHOD OF TEACHIN-O DEDUCTIVE OR 
 
 FORMAL LOGIC TO CHILDREN BY MEANS OF 
 WORDS AND COUNTERS, 8 May, 1872. 
 
 2. ON SCHOOL INDUCTIONS, 12 Feb., 1873. 
 
 BY 
 
 ALEXANDER J. ELLIS, 
 
 F.R.S., F.S.A.. F.C.P.S., F.C.P., ' 
 
 PEKSIPENT (FOR THE SECOVn TIME) OF TITE PniLOLOf^rrCAL SOCIETY. MEMBER OP TUB 
 
 MATHEMATICAL SOCIETY, FORMERLY A VICE-PRESIDENT OF THE COLLEGE 
 
 OF PRECEPTORS, AND F 'RMERLY SCHOLAR OF TRINITY 
 
 COLLLGE, CAMBRIDGE, B.A. 1837. 
 
 The first lecture reprinted from the Educational Times for June, July, and 
 August, 1872 ; the second from the same for March, April, and May, 1873 j 
 both with numerous additions. 
 
 Printed off in 1872 ani> 1873, and Published in 1882. 
 
 LONDON: 
 C. F. HODGSON & SON, 1, GOUGH SQUARE, 
 
 FLEET STllEET, 
 
• • • 
 
 • • • 
 
 • ••• 
 
 •• • 
 
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 NOTICE. 
 
 B I 1 ' 
 
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 • • 
 
 These pages are for Teachers only, or for Adult Students. 
 Ihey should on no account be put into the hands of children 
 Their object is to shew Teachers who have studied other 
 works (as those cited on p. 46, note 6, to which I would now 
 add : W. Stanley Jevons, Studies in Deductive Logic, a Manual 
 tor Students. Macmillan, 1880, pp. 304) how to teach Deduc- 
 tive Logic with the greatest amount of simplicity, combined 
 with scientific accuracy, and to instil the principles of In- 
 ductive Logic into the minds of the youngest children. The 
 notes are for the further guidance of teachers or others who 
 wish to pursae the study for themselves. 
 
 The Deductive Logic was printed off in 1872, and has re- 
 mained untouched ever since. In 1873 an Appendix on some 
 points of Higher Deductive Logic, applying my methods to' 
 everything in De Morgan's and Boole's works, was written 
 and the delay in publication arose from an intention to revise 
 this Appendix. But so much has since appeared on the sub- 
 ject which ought to be included, that the idea of printing 
 such an Appendix has been definitively abandoned, and the 
 tract IS published as it stood in 1872. It is believed that the 
 exemplifications by the vowels in words, notation, diagrams 
 and working of the syllogism, here explained, are original, 
 while they appear to be the simplest and most complete 
 hitherto proposed. In the abandoned Appendix it was shewn 
 that the same methods were applicable for the solution of the 
 highest logical problems. 
 
 The Inductive Logic, printed in 1873, was published sepa- 
 rately in that year, with some slight alterations, under the 
 title " On School Inductions, or how to familiarise school 
 children with the principles of Inductive Logic." It now 
 appears as it was originally written. 
 
 ALEX. J. ELLIS. 
 
 25, Argyll Road, 
 
 Kensington, London, W. 
 ■Aj>n/, 1882. 
 
 
*>• • 
 
 It •■ 
 a •• fr 
 
 A 
 
 • • 
 
 '\ 
 
 ERRATA. 
 
 p. 12, line last, diagram (2), for ay i° the left-hand square, read ^p. 
 p. o3,linel,/or:,(;r.y, x.!/) read t,{x.Y, x.y). 
 
 p. 91, foot-note, last lines of paragraphs 10, 11, and 12, for p. 26 .cad 
 p. 86 ; and last line of paragraph 14, for p. 24 read p. 84. 
 To prevent trouhle, the Reader is requested to correct these trifling, 
 but very confusing, errors before reading the pages in question. 
 
 A. J. E. 
 
 I 
 
 CONTENTS. 
 
 Part L— DEDUCTIVE LOGIC. 
 
 ART. 
 
 1. Goethe on Logic, p. 5. 
 
 2. Necessity of Logic for Children, p. 5. 
 
 3. The Old Logic unsuitable for 
 
 Children, p. 6. 
 
 4. The Problem of Deductive Logic, 
 
 p. 7. 
 
 5. The Value of Deductive Logic, p. 7. 
 
 6. The Laws of Attributes, p. 7. 
 
 7. Sorting words for one vowel. 
 
 General condition that the same 
 vowel does not occur in all the 
 words sorted, p. 8. 
 
 8. Sorting words for a second vowel, 
 
 p. 9. 
 
 9. The 4 sets resulting from the 
 
 double sorting, p. 9. 
 
 10. Compounds and Counters, p. 9. 
 
 11. Effect of one sot being absent. 
 
 Expression of presence, absence, 
 and doubt, with Counters, p. 10. 
 
 12. Expression for the presence, absence, 
 
 doubt, and limitation of doubt in 
 writing, p. 11. 
 
 13. The 7 complete Assertions or 
 
 possible arrangements of the 4 
 sets of Art. 8, p. 13. 
 
 14. The 8 usual arrangements (or In- 
 
 complete Assertions), p. 14. 
 
 15. Inconsistency, p. 17. 
 
 16. The transition to ordinary propo- 
 
 sitions purposely delayed, p. 18. 
 
 17. The 8 sets arising from sorting for 
 
 3 vowels, p. 19. 
 
 18. Effect of the sorting by two vowels 
 
 on the sorting by three — or of 
 assertions respecting two vowels 
 on conditions respecting three, 
 p. 19. 
 
 19. From the sorting with respect to 
 
 three vowels to determine the 
 effect of sorting with respect to 
 any two, p. 20. 
 
 20. AVlienihe general condition (Art. 7) 
 
 is not fulfilled. Singu]arity,p. 2 1 . 
 
 ART. 
 
 21. 
 
 22. 
 23. 
 
 24 
 25. 
 
 26. 
 
 27. 
 28. 
 29. 
 
 30. 
 
 31. 
 
 32. 
 33. 
 
 34. 
 
 35. 
 
 36. 
 37. 
 
 38. 
 
 Syllogisms, their nature and solu- 
 tion, p. 21. 
 First kind of Syllogism, p. 22. 
 Its eight cases, and skeleton rule 
 
 for the same, p. 24. 
 Second kind of Syllogism, p. 25. 
 Its eight cases, and skeleton rule 
 
 for the same, p. 26. 
 Third kind of Syllogism, and the 
 use of Indices in the case of 
 numerous limitates, p, 26. 
 Its sixteen cases, and skeleton rule 
 
 for the same. p. 27. 
 Syllogisms with doubtful con- 
 clusions, p. 28. 
 Skeleton rules for the thirty- two 
 cases of syllogisms already con- 
 sidered, p. 29. 
 The eleven Syllogisms of the Old 
 Logic explained, with foot-note 
 on the actual 676 possible 
 syllogisms, p. 29. 
 Use of the skeleton rules in solving 
 complex syllogisms, with com- 
 pleteas8ertionsaspremi8ses,p.32. 
 Use of the skeleton rules in solving 
 
 enthymemes, p. 33. 
 Use of the skeleton rules for finding 
 all the syllogisms with a given 
 conclusion andgivenmean, p. 33. 
 Use of the skeleton rules to deter- 
 mine whether any syllogism is 
 erroneous, p. 33. 
 A conclusion not disproved by 
 shewing that either or both of the 
 premisses is or are false, p. 35. 
 Opponent syllogisms, p. 34. 
 The conclusion combined with one 
 premiss gives a new conclusion 
 consistent with, not the same as, 
 the other premiss, p. 34. 
 Given the resultant, to find pre- 
 misses and the conclusion and 
 other conclusions also, p. 35. 
 
IV 
 
 ART, 
 
 39. 
 
 40. 
 
 41. 
 
 42. 
 43. 
 
 44. 
 
 45. 
 46. 
 47. 
 
 48. 
 49. 
 
 60. 
 
 61. 
 62. 
 
 CONTENTS. 
 
 Written form for contrasting cases 
 of syllogisms, p. 36. 
 
 Transition, from dealing with 
 vowels of words, to dealing with 
 attributes of objects, p. 37. 
 
 Transition to ordinary propositions 
 p. 38. 
 
 Connotation and Denotation, p. 39. 
 
 Analysis of assertions, with a note 
 on Sir William Hamilton's, p. 39. 
 
 Diversity of ordinary expressions 
 for the same assertion, with note 
 on Subject, Copula, and Pre- 
 dicate, p. 43. 
 
 Definitions, p. 45. 
 
 Disjunctive Assertions, p. 46. 
 
 Elementary Text Books to supple- 
 ment this lecture, note, p. 46. 
 
 Other kinds of assertion, p. 47. 
 
 Assertions on the consistency of 
 other assertions, p. 48. 
 
 Notation and Counters for these 
 assertions, p. 49. 
 
 Inconsistent assertions, p. 49. 
 
 Unicates, p. 50. 
 
 ART. 
 
 53. 
 54. 
 55. 
 56. 
 
 57. 
 
 58. 
 59. 
 
 60. 
 61. 
 
 62. 
 63, 
 
 68. 
 
 69. 
 70. 
 
 Complexes, p. 50. 
 Sets of Complexes, p. 51. 
 Complexes of three truths, p. 51. 
 If one complex is present in a set, 
 
 all the rest are absent, p. 51. 
 If several complexes form a unicate, 
 
 all the rest are absent, p. 51. 
 If several complexes are absent, all 
 
 the rest form a unicate, p. 52. 
 Study of the eflfect of one or more 
 
 absent complexes from a set of 
 
 four, p. 52. 
 Ex Absurdo proof, p. 53. 
 Study of eflfect of absent complexes 
 
 in sets of three or four truths, 
 
 p. 53. 
 Method of solving problems, p. 54. 
 64, 65, 66, 67. Examples of di- 
 lemmas from Fowler, pp. 55, 66. 
 Ambiguities, story of Eualthus 
 
 and Protagoras newly completed, 
 
 with a Moral, p. 56. 
 Abridgements, p. 58. 
 Conclusion, with note on the 
 
 character of the Old Logic, p. 58. 
 
 Part II.— INDUCTIVE LOGIC. 
 
 ART. ART. 
 
 71 . Reasons for adding this Part, p. 61. 
 
 72. Inductive Logic deals with the 
 
 Formation of Assertions which 
 are taken for granted in Deduc- 
 tive Logic, p. 61. 
 
 73. Object of this Part, to show Teach- 
 
 ers how to put children on the 
 right track, p. 62. 
 
 74. Limitation to Verifiabilities, pp. 
 
 63-66. 
 
 75. The Object of Inductive Processes 
 
 is from the Known Past and 
 Present to predict the Unknown 
 Future, p. 66. 
 
 76. Inductive Reasoning is Regulated 
 
 Guessing, p. 67. 
 
 77. Its Basis is the assumed Uniformity 
 
 of Nature, p. 68. 
 
 78. Mode of directing a child to feel 
 
 that Relations are fixed, and 
 Conditions variable, p. 70. 
 
 79. Explanations and Illustrations of 
 
 the words: — ^i. Law, p. 71; ii. 
 Chance and Averages, p. 73; 
 iii. Restraint, p. 75; iv. Cause 
 and Eflfect, p. 76 ; v. Empirical 
 Laws, p. 77 ; vi. Popular Reason 
 Why, p. 78; vii. Scientific 
 
 Reason Why, p. 79 ; viii. Nature, 
 p. 81. 
 
 80. Phenomena, Noumena, Observa- 
 tion, Experiment, Test of 
 Authority, Classification, pp. 
 82-85. 
 
 81. Canons of Induction, Plurality of 
 Causes, Three General Laws, 
 Analogy, pp. 85-87. 
 
 82. Inductive Methods, — i. of Agree- 
 ment, p. 87 ; ii. of Diflference, 
 p. 87 ; iii. of Concomitant 
 Variations, p. 88 ; iv. of Residues, 
 p. 88. 
 
 83. The two Main Principles to be 
 taught to children, p. 89, with 
 foot-note on Comte's Fifteen 
 Laws of Primary Philosophy, 
 p. 90. 
 
 84. How children are to be led to form 
 Inductions, p. 91. 
 
 85. Suggested Scholastic Processes in- 
 vohdng — i.AccurateObservation, 
 p. 92 ; ii. Invariable Sequences, 
 p. 92 ; iii. Accidents, p. 93 ; iv. 
 Inquiries, p. 93 ; v. Induction, 
 pp. 93-^94 ; Conclusion, p. 94. 
 
 ••• ,•• 
 
 
 -• 0-' i ' 
 
 • . • • i * 
 
 
 LOGIC FOE CHILDEEN. 
 
 1. Goethe, in that biting satire upon University studies, to which 
 he ffives the shape of advice from Mephistopheles to a btudent, 
 makes his fiend say :-" My dear friend, I advise you, first of all, 
 to attend a course of logic. There your mind will be properly 
 drilled, and laced up in the tightest of boots, so that it may hence- 
 forth slink more circumspectly along the path of thought, and not 
 perchance will-o'-the-wisp-it hither and thither, through thick and 
 thin. Then you'll be taught for many a day, that what you ve 
 been in the habit of doing in a moment, as easy as eating and drink- 
 ing, must have its One ! Two ! Three ! To be sure, the manufacture 
 of thought is like a masterpiece of weaving, where one treadle 
 moves a thousand threads. The shuttles fly hither and thither, the 
 threads flow unseen, a single stroke strikes a thousand combi- 
 nations. Your philosopher comes up, and proves to you that it 
 can't help being so. * The first was so, the second so, and hence 
 the third and fourth were so ; and if the first and second had not 
 been there, the third and fourth would have never come to pass. 
 The students go into ecstacies over his explanation,— but none ot 
 them have become weavers." 
 
 2. Now this is very wittily and pungently expressed, and the 
 
 truth at the bottom of it all is incontestable. Thinking by rule 
 
 will not make an original thinker. But then, however original a 
 
 thinker may be, he can only think according to rule, for the rule is 
 
 merely the register of the processes actually followed by original 
 
 thinkers. We don't expect, and we don't desire, every human 
 
 being born into the world to have the insight of a Cxoethe — not to 
 
 mention an Aristotle, who, himself the towering mind of his day, 
 
 first bethought himself of the advantage that would accrue trom 
 
 accurately stating the processes and laws of reasoning. But eveiy 
 
 one of us, from the grandson to the grandsire, has to go through 
 
 small processes of reasoning every hour of his life, ^very one 
 
 of us, every time he speaks, has to make an assertion. He seldom 
 
 does speak without making several assertions, which m some way 
 
 modify and limit one another. And in point of fact, many ot us 
 
 are often apt to make assertions without knowing precisely what 
 
 B 
 
 » » 
 * 1 
 
 ' * » • • • i^ . 
 
 i 
 
6 
 
 LOGIC FOR CHILDREN. 
 
 [arts. 2, 3. 
 
 ^^ecch cf theiQ meais separately, and still more often without know- 
 ing how they limit previous assertions, or even how to find out the 
 e'ifent^of liyiita'tibn. Hence we are all prone to make blunders, 
 
 -,- some^ya^ips simply ridiculous, often mortifying, and not unfre- 
 quently entailing other disagreeable consequences. Now if there 
 aie.a; Vew simple and general considerations which will help us out 
 
 . of this difficulty, should we grudge the labour of mastering them ? 
 To say that when reasoners have thought out processes of reason- 
 ing, they should not be studied by those who wish to reason, seems 
 about as wise as to say that because people can make themselves 
 understood by picking up so much of a language as strikes their 
 ear, they should not make use of a grammar and a dictionary, or 
 that because all the rules of composition will never make a poet, 
 no simple-minded soul who wishes to convey his thoughts intel- 
 ligibly, and flounders awfully in the attempt, should avail himself 
 of rules of composition. 
 
 3. Now, the College of Preceptors seems to have plainly stated 
 its own opinion, by making a certain (albeit very small) amount of 
 logical knowledge indispensable for passing their diploma exami- 
 nations ; and though their main thought was to guide the teacher 
 himself in the application of methods of teaching, yet there must 
 necessarily have been a secondary object in view referring to the 
 matters taught. Clearly, of all others, children should be led to 
 comprehend the expression of thought, and hence teachers should 
 be put into a condition to train them. Of course, no one supposes 
 that the barbarous terminology and unwieldy processes of scholastic 
 logic, which seem very like sledge-hammers for killing flies, should 
 be presented to the mind of a child, already so much overburdened 
 with the pedantry of grammar. But are these necessary ? Do we 
 require more distinctions than objects distinguished ? more classes 
 than things classified ? a dozen sesquipedalian words to express a 
 single simple notion ? difficult and subtle discriminations, in which 
 the difficulty and subtlety mainly arise from viewing but half the 
 battle ? This is what shocks one in the old logic, which school- 
 men have petted into a monster. 'Tis well that in recent times, so 
 recent that they are within the experience of many here present, 
 two s^reat thinkers, whose loss we have had recently to deplore. 
 Dr. Boole and Professor De Morgan, have shown us a way out of 
 this muddle. It is from these two writers principally that 1 have 
 drawn the materials for my own view of the real scope of so-called 
 deductive logic. On working out a system in detail,* it struck me 
 that the method was eminently adapted for school instruction, and 
 that it could really be adapted, in its early stages, as a method of 
 teaching logic to children. Of course it is impossible, during early 
 school work, to do more than scrape the surface of so great a sub- 
 ject; but the soil is naturally so fertile, that I do not despair of a 
 crop with even such shallow cultivation. 
 
 * In ray paper, entitled " Contributions to Formal Logic," read 
 Royal Society on Thursday, 25th April, 1872. See ' Proceedinas of 
 Society," vol. 20, No. 134, p. 307. 
 
 before the 
 the Royal 
 
 ,u< 
 
 ARTS. 4 — 6.] 
 
 LOGIC FOR CHILDREN. 
 
 4. Let me first confide to you, as a secret which you must 
 strictly keep from youi- pupils, that " the intention of deductive 
 logic IS to determine the precise meaning of any number of given 
 assertions— that is, to determine precisely what they affirmT pre- 
 cisely what they deny, and precisely what they leave in doubt, 
 separately and jointly." This is the whole problem of deductive 
 logic. It has nothing to do with the formation of assertions, 
 i hat IS the special object of induction. It has nothing directly to 
 do with the correctness or incorrectness of the assertions. It takes 
 them as they come, good, bad, and indifferent, and determines 
 precisely what ground they cover, singly and in combination. 
 Hence it must not be imputed as a fault, that logic leads to no 
 new discovery; that the conclusion is involved in the premisses- 
 that It merely teUs us in other words, as a final assertion, what has 
 been already sufficiently told in the preliminary assertions. This 
 IS no fault m logic; this is its aim. It is just because people do 
 not in general know the full force of each assertion they launch the 
 number of cases which it settles, and more especially the number of 
 cases which it leaves entirely unnoticed ; it is just because people 
 do not see the combined effect of their assertions, do not know 
 what the premisses do and do not contain or imply; it is iust for 
 these reasons that it becomes of the last importance to have an 
 easy, straightforward, infallible, and general method of finding all 
 this out. Herein hes the raison d'etre of deductive logic. 
 
 5. The value of this method, in indirectly determining the cor- 
 rectness of assertions, cannot be over-estimated. It is only when 
 we see every case which an assertion does and does not cover, that 
 we can confront it with fact. Hence, in every stage of induction, 
 deduction IS imperative. All induction consists in the formation of 
 successive hypotheses, which have to be stated as assertions, and 
 these hypotheses have to be put to the test of observation and ex- 
 periment in the well-known methods. By deduction, and by de- 
 duction alone, can we be sure of conducting this test with the requi- 
 site thoroughness. Of course such applications imply a very much 
 deeper knowledge of the subject than can be presented to children ; 
 \^}l f J^^^^^^^ "Ot, sufficient foundation can be laid in very 
 youthful days for a superstructure which will prove advantageous 
 to the most adventurous investigator of nature. To-night I shall 
 only deal with two kinds of assertion, and I shall not advance far 
 111 their consideration ; but the method I shall explain is general, 
 and It will be found that most other problems of deduction differ 
 Irom those now adduced in complication only, not in principle. 
 
 6. Language has become possible solely because "different at- 
 tributes frequently reside at the same time in the same thing, and 
 the same attribute generally resides at the same time in different 
 things, where the word "things" is used with the utmost gene- 
 rality, for all objects of thought, such as material objects, their 
 qualities and actions, states, conceptions, feelings, relations, in 
 short, whatever can be named. Now, of course, no teacher would 
 be mad enough to present children with such excessively abstract 
 general propositions. But if he wishes to make his pupils think, - 
 
 B 2 
 
B 
 
 LOGIC FOE CHILDREN. 
 
 [arts. 6, 7. 
 
 he must contrive some method of making them perfectly familiar 
 with this fact, which pervades all language, and is the foundation 
 of the first class of assertions that they will have to analyze, a 
 class which occupies by far the greater part of all treatises on logic. 
 Of course this must be effected by presenting the pupils with con- 
 crete objects of thought ; and the method which occurs to me as 
 most convenient to the teacher, and at the same time as the best 
 introduction to the future expression of abstract logical relations, 
 is the following. It is based on the supposition that a teacher 
 would not be called upon to give lessons on logic to pupils who 
 cannot read and write. 
 
 7. The teacher requests each pupil in his class to write down 
 about twenty English words, each on a separate piece of paper. 
 The choice of words should be left to the pupils, but for conveni- 
 ence of writing they should be recommended to choose " short" 
 words. This done, let each pupilsort his words, putting on the left 
 hand all those containing A, and all the rest on the right. Let this 
 right-hand heap be well looked through to ascertain that not one 
 of them contains A. All the words are therefore divided into two 
 heaps, which we will call A-words, and non-A-words, and in writ- 
 ing we will employ a little a in place of the words " non-A'*, so 
 that the second heap contains a-words only. If each pupil has been 
 able to make this division, it will be felt that it can always be 
 made. But perhaps one or two pupils will have no A-words, or no 
 a-words. If this should not happen to be the case, make instances 
 by requesting one pupil to throw away his A-words, and another 
 pupil his a-words. Then an opportunity will be offered for show- 
 mg that this was not a peculiarity of our language, because other 
 pupils have found words of the missing kinds, but that it merely 
 depended upon the choice of words, that is, upon the words thought 
 of, or upon the " range of thought," a most useful and important 
 conception. Then explain that, to get the most common, or the 
 most general case, we ought to lay down as a rule or general con- 
 dition, that when we have to look alter A-words, we should have " at 
 least one" A-word, and at least onea-word word, within the "range of 
 thought," because the exceptional cases are very easily hunted up 
 when the general cases are known (art. 20). Perhaps some word, 
 as " papa, avail, abase," may contain more than one A. If no such 
 words occur, let the teacher supply some on the black board, and 
 ask to which set they belong. Clearly to the A-words, because 
 they contain " at least one" A, and that is all we mean by an A- 
 word. Hence, if we suppose the ** general condition" just stated 
 to be fulfilled, — and ivhatever letters we taJJc about we shall hereafter 
 suppose it to be fulfilled — all the words within the range of thought 
 can be separated into two sets, the first or A-words, containing 
 *' at least one" A, and the second or a-words, containing ** not even 
 one" A. No word can belong to both sets, which are perfectly ex- 
 clusive one of the other. It is worth while bestowing great pains 
 on this elementary conception, which is the basis of all classifica- 
 tion, and which, expressed in the usual abstract language of logic, 
 is intensely perplexing to a child's mind. 
 
 t ! 
 
 ^ 
 
 1 i 
 
 ( 
 
 K 
 
 ARTS. 8 — 10.] LOGIC FOR CHILDREN. 
 
 9 
 
 8. We now advance a stage. Take each of the two sets of A- 
 words and a-words, and examine them for E. Observe that 
 vowels are selected by the teacher because all words have some 
 vowel or other. The A-words can then be divided into two heaps, 
 containmg at least one E, and containing not even one E. These 
 will then be E-words and e-words. But in order to show that they 
 have been formed out of the single set of A-words, prefix A to 
 each and call the sets AE-words, (read "A, E, words") and Ae- 
 words (read "A, non-E words"). Do the same with the a-words, 
 and thus find two other sets of aE-words, (read "non-A, E 
 words, ) anda^-words (read "non-A, non-E words"). Of course 
 the general condition has been complied with, so that there is at 
 least one E-word and one e-word. But it will not always happen, 
 even then, that each pupil will have all four sets of AE Ae aE 
 ae words. Having ascertained whether any one has got all four 
 sets, begin with him. ° 
 
 9. Then the AE-words and aE-words are two sets which could 
 have been formed out of the one set of E-words, supposing that 
 the list of words had been examined for E first. And the Ae and 
 ae are two sets which could have been formed out of the one set 
 ot e-words. This must not merely be acknowledged, but prac- 
 tically done. Let one pupil that has all the four sets of words 
 read out his list, and let the teacher write it on the black board 
 thus, — 
 
 ^E Ae aE ae 
 
 E 
 
 1 
 
 
 
 i 
 
 
 
 got 
 
 put 
 
 mutton 
 
 slit 
 
 round 
 
 ale 
 
 bate 
 
 seat 
 
 have 
 
 amen 
 
 ail 
 
 papa 
 
 cat 
 
 sham 
 
 lady 
 
 feet 
 
 sieve 
 
 seize 
 
 conceive 
 
 people 
 
 The continuous lines group together all the A- and all the E- 
 words ; the dotted lines group together all the a- and all the e-words. 
 Ihis being understood, merely writing such Hues with A and E at 
 
 rlil ^^'."""Tf •' ^-^^ ^""^""l *° '^'^^' ^y overlapping, how the two 
 different subdivisions can be made. Thus a mode of forming logical 
 diagrams is suggested, which will be found of the greatest use. 
 
 10. Now observe that, instead of selecting words, it would have 
 been sufficient to have written upon at least one of the slips of 
 paper the letter A, implying that at least that one slip con- 
 tained or represented a word containing A, and to have written on 
 all the rest the letter a, implying that all these represented words 
 without A, and then to have written E upon at least one of all these 
 Slips, and e upon all the rest, implying respectively that the slips 
 represented words containing or not containing E. In this way 
 every one of the slips would bear one or other of the marks AE, 
 Ae, aJli, ae, and these marks would represent the " names" of these 
 
10 
 
 LOGIC FOR CHILDREN. [aRTS. 10,11 . 
 
 words considered in relation to their containing or not containing 
 the letters A and E. The names A and a would be " simple" 
 names; but AE, Ae, aE, ae, would be "compound" names, and 
 may be briefly termed compoimcU. This being well understood 
 and tried with various slips of paper,* will form the basis for a 
 mechanical arrangement of great value in teaching. The number 
 of slips of paper containing each ** compound" is evidently unim- 
 portant for ascertaining the fact that there h at least one such 
 compound. Hence four slips of paper — or, for greater convenience 
 of handling, four slips of pasteboard, or " counters" — may be used, 
 marked in bold letters with the names of the compounds AE, Ae, 
 aE, ae, respectively. For class teaching these counters should be 
 six inches long and three high, and the capital letters should be 
 more than two inches high, and very clear and bold. These are 
 readily drawn with a camel-hair pencil dipped in hlach ink. The 
 back of each counter, for a purpose immediately to be explained, 
 should be marked with the same letters in red ink. If four other 
 counters be mai-ked with A, a, E, e, respectively, and little ledges 
 be arranged for resting them on, the double arrangement may be 
 
 ehown thus, — 
 
 A a I E 6 
 
 AE Ae «E oe I AE aE Ae ae 
 So that, in fact, we have only to transpose Ae, aE, in order to con- 
 vert the first into the second. Be careful that the pupil under- 
 stands that each of these counters stands for a set of words con- 
 sisting of at least one word. Each counter therefore represents a 
 column, Yrith at least one word in it.§ 
 
 11. Now suppose, as is most likely, that some of the pupils have 
 not supplied words which can be fitted into every column, so that 
 at least one column is empty. The question then arises (and it is 
 one of primary importance), in how many different ways is it pos- 
 sible to have empty columns, consistent with the fulfilment of the 
 "general condition"? Tliis the pupils must be led to answer for 
 themselves. Take a pupil that has only three sets. Suppose AE 
 to be " absent," that is, that there is not even one word in the AE 
 
 ♦ Beoin by writing the compound names on the front or back of the 
 actual slips of paper containing words ; as, mate AE, mot aE, &c. 
 
 § A complete set of counters for the first kind of assertions consists of 
 twenty-six pieces, namely A, a, E, ^, I, t ; AE, A^, aE, ae ; AI, A«, al, ai ; 
 EI, Ej, el, ei ; AEI, AEj, A^I, A^», aEI, aEt", a^I, aei ; all with black 
 letters on one side, and red on the other. Also a set of eleven indices, three 
 inches high and one and a-half inch wide, black on one side only, namely 
 2i, 2i, %y 22, 23, 23, 24, 24, 3i, 82, 83. For the second set of assertions thirty 
 pieces are "required, namely, twenty-six black on one side and red on the 
 other, X, x, Y, y, Z, 2 ; X. Y, X.y, ir.Y, x.y ; X.Z, X.«, x.Z, x.z ; Y.Z, Y.2, 
 y.Z, y.z ; X.Y.Z, X.Y.z, X.y.Z, X.y.r, a;. Y.Z, x.Y.z, x.y.Z, x.y.z ; and four 
 black on one side only, X', X'', x', xf'. The uses of these forms will be 
 explained in due course. For classes of two or three persons, smaller 
 counters, made on blank address caids, will suffice ; and as these may be 
 laid on a table, no ledges to support them are necessary. TJnlcBs the reader 
 will cmstrnct caunters of some hind, and use them as directed, he cannot pro- 
 perly appreciate the method. 
 
 { 
 
 ARTS. 11, 12.] LOGIC FOR CHILDREN. 
 
 11 
 
 / 
 
 column. Show this by turning the red side of the AE counter, 
 which will therefore mean an empty column. Now is it quite 
 necessary that when AE is absent, Ae, aE, ae, should be all "pre- 
 sent, ' that IS, that there should be at least one word in each of 
 these columns ? Try this. What does the general condition de- 
 mand f I hat there should be at least one A-word and one a-word 
 But what is implied by there being at least one A-word? Ali 
 A-words, as we have seen, must be AE-words or Ae-words. There 
 ^e no AE-words, there must be, therefore, at least one Ae-word 
 Hence Ae must be present. The Ae counter is therefore ranged 
 with the black side out, beside the red AE. Again, there must be at 
 least one E-word. But all E-words are either AE-words or aE- 
 words. There are no AE-words. There must, therefore, be at 
 least one aE-word. Place a black aE on the ledge. Then observe 
 that, as we have Ae and aE, we have already satisfied the general 
 condition for e-words and a-words, and hence cannot tell whether 
 there are any ae-words or not. There may be, or may not be, any ; 
 there must be at least one, or else not even one ; but when we say 
 there are no AE-words, we say nothing about the ae-words. 
 Their presence is therefore " doubtful." Mark this by putting the 
 ae counter on the ledge, black side out, but inverted, thus ["^ 
 
 9D 
 
 \ 
 
 This doubtful compound must be well understood. Revert to the 
 general condition. There must be at least one a-word All 
 a-words are aE-words, or ae-words. We know that there is at 
 least one aE-word, and this gives at least one a-word, because we 
 only want a word without A, and whether it contains E or not is 
 quite indifferent. Hence we cannot tell whether there are any 
 ae-words from this consideration. Similarly from knowing that 
 there must be at least one e-word, and that there is at least one 
 Ae-word, we cannot tell whether there is any ae-word or not 
 This is very important for all future applications of reasoning. 
 
 We see then that absent AE gives present Ae, and present aE, 
 but doubtful ae ; but that in any real case that can be formed, the 
 doubt will be resolved when all the words are known, and 'that 
 there will be either present ae or absent oe. Cases, however, may 
 well be imagined where all the words are not known, but only the 
 fact that none contain both A and E, and these, in the applications 
 are the usual cases. ' 
 
 12. Now the counters are very convenient, nay, almost indis- 
 pensable, in teaching, but for registering the results we require 
 other signs. We cannot use the distinction of upright black 
 letters for present, inverted black for doubtful, and red for absent 
 compounds. Hence it is convenient to use abbreviations for these 
 words. In my Complete System of Logical Notation, I have 
 found it most convenient to use — 
 
 J for present, P for doubtful, and f for absent,— 
 as shown in the following cases, where the exact meaning of each 
 symbol should be rendered quite plain, and no use of " some" for 
 "at least one," or "no" for "not even one," should be allowed, as 
 the words " some, no" have occasioned groat confusion. 
 
12 
 
 LOGIC FOR CHILDREN. 
 
 [art. 12. 
 
 ARTS. 12, 13.] LOGIC FOR CHILDREN. 
 
 13 
 
 JAB, read " present, A,E," means that " there is at least one 
 word within the range of thought which contains at least one A 
 and at least one E." 
 
 JAe, read " present, A, non-E," means that " there is at least 
 one word within the range of thought which contains at least one 
 A and not even one E." 
 
 JaE, read " present, non-A, E," means that " there is at least 
 one word within the range of thought which contains at least one 
 E and not even one A." 
 
 Joe, read "present, non-A, non-E," means that "there is at 
 least one word within the range of thought which contains not 
 even one A, and also not even one E." 
 
 fAE, fAe, t^E, t^xe, read respectively "absent, A, E," 
 " absent. A, non-E," " absent, non-A, E," and " absent, non-A, 
 non-E," mean that " there is not even one word within the range 
 of thought which contains, respectively, at least one A and at least 
 one E ; at least one A and not even one E ; at least one E and not 
 even one A; and not even one A and not even one E." 
 
 While PAE, PAe, PaE, ?ae, would imply a doubt arising from 
 the absence of information as to whether J AE or fAE, whether 
 JAe or fAe, whether JaE or faE, whether Jae or fae respectively. 
 
 To embrace the cases where the presence or absence of every 
 single one of the four compounds is declared, it is convenient to 
 compound these signs, thus — 
 
 tj for all present with, or pre-present, 
 Jt for all present hut, or pre-absent, 
 tt for one absent with, or ab-absent, 
 the precise use of which is shown in the next article. 
 
 We have already seen that the general condition that there 
 should be at least one A-word, obliges us to acknowledge that 
 there must be at least one AE-word, or else at least one Ae-word, 
 and that there may be more than one of either or both sets. That 
 is, at least one of the compounds AE, Ae must be present, and both 
 may be present. That is, we may have JAE and JAe, or JAB 
 and fAe, or fAE and JAe, but cannot have fAE and fAe. This 
 very curious limitation is constantly occurring in logic, and 
 requires a sign. It is expressed by — 
 
 J' (AE, Ae) read "present at least one of AE and Ae." 
 With the counters it is expressed by inverting both, and placing 
 them close together thus : | gy I ay 
 
 Observe that the general conditions give four such combina- 
 tions, called limitates, namely, 
 
 JHAE, Ae), +HAE, aE), t^(Ae, ae), J^(aE, ae,) 
 
 Combinations of these may be indicated by groupings of the 
 counters. Thus the two first might be (1), and the two last might 
 be (2), while the whole four sets might be (3). 
 
 (1) (2) (3) 
 
 ay 
 
 a» 
 
 ay 
 
 ay 
 
 ay 
 
 9V 
 
 ay 
 
 9Y 
 
 a» 
 
 9V 
 
 if 
 
 i \ 
 
 The inversion shows the really doubtful nature of each indi- 
 vidual compound, and the grouping shows the limitation of that 
 doubt. Another method of marking this fact is given in art. 26. 
 
 The following signs are also useful in writing : 
 II for means, and 
 ( • ) for and, with, together with. 
 
 These signs must only be introduced gradually as a means of 
 easily writing what the counters indicate, and the pupils must be 
 exercised in passuig from the written forms to the counters, and 
 thence to the sets of words themselves, as the real things under 
 consideration. 
 
 13. We may now note all the cases which can possibly happen 
 regarding the separation of vrords into four sets, and each case 
 must be rigorously demonstrated by the pupils themselves, in the 
 manner described in art. 11. The continuous and dotted lines show 
 immediately how the columns would overlap, and what columns 
 would be absent ; but each case should also be illustrated by words, 
 and, if possible, the actual cases which occur among the pupils 
 should be taken first ; if these are not enough, the pupils, rather 
 than the teacher, should be led to furnish the rest. Pupils should 
 also be exercised in constructing diagrams for themselves, giving 
 various lengths to the dotted and continuous lines, interrupting 
 dotted by continuous lines, and so on, till they feel that the length 
 of these lines is of no consequence, (because they merely indicate 
 the breadth of columns containing words ; and as the length of 
 those columns is indefinite, any number of words can be written 
 in any column, however narrow, ) but that the overlapping of the 
 dotted and continuous parts in the two lines is everything, (be- 
 cause each overlapping shows the presence of a certain set of 
 words). 
 
 The Seven Possible Arrangements (or Complete Assertions). 
 
 i. JJAB, "all present," also written JJAe, JJaE, or JJoe, 
 means present AE, and present Ae, and present aE, and present 
 ae, or in the abbreviated form, || JAE • JAe • JaE * Jae ;. 
 
 f A that is, at least one word occurs in 
 
 ( jj each one of all the four sets. 
 
 { 
 
 'K 
 
 ii. JfAE, " aU present but AE," || fAE • JAe • JaE • Jae ; 
 
 A there is not even one word of the 
 
 B first set, but there is at least one 
 
 word in each of the other three sets. 
 
 iii. JfAe, " all present but Ae," || J AE • fAe • JaE • Joe ; 
 
 A there is not even one word of the 
 
 J] second set, but there is at least one 
 
 word in each of the other three sets. 
 
 iv. JfaE, " all present but aE," || JAE • JAe • faE • Jae ; 
 
 A there is not even one word of the 
 
 jg third set, but there is at least one 
 
 word in each of the other three sets. 
 
14 
 
 LOGIC FOR CBILDREN. [ARTS. 13, 14. 
 
 { 
 
 V. Ifae, " all present but ae," || {AE • J Ae • JaE • fae ; 
 
 A there is not even one word of the 
 
 E — fourth set, but there is at least one 
 
 word in each of the other three sets. 
 
 vi. tfAE, "one absent with AE," also written ffae, "one 
 absent with ae" (because AE and ae can be absent together, 
 and neither Ae nor oE can be absent at the same time with 
 either AE or ae, consistently with the "eceneral condition.") 
 (I tAE-JAe-JaE-fae; 
 
 (A there is not even one word of the 
 
 i^ first and fourth sets, but there is at 
 
 . . least one word in the second and 
 
 also m the third set. This is the arrangement of complete diversity, 
 for (1) no A-words are E-words, (2) no E-words are A-words, 
 (3) no a-words are e-words, and (4) no e-words are a-words. 
 
 vii. tfAe, "one absent with Ae," also written ffaE, "one 
 absent with aE," (because Ae and aE can be absent together, 
 and neither AE nor ae can be absent at the same time with 
 either Ae or aE, consistently with the "general condition") 
 II JAE-fAe-faE'+ae; ' 
 
 [ 
 
 A 
 E 
 
 there is not even one word of the 
 
 second and third sets, but there is 
 
 least one word of the first and also 
 of the fourth set. This is the arrangement of complete identity, 
 for (1) every one of the A-words is an E-word, (2) every one of the 
 E-words is an A-word, (3) every one of the a-words is a e-word, 
 and (4) every one of the e-words is a a-word. Hence, so far as 
 the words thought of are concerned, it would be indifferent 
 whether we said any given word were an A-word or an E-word, 
 a a-word or a e-word. But the possession of an A is a very dif- 
 ferent mark from the possession of an E. Hence ffAe or ffaE 
 points to identical groups formed by means of different marks A 
 and E. This is really a point oi great importance. 
 
 These seven arrar-T'ements are the only arrangements that can 
 possibly occur consistently with the general condition. This 
 fact should be made perfectly clear to the mind of the pupils, and 
 not a step further should be taken until every pupil is familiar 
 with it, and has tested it in every possible way, because it is the 
 fundamental fact of all deduction. 
 
 14. Now make the following observations, leading to — 
 
 The Eight Usual Arrangements {or Incomplete Assertions). 
 i. t AE (read strictly according to the interpretation given in 
 art. 12) occurs in 5 of the 7 arrangements, namely, 1, 3, 4, 5, 7, 
 and does not occur in 2, 6. That is, if I know that one of the 
 pupils has an AE-set, I know that he must have one of those five 
 arrangements ; but, not knowing which of the five, I can't tell 
 where he has an Ae-set (present in 1, 4, 5, but absent in 3, 7) or 
 an aE-set (present in 1, 3, 5, but absent in 4, 7), or an ae-set (pre- 
 
 Mifc- 
 
 i 
 
 ( 
 
 ( 
 
 ART. II.] 
 
 LOGIC FOR CHILDREN. 
 
 16 
 
 sent in 1, 3, 4, 7, but absent in 6). The limited knowledge, JAE, 
 consequently leaves the presence or absence of each of the other 
 three sets of words, Ae, aE, ae, in doubt, but not in unlimited 
 doubt. This we gather also from the four limitates given in art. 12. 
 Thus JAE in conjunction with the two JM^E, Ae) and 
 J'(AE, aE), leaves Ae, aE in unlimited doubt. But the two other 
 limitates l^{Ae, ae) and JK^E, ae), limit this doubt, by showing 
 that there are only five possible cases of the presence and absence 
 of Ae, aE, ae, which are easily deduced, and seen to be the same 
 as those pointed out by the seven complete arrangements. This 
 is very easily done with the counters and then registered, and 
 every pupil should be made to do it for himself, and write out his 
 process at length. It is an important and very easy exercise. 
 Hence JAE implies J* (Ae, ae) ' J» (aE, ae). 
 
 /-^ The diagram will have an incom- 
 
 ? jj plete line for E, and this line can 
 
 be filled up so as to make any one 
 of the diagrams 1, 3, 4, 5, 7, and hence satisfy the two limitates, 
 but can not be filled up so as to form either of the diagrams 2, 6. 
 The teacher must see that the pupil actually fills up the diagram 
 in all these ways for himself, and feels that there are no other ways 
 in which it can be filled up, and so on in each of the following cases. 
 
 ii. JAe occurs in the five arrangements 1, 2, 4, 5, 6, and does not 
 occur in 3, 7. And by the same process as before we see that JAe 
 implies X\A.E, aE) • t^(aE, ae), and may be represented thus : 
 
 4 A which may be filled up into the 
 
 I E diagrams of 1, 2, 4, 6, 6, only. 
 
 iii. JaE occurs in the five arrangements 1, 2, 3, 5, 6, and does 
 not occur in 4, 7; so that JaE implies J^AE, Ae) * JHAe, ae), 
 and may be represented thus : 
 
 VA which may be filled up into the 
 
 i E diagrams of 1, 2, 3, 5, 6, only. 
 
 iv. Jae occurs in the five arrangements 1, 2, 3, 4, 7, and does 
 not occur in 5, 6 ; so that Jae implies J^AE, Ae) • J^(AE, aE), and 
 may be represented thus : 
 
 J A which may be filled up into the 
 
 i E diagrams of 1, 2, 3, 4, 7, only. 
 
 V. fAE occurs in the two arrangements 2, 6, and does not occur 
 in 1, 3, 4, 5, 7, so that fAE implies JAe • JaE, but gives Poe, where 
 the doubt is absolutely unlimited, and may be represented thus : 
 
 CA which may be filled up into the 
 
 <B diagrams 2 or 6, but into no other. 
 
 Hence we see that (1) not even one 
 of the words within the range of thought that contains A also 
 contains E, but that (2) every one of the words containing A is 
 without E, (3) every one of the words containing E is without A, 
 and (4) it is doubtful whether any one word without A will also 
 be without E, and any one word without E will also be without A. 
 
16 
 
 LOGIC FOR CHILDREN. 
 
 [art. H. 
 
 All these conclusions to be drawn from fAE should be made per- 
 fectly familiar. It should also become manifest that fAE is 
 implied by any one of the first three of these four assertions fin 
 conjunction with the " general condition"), but that the fourth 
 assertion makes no addition to our knowledge, because it would 
 be applicable to any one of the seven arrangements, one of which 
 we know must happen, even when no statement at all has been 
 made. 
 
 vi. fAe occurs in the two arrangements 3, 7, and does not occur 
 in 1, 2, 4, 5, 6, so that fAe implies JAE • Jae, but gives ?aE, and 
 may be represented thus : 
 
 f A which may be filled up into the 
 
 (E diagrams 3 or 7, but into no other. 
 
 Hence we see that (1) not even one 
 of the words within the range of thought, that contains A is 
 without E, but that (2) every one of the words containing A also 
 contains E, and (3) every one of the words without E is also with- 
 out A, and (4) it is doubtful whether any one word without A will 
 have an E, and whether any one word which has an E will be 
 without A. All these conclusions from fAe should also be quite 
 familiar, as also that fAe is the one sufficient representation of 
 the first three of these four assertions, and that the fourth asser- 
 tion makes no addition to our knowledge. 
 
 vii. faE occurs in the two arrangements 4, 7, and does not 
 occur in 1, 2, 3, 5, 6, so that faE implies JAE • ae, but gives ?Ae, 
 and may be represented thus : 
 
 £A which may be filled up into dia- 
 
 JE grams 4 and 7, but into no other. 
 
 . 1 • , Hence we see that (1) not even one 
 
 ot the words within the range of thought, that contains E is with- 
 out A, but that (2) every one of the words containing E also con- 
 tains A, and (3) every one of the words without A is also without 
 E, and (4) it is doubtful whether any one word which has an A 
 will be without E, or any one word without E will have an A. 
 Also observe that faE is the one sufficient representation of the 
 first three of these four assertions, and that the fourth assertion 
 makes no addition to our knowledge. 
 
 viii. fae occurs in the two arrangements 5, 6, and does not 
 occur in 1, 2, 3, 4, 7, so that fae implies JAe • JaE, but gives PAE, 
 and may be represented thus : 
 
 S^ which may be filled up into dia- 
 
 l^ grams 5 and 6, but into no other. 
 
 „^, - . , . , Hence we see that (1) not even one 
 
 ot the words within the range of thought which is without A is 
 without E, or which is without E is without A, but that (2) every 
 on-e of the words without A contains E, and (3» every one of the 
 words without E contains A, and (4) it is doubtful whether any 
 word containing A contains E, or any word containing E contains 
 A. Also observe that fae is the one sufficient representation of 
 
 I 
 
 ^ 
 
 ) h 
 
 \ 
 
 ( 
 
 AETS. 14, 16.] LOGIC POE CHILDEEN. 
 
 17 
 
 the first three of these four assertions, and that the fourth asser- 
 tion makes no addition to our knowledge.* 
 
 15. Let the teacher now draw the pupil's attention (by ques- 
 tions) to the facts (1) that no set of words can be both present and 
 absent, (2) that every one of the 7 arrangements asserts the pre- 
 sence (or absence) of some set of which the absence (or presence 
 respectively) is asserted in every one of the other arrangements, 
 and hence (3) that no two of the 7 arrangements can hold good for 
 the same original selection of words or range of thought, and (4) 
 that this fact is expressed by saying that assertions which imply 
 that any twO of the arrangements hold at the same time are incon- 
 sistent, so that one or the other of them must he false, but (5) only 
 the arrangements of 6 and 7 are totally inconsistent, or contradic- 
 tory, because of only these two can it be said that one asserts the 
 presence of every set of which the other asserts the absence. 
 
 A s regards the 8 incomplete (in relation to the 7 complete) ar- 
 rangements, any one of the first 4, (jAE, lAe, JaE, Xae,) is, as 
 appears in art. 14, consistent with 6, and inconsistent with 2, of 
 the 7 complete arrangements, and any one of the last 4, (fAE, fAe, 
 
 * There are 11 other incomplete arrangements, of a complex character, 
 which are here added for the convenience of the teacher : — 
 
 i. :(A) II JAE 'XAe't' (aE, ae). | 
 
 ii. J(E) II tAE-taE-Ji(Ae,ae). |^ 
 
 iii. t(a) II JaE • Xae • X' (AE, Ae). |^ 
 
 iA 
 E- 
 
 A 
 
 B- 
 
 iv. X{e) II tAe-ta^-Ji(AE,aE). | 
 
 V. J(AE, ae) \\ JAE • ?Ae • PaE • Jae. \^ 
 
 (E- 
 
 vi. J(Ae, aE) H PAE • JAe • JaE • ?ae. \^ 
 
 »E- 
 
 vii. J(A, E) II JAE • JAe • JaE • ?ae. \^' 
 viii. t(A, e) II JAE • JAe • PaE • Jae. \^ 
 
 ix- JK E) II JAE • PAe • JaE • Jae. l^' 
 X. J(a, e) II PAE • JAe • JaE • Jae. | 
 
 E 
 
 A 
 
 E 
 
 xi. PjAEI II Ji(AE, Ae)-J'(aE, ac) 5A 
 
 •J'(AE,aE).J»(Ae,ae).^E 
 
 These 26 assertions (7 complete, 8 simple incomplete, 1 1 complex in- 
 complete) are all that can be made respecting two letters (or names). 
 The importance of this complete enumeration will appear in art. 30, note. 
 
18 
 
 LOGIC FOR CHILDREN. [ARTS. 15, 16. 
 
 frtE, fae,) is consistent with 2, and inconsistent with 5, of the 
 same. This consistency arises from the incompleteness of these 
 assertions, by virtue of which all the inconsistent portions are left 
 out of consideration. The teacher will observe for subsequent use, 
 the bearing of this last remark upon all cases of general asser- 
 tions, and on the distinction between abstract and concrete. It is 
 only by dwelling on points of known resemblance, and passing 
 over points of known or possible dissimilarity, that we are able to 
 arrive at the notion of general laws, or invariable relations appre- 
 hended in the midst of infinite variability. 
 
 Lastly, comparing the 8 incomplete arrangements with each 
 other only, observe that any one of the first four, (JAE, jAe, JaE, 
 Jae,) is consistent with 6, and inconsistent with only one of the 
 rest. Thus JAE is consistent with JAe, JaE, Jae, fAc, faE, 
 fae, and inconsistent with fAE alone. Also, any one of the last 
 four, (fAE, fAe, faE, fae,) is consistent with 4, and inconsistent 
 with 3, of the remainder, (admitting the general condition). Thus 
 fAE is consistent with JAe, JaE, (both of which it implies,) and 
 also with Xae, fae, (neither of which it implies,) and inconsistent 
 with JAE, fAe, faE. Hence we disprove JAE, by proving fAE, 
 and we disprove fAE by proving JAE, or else either fAe or faE, 
 admitting the general conditions ; for if they are not admitted, so 
 that we might have fA or fE, there would be nothing to prevent 
 fAE coexisting with either fAe or faE. But there is this differ- 
 ence between these disproofs, that JAE disproves fAE directly, 
 without reference to the general condition; and either fAe or 
 faE disproves fAE by implication only, in consequence of the 
 general condition, and this distinction has been recognized in the 
 old logic, which did not explicitly introduce the general condition. 
 Thus, according to the old logic, fAE and JAE, or fAe and JAe, 
 or faE and JaE are contradictories, but fAE and fAe, or fAE 
 and faE, are contraries. The old logic also calls JAE and JAe or 
 JAE and JaE, suhcontrarics, the real relations being, as we have 
 already seen J^AE, Ae) and JHAE, aE). 
 
 Observe that JAE and fAE, if the general condition be ad- 
 mitted, do not affirm and deny the existence of precisely the same 
 sets of words, but merely of one particular set of AE-words. 
 Written at full, they are JAE • J»(Ae, ae) ' J»(aE, ae), and fAE ' JAe * 
 J aE • ? ae ; whence it appears at once that fAE is perfectly con- 
 sistent with 2 out of the 5 arrangements of J'( Ae, ae) • J'(aE, ae), 
 namely, either jAe ' JaE ' lac or J Ae ' JaE • fae, but not with the 
 3 remaining arrangements, which involve either fAe or faE, or 
 both fAe, and faE. 
 
 A clear comprehension of these relations of consistency and in- 
 consistency is of great value in actual reasoning, and hence it is 
 worth the teacher's while to dwell on them. However perplexing 
 they may be as abstract propositions, they become simplicity it- 
 self when referred to the concrete case of words grouped accord- 
 ing to the vowels A and E. 
 
 16. The above observations are all so simple, and evident when 
 made upon the letters forming words in the manner suggested 
 
 I 
 
 I 
 
 •a 
 
 I M 
 
 \ [t 
 
 1 
 
 ARTS. 16 — 18.] LOGIC FOR CHILDREN. 
 
 19 
 
 that they may appear merely play to the child, and the teacher 
 may run the risk of hurrying over them too fast. Let me impress 
 upon any one who adopts the method I am explaining, to guard 
 against any such error. Every particle of what has here been sug- 
 gested is of extreme importance afterwards, and a correct general 
 view of the usual logical relations and assertions cannot be ob- 
 tained without a perfect mastery of such apparently self-evident re- 
 lations, which can be best acquired by confining attention to one 
 simple but sufficient class of phenomena, such as the vowels con- 
 tained in English words ; and although the step to general propo- 
 sitions referring to matters usually spoken of, might now be easily 
 taken, I believe it will be best to postpone their consideration until 
 a certain amount of dexterity of manipulation has been acquired in 
 the use of the preceding symbols or counters, by proceeding to 
 syllogisms, still solely relating to the presence or absence of certain 
 vowels in a certain set of English words chosen at random, or sub- 
 ject to certain conditions of presence and absence. This is the 
 course which I proceed to explain. 
 
 17. Consider words in respect to three vowels. A, E, I. First 
 
 form AE, Ae, aE, ae as before. Then words c^-jtaining both A and 
 
 E, must contain I or not, and hence if JAE. there must be J^AEI, 
 
 AEi). Thus increase is an AEI-word, hut ;please is an AEi-word. 
 
 Similarly if JAe, then J^ (Ael, Aei.) Also if JaE, then J^ (aEI, 
 
 aEi), and if Jae then J^ (ael, aei). Now, when we do not know 
 
 what words have been selected, we only know that all or some of 
 
 the four compounds AE, Ae, aE, ae may be present, and hence 
 
 that all or some of the eight compounds AEI, AEi, Ael, Aei, aEI, 
 
 aEi, ael, aei may be present; and no others. Now begin with EI, 
 
 Ei, el, el, and show that if JEI, then J* (AEI, aEI) and so on. The 
 
 same eight compounds will result and no others. Agam begin with 
 
 A I, Ai, al, ai, and show that if JAI, then J^ (AIE, Ale) and so on. 
 
 The same eight compounds again result. All this is well shown 
 
 by counters marked with these compounds, thus — 
 
 AE 
 AEI, AEi 
 
 Ae aE ! ae 
 
 Ael, Aei aEI, aBi ael, aei 
 
 And then altering the top line, show that you have only to shift 
 the counters thus : 
 
 AI 
 AEI, Ael 
 
 Or thus : 
 
 EI 
 AEI, aEI 
 
 Ai 
 AE/, Aei 
 
 Ei 
 AEi, am 
 
 al 
 oEI, ael 
 
 el 
 Ael, ael 
 
 ai 
 aEi, aei 
 
 ei 
 Aei, aei 
 
 18. Then observe that we don't know until we are told whether 
 any of the smaller (or inferior) compounds, AE, Ae, aE, ae, or AI, 
 Ai, al, ai, or EI, Ei, el, ei are absent or present. But if any one 
 is present, then at least one of the two larger {or superior-) compounds 
 under it in these forms must be present, and both may be ; but if 
 any one is absent, both of the corresponding larger compounds must 
 he absent. Thus if we know that there is no word containing E 
 without I, or fEi, then we know that there is no word containing 
 
20 
 
 LOGIC FOR CHILDREN. 
 
 ( 
 
 [arts. 18, 19. 
 
 A and E without I, or containing E without either A or I. That 
 is, if fEi then t(AEi, a'Ei). But we know that if fEi then JEI 
 •Jei-Pel, hence we know J» (AEI, aEI)- J» (Aei, aei) '?{Ael, ael). 
 All these consequences of fEi should be very distinctly seized and 
 well exemplified by words. Get the whole class to write down 
 twenty words a-piece, each having 2 or 3 vowels, and none having 
 E without I, taking care that the general condition is fulfilled ; as- 
 certain that the above results are obtained, by actually sorting first 
 with respect to E, e; then each bundle with respect to I, i; and 
 finally each with respect to A, a. Some may have JAel, others 
 fAfil ; some may have Jael, others fael. None will have JAEi or 
 J«Ei. Some may have JAEI-JaEI, others JAEI-faEI, others 
 fAEI • JaEI, but none fAEI • faEI, and so on. By this means the 
 real meaning of the signs and processes will become evident to 
 them, and they will feel that they are dealing with a law of thought. 
 Any set of words being given, the pupils should be exercised in 
 markmg what compounds are present or absent. Thus avail, Italy, 
 debasement, conceive, seem, give fAEI • JAEi * JAel * fAei ' JaEI 
 •JaEi • fael ' faei. This is best done by writing the compounds on 
 the black board, and writing against them each word as it arises, 
 thus, — 
 
 aEI conceive 
 debasement aEi seem 
 
 avail, Italy ael 
 
 aei. 
 
 AEI 
 AEi 
 Ael 
 
 Aei 
 
 AH the words being used up, the blanks show the absent sets. 
 This exercise is really important. 
 
 19. Next suppose we happened to know which of the larger com- 
 pounds were present or absent or doubtful, see if we could find out 
 what is the case with the smaller. 
 
 ^ It is quite clear that if JAEI, then JAE, +AI, and +EI. Thus 
 tnci-ease being an AEI-word, is also an AE-word, and an Al-word, 
 and an E[-word; so that if there is at least one such word as the 
 first, there is by that means at least one such word as each of the 
 three last. 
 
 Again, if +» (AEI, AEi), then certainly JAE ; because in that 
 case either JAE I, or JAEi, or both, and in either case, as we have 
 seen, there must be JAE. Thus if we know that either increase or 
 mate, or both, are among the words thought of, we know that there 
 must be at least one AE-word. 
 
 If JAEI • fAEi, or if JAEI' ?AEi, still there must be JAE on ac- 
 count of JAEI. 
 
 If fAEI • fAE?', then, and in no other case, fAE, That is, if no 
 word thought of contains both A and E, either with or without I, 
 then no word at all contains both A and E. 
 
 If PAEI- PAEi, or if PAEIfAEi, or if fAEI' PAE/, then PAE. 
 For if all the P were f, then fAE, by the last case ; but if only one 
 of them were J, then JAE, by the former cases. But we do not 
 know which is the case, hence PAE. 
 
 If three compounds all containing the same letter are f, then 
 the fourth will be J. Thus if fAE I -fAE i fAel, then J Aei. Be- 
 
 > 
 
 J 
 
 > li 
 
 i 
 
 I \ 
 
 ARTS. 19 — 21.] LOGIC FOR CHILDREN. 
 
 21 
 
 cause by the general condition there must be at least one com- 
 pound present containing A, and only these four compounds can 
 contain A. This is often very useful. 
 
 All these cases must be worked out over and over again, with' 
 all varieties, and for any two or three of the 8 larger compounds, 
 and results deduced with ease and certainty, just as in simple 
 arithmetic. 
 
 Where no doubtful compounds occur, the pupils should give 
 words completely illustrating given cases. Where doubtful com- 
 pounds occur, they should give sets of words answering to each 
 case included in the doubt, not omitting any one included, but not 
 giving any one excluded. 
 
 20. But suppose the general condition is not fulfilled, what hap- 
 pens P Suppose there is at least one word containing A, but no 
 a- word, that is no word without A. Then of course all the words 
 are A words, and we have faE and fae and hence t«EI • faEi and 
 ■fael' faei. This occasions no difiiculty. But it will be seen that 
 the only compounds which can be present are AEI, AEi, Ael, Aei, 
 all of which contain A, and are in fact the compounds EI, Ei, el, 
 ei, with A prefixed. Hence there is really no use in mentioning 
 A at all. The range of thought is confined to A- words. 
 
 But suppose there is 07ily one word containing A, what happens ? 
 One of the two compounds AE, Ae must be present, and only one 
 can he present. In counters show this by grouping | AE | Ae'j 
 not inverted. In writing prefix Ji, with the little 1 below, thus, 
 Ji (AE, Ae), and read " present only one of AE, Ae." We have 
 then also Ji (AEI, AEi, Ael, Aei). In writing, it is often conve- 
 nient to put a little i below the letter which occurs in only one 
 word. Thus JAj E shows that there is only one word containing 
 A, and that that word also contains E, so that fA^e. This is a case 
 of common occurrence in practice, when general propositions are 
 used, and should be carefully studied in every case. A little rider 
 of paper, or a clip, placed across the letter on the counter, or an 
 elastic band strung round it, will sufficiently mark A^, &c., as the 
 case may be. 
 
 21. Every preparation has now been made for considering syl- 
 logisms, and indeed much more complicated cases. Considered as 
 problems upon words containing letters, the syllogism is as fol- 
 lows -.—Given the state of a certain group of ivords ivith respect to 
 containing A and E ; and also their state with respect to containing 
 E and I ; to find their state ivith respect to containing A and I. 
 These states are given by one of the complete or incomplete 
 arrangements of Art. 13 or 14. The two first are called the pre- 
 misses, and the last the conclusion. The letters A and I are called 
 the extremes. The letter E, which occurs in both the premisses, 
 but does not occur in the conclusion, is called the mean. The 
 process is as follows, and may bo conducted with counters or on 
 paper, but for children counters are far best. 
 
 First, state the two premisses, in the abbreviated form of Arts. 
 13 and 14, forming the upper line of counters, remembering all 
 the implied present, absent, or doubtful parts. 
 
 C 
 
20 
 
 LOGIC FOR CHILDREN. 
 
 [arts. 18, 19. 
 
 ARTS. 19 — 21.] LOGIC FOR CHILDREN. 
 
 21 
 
 A and E without I, or containing E without either A or I. That 
 is, if fEi then t(AEi, crEi). But we know that if fEi then JEI 
 •JerPel, hence we know J' (AEI, aEI)' J» (Aei, aei) '?{Ael, ael). 
 All these consequences of fEi) should be very distinctly seized and 
 well exemplified by words. Get the whole class to write down 
 twenty words a-piece, each having 2 or 3 vowels, and none having 
 E without I, taking care that the general condition is fulfilled ; as- 
 certain that the above results are obtained, by actually sorting first 
 with respect to E, e; then each bundle with respect to I, t ; and 
 finally each with respect to A, a. Some may have JAel, others 
 fAfil ; some may have Jael, others fael. None will have JAEi or 
 J«Ei. Some may have JAEI'JaEI, others JAEI-faEI, others 
 fAEI • JaEI, but none fAEI * faEI, and so on. By this means the 
 real meaning of the signs and processes will become evident to 
 them, and they will feel that they are dealing with a law of thought. 
 Any set of words being given, the pupils should be exercised in 
 marking what compounds are present or absent. Thus avail, Italy, 
 debascvient, conceive, seem, give fAEI' JAEi * JAt'I'fA^'i* JaEI 
 'Xa'Ei'fael'faei. This is best done by writing the compounds on 
 the black board, and writing against them each word as it arises, 
 thus, — 
 
 AEI aEI 
 
 AEfc debasement oEi 
 
 Ael avail, Italy ael 
 
 Aei 
 
 conceive 
 seem 
 
 aei. 
 
 All the words being used up, the blanks show the absent sets. 
 This exercise is really important. 
 
 19. Next suppose we happened to know which of the larger com- 
 pounds were present or absent or doubtful, see if we could find out 
 what is the case with the smaller. 
 
 ^ It is quite clear that if JAEI, then JAE, JAI, and JEI. Thus 
 increase being an AEI-word, is also an AE-word, and an Al-word, 
 and an El-word; so that if there is at least one such word as the 
 first, th ere is by that means at least one such word as each of the 
 three last. 
 
 Again, if J^(AEI, AE/), then certainly JAE; because in that 
 case either J AEI, or JAEt, or both, and in either case, as we have 
 seen, there must be JAE. Thus if wo know that either increase or 
 mate, or both, are among the words thought of, we know that there 
 must be at least one AE-word. 
 
 If JAEI • fAEt, or if JAEI- ?AE/, still there must be JAE on ac- 
 count of JAEI. 
 
 If fAEI • fAE?', then, and in no other case, fAE, That is, if no 
 word thought of contains both A and E, either with or without I, 
 then no word at all contains both A and E. 
 
 If PAEI- ?AE^, or if PAEIfAEi, or if fAEI' PAE/', then PAE. 
 For if all the P were f, then fAE, by the last case; but if only one 
 of them were J, then JAE, by the former cases. But we do not 
 know which is the case, hence PAE. 
 
 If three compounds all containing the same letter are f, then 
 the fourth will be J. Thus if fAE I -fAE i -fA^I, then J Aei. Be- 
 
 ) 
 
 cause by the general condition there must be at least one com- 
 pound present containing A, and only these four compounds can 
 contain A. This is often very useful. 
 
 All these cases must be worked out over and over again, with' 
 all varieties, and for any two or three of the 8 larger compounds, 
 and results deduced with ease and certainty, just as in simple 
 arithmetic. 
 
 Where no doubtful compounds occur, the pupils should give 
 words completely illustrating given cases. "Where doubtful com- 
 pounds occur, they should give sets of words answering to each 
 case included in the doubt, not omitting any one included, but not 
 giving any one excluded. 
 
 20. But suppose the general condition is not fulfilled, what hap- 
 pens P Suppose there is at least one word containing A, but no 
 a- word, that is no word without A. Then of course all the words 
 are A words, and we have f^E and fae and hence faEI • faEi and 
 frtel-faei. This occasions no difficulty. But it will be seen that 
 the only compounds which can be present are AEI, AEi, Kel, Aei, 
 all of which contain A, and are in fact the compounds EI, Ei, el, 
 ei, with A prefixed. Hence there is really no use in mentioning 
 A at all. The range of thought is confined to A- words. 
 
 But suppose there is only one word containing A, what happens ? 
 One of the two compounds AE, Ae must be present, and only one 
 can he present. In counters show this by grouping | AE | Ae \ 
 not inverted. In writing prefix Ji, with the little 1 below, thus, 
 ti (AE, Ae), and read " present only one of AE, Ae." We have 
 then also Ji (AEI, AEi, Ael, Aei). In writing, it is often conve- 
 nient to put a little i below the letter which occurs in only one 
 word. Thus JAi E shows that there is only one word containing 
 A, and that that word also contains E, so that f^ie. This is a case 
 of common occurrence in practice, when general propositions are 
 used, and should be carefully studied in every case. A little rider 
 of paper, or a clip, placed across the letter on the counter, or an 
 elastic band strung round it, will sufficiently mark Ai, &c., as the 
 case may be. 
 
 21. Every preparation has now been made for considering syl- 
 logisms, and indeed much more complicated cases. Considered as 
 problems upon words containing letters, the syllogism is as fol- 
 lows : — Given the state of a certain group of words with respect to 
 containing A and E ; and also their state tvith respect to containing 
 E and I ; to find their state ivith respect to containing A and I. 
 These states are given by one of the complete or incomplete 
 arrangements of Art. 13 or 14. The two first are called the pre- 
 misses, and the last the conclusion. The letters A and I are called 
 the extremes. The letter E, which occurs in both the premisses, 
 but does not occur in the conclusion, is called the mean. The 
 process is as follows, and may be conducted with counters or on 
 paper, but for children counters are far best. 
 
 First, state the two premisses, in the abbreviated form of Arts. 
 13 and 14, forming the upper line of counters, remembering all 
 the implied present, absent, or doubtful parts. 
 
 C 
 
22 
 
 LOGIC FOR CTTTLDREK. [aKTS. 21, 22. 
 
 Secondly, arrange the 8 compounds below. 
 
 Thirdly, work with one premiss upon those coraponnds, as 
 shown by art. 18, and then with the other upon the- result. The 
 final result is called the resultant. 
 
 Fourthly, from this resultant, deduce the conclusion as in 
 art. 19. 
 
 Fifthly, place this conclusion in the top line. The examples will 
 make this quite easy. Every case given should be worked over 
 and over again, to gain ease, rapidity, and certainty. Be sure not 
 to hurry. If properly conducted, the work will be found very 
 amusing. For convenience of printing, only written signs are 
 given. For tAEI the teacher will show black side of counter 
 erect ; for P AEI, black side inverted ; and for f AEI, red side. 
 Before being thus arranged so as to indicate that any one is pre- 
 sent, absent, or doubtful, the eight counters of the larger com- 
 pounds will be all put on their sides, in two rows, thus — 
 
 HH 
 
 
 
 1— ( 
 
 
 
 t— 1 
 
 
 
 i-i 
 
 
 ^ 
 
 
 «> 
 
 
 p&) 
 
 
 <» 
 
 
 -< 
 
 
 
 <1 
 
 
 
 e 
 
 
 
 e 
 
 
 
 
 
 
 
 
 
 •<* 
 
 
 
 •«>» 
 
 "» 
 
 
 < 
 
 
 
 <1 
 
 
 
 e 
 
 
 
 e 
 
 This may be indicated by writing 
 (AEI) (Ael) 
 (AEi) {Aei) 
 
 (aEI) 
 (aEi) 
 
 (ael) 
 {aei) 
 
 22. First hind of Syllogism. 
 
 " No words have A without E or E without I ; then no words 
 have A without I." 
 
 First premiss. " No words have A without E," or fAc, or "all 
 words having A have E," This implies J AE * Jnc and PaE. 
 
 Second premiss. "No words have E without I," or fEi, or 
 "all words having E have I." This implies JEI * Jei and Pel. 
 
 Resultant. First work for f A e, and get f Ael and f Aei. Then 
 work forfEt, and get fAEi and faEt. Hence, in counters, the 
 resultant will stand thus — 
 
 (AEI) tAel (aET) (ael) 
 tAEi fAei t«Ei (aei) 
 
 Rule : whenever there are f compounds in the premisses, work 
 with them first. 
 
 Next go to the implied P compounds, and work first with PaE. 
 This gives only PaEI, for faEi is already determined. Next work 
 with Pel, and get Pael, since fAel is already determined. The 
 resultant now stands in counters — 
 
 (AEI) tAel PaEI Pael 
 fAEt fAei faEi (aei) 
 
 Next work for the J compounds of the premisses. Then JAE 
 gives JH AEI, AEi), but we have already got fAEi, hence we must 
 have J AEI, in order that one of the two compounds should be 
 present. Next try tae, giving J'(ael, aei). But here we have 
 rael, and hence also raei, so far as this is concerned. But we have 
 
 ART. 22.] LOGIC FOR CHILDREN. 23 
 
 +i?l ^T t^o"^Po«".^s of the other premiss to try. Now JEI gives 
 + ( AlLl, aEI) and as we have already J A EI, we have PaEI, so that 
 we are not advanced Next lei gives J'(Ae/, aei), and here sinc6 
 
 ' m> T ^"/* ^^^® +^^^' so ^^^^^ t^e previous Paei becomes laei. 
 Ihe final form of the resultant is, therefore, 
 
 I AEI fAei PaEI Pael 
 
 fAEi fAei faEi Jaei 
 
 The two J compounds might have been found from the fact that 
 they are the only non-absent compounds containing A and i re- 
 spectively. 
 
 Now make a diagram of this syllogism, thus — 
 
 (A 
 
 !f== ;;:::: 
 
 and show how it agrees with the resultant. 
 
 Conclusix)7t. JAEI • fAEi give JAI ; f A Ei • fAei give fAi ; P aEI • 
 
 1a- ^at^ ' ^^^^ "^^^^ ' +^'^'' ^^^® +"*'• ^^"c® ^^^ conclusion is 
 fAi. Now place the counter fAi on the upper ledge, widely se- 
 parated from the premisses. In writing put .'., read " therefore," 
 before the conclusion. This conclusion shows that there are no 
 words having A without I. 
 
 Write the whole syllogism thus : fAc • fEi .*. fAi. 
 
 Illustrate by words. Get each pupil to do it separately. Pro- 
 bably the P compounds will be differently determined by different 
 pupils, as in the four following instances, 
 
 J'j^^^^^'^9* '^ecisuring, seeming, sin, fog. This gives JfAe and 
 +fEi as premisses, and JfAi as conclusion. Show that in such 
 a case, instead of PaEI, and Pael, we 7)iust have (as in the particular 
 instance selected) JaEI * Jacl. Compare art. 30, note, 14. 
 
 ii. ftea^m^. measuring, sin, fog. This gives ffAe and JfEi, with 
 +tiU as conclusion again, but PaEI and Pael are now faEI and 
 Jael. Compare art. 30, note, 16. 
 
 ni. heating, measuring, seeming, fog. This gives JfAe and 
 tt^'_, with the conclusion XfAi, as in the preceding cases, but now 
 fahl and Pael become JaEI and fa^I. Compare art. 30, note, 16. 
 
 iv. beating, measuring, fog. This gives tfAe and ffEi with 
 the conclusion ffAi, which differs from all the preceding, and now 
 f'aEIand Pael become faEI and fael. Compare art. 30, note, 17. 
 
 Observe that in working out each separately, it will be best to 
 display the premisses and conclusion in counters, thus— 
 
 (i.) fAe JaE fEi Jel fAi +al 
 
 (ii.) fAe faE fEi Jel fAi fal 
 
 (lii.) fAe JaE fEi fel fAi fal 
 
 (iv.) fAe faE fEi fel fAi fal 
 
 The pupils should see that it is only the P compounds that are 
 affected by these changes, and that, altliough the three first con- 
 clusions are the same, the corresponding resultants (which alone 
 fully represent the premisses) are different. All the different cases 
 must always be furnished with instances in words. 
 
 C2 
 
24 
 
 LOGIC FOR CHILDREN. 
 
 i- 
 
 I 
 
 I 
 
 *1 
 
 [art. 23. 
 
 23. There are eight different cases of this first kind of syllogisni, 
 namely — 
 
 fAEfel . 
 
 '. tAI 
 
 t«E • tel 
 
 .*. -t«I 
 
 fAEfei . 
 
 ■. tAi 
 
 taE'tet 
 
 .*. fai 
 
 fAefEI . 
 
 •. tAI 
 
 tcre'tEI 
 
 .'. fal 
 
 fAefEi . 
 
 \ fAi 
 
 tae'tEi 
 
 .*. fai 
 
 These are distinguished by having the short forms of the pre- 
 misses both t, and the mean E in one and e in the other, that is, 
 the contrary in one to what it is in the other. This gives the 
 " skeleton rule." 
 
 [1.] Means unlike. Both premisses f. Conclusion^ f extremes. 
 
 Tliis is a valuable rule. See how it applies in each case. Each 
 case has four subcases as before, and each should be worked out 
 separately, with counters, with diagrams, and with words. 
 
 The pupils should be exercised in drawing diagrams for each of 
 these syllogisms, and in showing all the four cases which can ariso 
 from filling up the blanks in each. In the diagram given above 
 the E-line, or line of the mean, has been completely filled, and the 
 doubts thrown on the other lines. The teacher should vary this, 
 and much vary the lengths of the lines, and show what the essen- 
 tial relations are. This will be excellent practice in understanding 
 the effect of combined assertions. 
 
 It will soon be found that it is possible to introduce into the 
 diagram a condition which is not contained in the premisses, al- 
 though the full incompleteness of the premisses is retained. Thus 
 the diagram — 
 
 {A satisfies JAE * tAc * PaE ' Jae, and 
 E also JEI'tEi- Pel- te<:. But it also 
 
 I manifestly gives JAI • tAi ' Jal * Jai 
 
 or JtAi instead of tAi, as a conclusion. Why sop Because a new 
 relation has been furtively introduced, limiting the absolute doubt- 
 fulness attachable to PrtE and Pel, by rendering it impossible for 
 them both to be absent at once, namely, JHaE, el), which gives in 
 the resultant JH^EI, aEi, Ael, oel), or since t«Ei*tAeI, only 
 J'(aEI, ael), whence Jal. Thus it is impossible to fill up the 
 blanks in the E line without introducing faEI, or Jael, or both. 
 In fact, both the A and I lines arc completely filled, and the 
 doubts are thrown on the single line E. It will be readily seen 
 that we cannot avoid introducing such an additional condition, if 
 we begin by filling up.the line of either of the extremes instead of 
 that of the mean, as in this case we should have also to fill up the 
 line of the other extreme completely. But if the lines of both ex- 
 tremes are filled up, the conclusion appears as a complete, instead 
 of an incomplete arrangement, limiting the absolute doubtfulness 
 of the two r compounds in the resultant. In actual argument, 
 such new conditions are often covertly introduced, even without 
 the knowledge of the arguer, especially when he is exemplifying 
 his argument, and are then very difficult to detect. But it is al- 
 ways necessary to detect them, as the new condition alters the 
 nature of the argument. In the present case, this new condition 
 
 ART. 23, 2 k] 
 
 LOGIC FOR CHILDREN, 
 
 25 
 
 'J 
 
 removes the argument from the class of syllogisms, because it in- 
 troduces a third premiis, namely +'(aE, el), which relates to all 
 three of the letters A, E, I at once. 
 
 2\. The Second hind ofSi/Uogism. 
 
 " If no words have E without A, and also no words have E 
 without I; then at least one word has both A and I." 
 First premiss. No words have E without A, or taE. 
 Second premiss. No words have E without I, or fM, 
 EesuUant. After using taE, tEi, 
 
 (AEI) (Ael) t«EI (ael) 
 
 r .^. ,+^^' (^^^) t«Et (aei) 
 
 In this case only three compounds are f. 
 
 Next work with PAe and Pel, and the resultant becomes 
 (AEI) PAel faEI ?ael 
 fAEi ?Aei fam {aei) 
 
 Now take |AE, which gives JM AEI, AEi), and hence, since 
 tAiii, determines JAEI. But Jae gives only l\ael, aei), and this is 
 the utmost limitation we can get. Next JEI gives t'^AEI aEI) 
 and as taEI, we have JAEI as before. Lastly, lei gives JHAe'i, aei) 
 and this is the utmost limitation we can get. Now observe that 
 the general conditions are satisfied, without taking any considera- 
 tion of the compound Ael, which was left quite doubtful by the 
 premisses. The final resultant is 
 
 JAEI PAcI t^EI l\ael, aei) ' J^Aei, aei) 
 tAEi faEi 
 
 The two limitates are indicated by arranging the counters thus— 
 
 >*- 
 
 
 
 ton 
 
 • 
 
 Conclusion. |AI from JAEI ; l\Ai, ai) from Ji(Ae., aei) ; and 
 + (ai, ai) from J'(ael, aei). Observe that PAel would give PAI 
 but as we have already found JAI from JAEI, this does not affect 
 the conclusion. The only certain thing, then, is, JAI, or that at 
 least one word has both letters A and I. 
 
 Write the whole syllogism thus t^E' fEi .'. JAI, the two limi- 
 tates of the conclusion being implied, as in art. 14. 
 This is rather a difficult case, especially for the diagram 
 
 A 
 
 E 
 
 ,1 
 
 This must also be furnished with words, and the four subcases 
 worked out, as in the first kind of syllogism. The results beinc^ 
 • Jt«E • JtEZ .-. JAI I ttAe-JtE^.-.JtAt 
 JtaE • ffEi .: Xfal \ ttAe • tt^^ /. tfAi. 
 Be very particular in obtaining the full resultant for each case, 
 bhow how the diagram and the first resultant can be altered for 
 each particular case without disturbing any of the fixed parts. 
 
 The first case may present a little difficulty, because the pupil 
 Will be apt to consider, that when the assertions are complete, he 
 
 1 
 
26 
 
 LOGIC FOR CHILDREN. [aRTS. 24 — 26. 
 
 i p 
 
 
 has to fill up the whole lines, and leave no empty parts. Wc aro 
 ignorant of the extent to which the lines should overlap, and the 
 diagram for Jt<*E'JtEi should present so many blanks that it 
 can be filled up into any one of the forms of the resultant which 
 would give any one of the 5 forms of the conclusion JAI * X^i-^^* 
 ai) ' JHal, cii)- The following will be found to be a mode of filling 
 up the original diagram so as to answer these conditions, but the 
 ingenuity of the pupil should be well taxed before any solution is 
 presented to him. 
 
 A 
 
 £ 
 
 \ 
 
 25. There are also eight different cases of this second kind of 
 syllogism, namely — 
 fAE • fEI 
 
 fAE-fEi 
 
 tAe fel 
 fAe 'fei 
 
 Jal 
 
 ai 
 
 Xal 
 
 faE • fEI 
 faE • fEi 
 foe 'fel 
 fae 'fei 
 
 +Ai 
 
 Jai 
 
 XAi 
 JAI. 
 
 These are distinguished by having the short forms of the pre- 
 misses both t (as in art. 23 •, but the mean is E or e in both^ (which 
 is quite different from art. 23). -This gives the " skeleton rule." 
 
 [2.] Means like. Both premisses f. Conclusion^ J contraries of 
 extremes. 
 
 See how this applies in each case. Each case has four subcases 
 as before, and each should be worked out separately with counters, 
 with diagrams, and with words. 
 
 26. The Third hind of Syllogism. 
 
 " If at least one word has both A and E, and not one word has 
 E without T, then at least one word has both A and 1." 
 
 First premiss. At least one word has both A and E, or JAB. 
 Second premiss. Not even one word has E without I, or f Et. 
 Hesultant. Working first with fEi, we obtain 
 
 (AEI) (Ael) (aEI) (ael) 
 tAEi (Aei) faEi (aei) 
 
 Then working for J*(Ae, ae), JKaE, ae), (which may be consi- 
 dered as ?Ae, PaE, ?ae, leaving the limitations to be recovered 
 from the general conditions,) and also Pel, we get — 
 
 (AEI) PA el PaEI Pael 
 fAEi ?Aei faEi ?aei 
 
 Next work with J AE, which gives J^AEI, AEi) ; whence, since 
 fAE?', we have J AEI, which has to be put for (AEI). Then work 
 with JEI, giving J\AEI, aEI), whence, as JAEI has already been 
 found, aEI remains doubtful. Next Jei, gives J' (Aei, aei), and 
 nothing further. We have now got J(A, E, I, e, i), but, to satisfy 
 the general condition, we must also have Ja, and this gives 
 J*(aEI, oel, nci)y leaving PAel. The complete resultant then is — 
 
 {AEI PAel J'(aEI, ael, oeO 
 
 tAEi iaEl {'(Aei, aei) 
 
 I 
 
 ^ I ( 
 
 I 
 
 ARTS. 26, 27.] LOGIC FOR CHILDREN, 27 
 The last limitates are most conveniently set up in counters, thus 
 
 
 m^ 
 
 h-l 
 
 
 
 FW 
 
 %9V 
 
 • 
 
 But when several limitates occur, and one compound enters 
 into several of them, it is often convenient to use a special mark 
 both in writing and with counters, called an index, which may be 
 placed before each compound separately. This consists of a large 
 number as 2 or 3, showing how many compounds there are in the 
 limitate, and a smaller number subscribed, showing to which set 
 of 2 or 3 the compound belongs. These should be prepared in 
 counters, about U inches wide, and 3 inches high, iu sets, thus 
 2] 2„ 22 22, 2^ 23, 24 2^, 3i 3i 3^, &c. 
 
 The resultant may now be more conveniently written, thus — 
 {AEI PAel 3iaEI 3, ael 
 
 fAEi 2iAei faEi di2i aei 
 
 Where 3i 2i aei shows that aei belongs to two limitates. 1 he 
 counters may remain inverted after these indices, to show their 
 doubtful character, but this is no longer essential, as the index 
 itself marks limited doubtfulness. 
 
 Conclusion. {AI from {AEI; {'( Ai, ai) from 2,(Aet, aei); and 
 {•(«I, ai) from 3i (o'EI, ael, aei). Hence the only thing certain is 
 that at least one word contains both A and I. 
 
 Write the whole syllogism thus ; 
 
 {AE-fEi .-. {AI 
 the limitates of the conclusion being implied as in art. 14 
 
 Diagrwm : — f A 
 
 Je 
 
 ^I 
 
 Now there are 6 different complete arrangements correspond- 
 ing to the first premiss, and 2 to the second, hence there are 10 
 syllogisms in which the premisses would be the same in the non- 
 doubtful parts, and each of these would lead to resultants and 
 conclusions, also the same in the non-doubtful parts. In the pre- 
 sent case these 10 syllogisms are — 
 
 {faE • {tE^ 
 JtAe•{tE^ 
 {fae • {fEi 
 {fAe-ffEi 
 {faE • tfEi 
 
 X^ai 
 
 {tA^ 
 
 {t«I 
 
 X\ai 
 
 {fAi 
 
 tfA* 
 
 {AT-{aX 
 
 {AI- {Ai-{al-{at. 
 
 {AI {fae •ttE^ 
 
 tfAi ttAe-{tEi 
 
 ttAe'tfEi 
 
 {{AE-{tEi 
 
 {{AE-ffE* 
 
 All these cases should be worked by counters (they are worked 
 on paper in art. 39 ^ and exemi)litied, and the blanks of the diagram 
 of the general case should be filled up so as to illustrate every one 
 of these cases. This will be found both instinictive and entertain- 
 ing, and will serve to make the process of deduction something 
 real and intelligible instead of abstract and hazy. 
 
 27. There are sixteen cases of this third kind of syllogism, eight 
 when the first, and eight when the second premiss is {, namely :— 
 
28 
 
 LOGIC FOR CHILDREN. 
 
 [arts. 27, 28. 
 
 I 
 
 I ! 
 
 +AE • fEI 
 
 jAE-fEi 
 lAe fel 
 jAe • fei 
 +aE -fEI 
 JaE -fEi 
 •f.I 
 
 Jae 
 
 lAi 
 
 lAI 
 
 JAi 
 
 JAI 
 
 pii 
 
 la\ 
 
 tal 
 
 fAE • +EI 
 tAE-+Ei 
 
 tAe • |el 
 
 • X^i 
 
 •tEI 
 
 • +1 
 
 fAe 
 
 fc^E 
 
 faE 
 
 ■fae 
 
 foe 
 
 ^Ei 
 
 tAI 
 
 tAt 
 
 Jai 
 
 JAi. 
 
 These are distinguished by having the short form of one pre- 
 miss J, and the other f, and having the means likp, that is, either 
 both E or both e. This gives the important " skeleton rule" : — 
 
 [3]. Means like. One premiss J, and one f. Gonclusiont J eX' 
 treme of the J, with contrary of extreme of the f prem iss. 
 
 Thus the J premiss in the first case above is JAE, and its ex- 
 treme is A. The f premiss is fEI, and its extreme is I, of which 
 the contrary is i. Hence the conclusion is JAi. 
 
 Each syllogism has 10 subcases, and each should be worked 
 out separately with counters, with diagrams, and with words. 
 Sec also art. 39. 
 
 28. Syllogisms with Dotihtful Conclusions. 
 
 The 32 syllogisms in the three kinds just considered (with their 
 cases), are the only syllogisms which give any certain conclusion. 
 All others leave every compound in the conclusion cither abso- 
 lutely or limitedly doubtful. It is easily seen that the 16 cases in 
 which both premisses are +, will leave every compound even in the 
 resultant, doubtful. But it is not so evident that the \Q cases in 
 which one premiss is J, and the other f, but the means imlihe, will 
 give only a doubtful conclusion, because two compounds in the 
 resultant will always be absent. Hence it is best to work out a 
 case. 
 
 " If at least one word has both A and E, and not one word has I 
 without E, nothing can be predicted." 
 
 First premiss. At least one word has both A and E, or JAE. 
 
 Second previiss. Not even one word has I without E, or fel. 
 
 Besultant. Worked for fel, it becomes — 
 
 (AEI) fAel (aEI) fael 
 (AEi) (Ae^) (aEi) (aei) 
 
 Now, working for JAE, we get t*(AEI, AEi), with nothing 
 further. Working for JEI, we get J' (AEI, aEI.) Working for 
 Jet, we have J'(Aet, aei). Working for Ja, which is still undeter- 
 mined, we get J'(aEI, aEt, aei). So that, using the indices, the 
 complete resultant is — 
 
 232, AEI tAel 3i22aEI faeJ 
 2i AE* 23Aei 3i aEi 3i ^soei 
 
 Conclusion, ^or Al, either JK AI, Ai) from 2i (which may be used 
 as an abbreviation for all the compounds bearing that index) or 
 JHAI, al) from 2:. For At, either JHAI, Ai) from 2i or J>(Ai, ai) 
 from 23. For al, either J'(AI, al) from 2^ or J'(al, ai) from 3i. 
 For ai, either JH Ai, ai) from 23, or J'(al, ai) from 3i. But these four 
 limitatcs JHAI, Ai), J'(«I, ai), JHAI, al), J'(Ai, ai) result in every 
 case fjcom the general condition JA, Ja, JE, Je, (art. 12,) and hencQ 
 
 ARTS. 28 — 30.] LOGIC FOR CHILDREN. 
 
 29 
 
 s 
 
 < 
 
 i 
 
 add nothing to our knowledge. That is, the premisses tell us no- 
 thing as respects the possession of A and I ; or the conclusion is 
 doubtful. 
 
 Write JAE-f^I .-. ? 
 
 Observe, however, that the resultant has a very determinate 
 form, and that we know from fAel that no words contain A and 
 I without E, and from fael, that no words contain I without either 
 Aor E. 
 
 29. We may comprise all these 32 syllogisms under two skeleton 
 rules, thus — 
 
 [4.] Means unlike. One premiss J, and one f. Conclusion 
 doubtful. 
 
 [5.] Means like or unlike. Both prernisses J. Conclusion doubtful. 
 
 30. The 64 cases of syllogism here considered are all admitted 
 by Prof. De Morgan.* The old logic only admitted 11 of them, 
 
 ^ ♦ Teachers wiU perceive that the number of different syllogisms is 
 limited by the number of different assertions. As there are 26 of these 
 (art. 14, note,) there must be 26 x 26 = 676 possible syllogisms, each of 
 which will give a different resultant. Of these 2x15x11 = 3 30 will have 
 one of the 1 1 complex incomplete as one of their premisses, the other being 
 complete or else simple incomplete; and llxllorl21 will have both 
 premisses complex incomplete. These syllogisms have very httle inte- 
 rest, but may all be worked out on the model of the third kind of syllo- 
 gism, and as such form occasionally useful exercises. The remaining 225 
 may be distributed into 3 classes ; 8 x 8 = 64 having both premisses 
 incomplete, which are those considered in the text ; 2x7x8 = 112 hav- 
 ing one premiss incomplete and one premiss complete ; and 7 x 7 = 49 
 having both premisses complete, which comprise all the subcases mentioned 
 in the text under each kind of syllogism. To embrace all these 225 cases, 
 20 " skeleton rules" are necessary, which are here added, with the numbers 
 of cases comprised in each, and one example in the short form, as they 
 may be useful in suggesting exercises to teachers ; each has been specially 
 considered and worked in my Contributions to Fonnal Logie. Rules 1 
 to 5 belong to the first class, 6 to 13 to the second, and the remainder to, 
 the third. 
 
 1, 2, and 3 are the first, second, and third kind of syllogisms in the text. 
 8, 8, and 16 cases respectively. See arts. 22, 24, 26. 
 
 4. One premiss f, and one J. Means unlike. Conclusion, ?. Ex. fAE • lei 
 .-. ? I AI |. 16 cases. See [4], art. 29. 
 
 5. Both premisses J. Means indifferent. Conclusion, ?. Ex. JAE • JEI 
 .-. ? I All, JAE • Xel .'. ? I AI|. 16 cases. See [5], art. 29. 
 
 6. One premiss Jf, and one f. Means unlike. Conclusion, Jf extremes.. 
 Ex. JtAE • t«X .*. JtAI. 16 cases. 
 
 7. One premiss Jf, and one f. Means like. Conclusion, J contrariea 
 of extremes. Ex. JfAE • f EI .*. Ja». 16 cases. 
 
 8. One premiss Jf, and one J. Means like. Conclusion, J extreme of J 
 premiss with the contrary of extreme of Jf premiss. Ex. Jf AE • J EI .*. 
 Jrtl. 16 cases. 
 
 9. One premiss Jf, and one J. Means unlike. Conclusion, ?. Ex.. 
 Jt AE • %el .'. ? |_AI |. 16 cases. 
 
 10. One premiss ft* aiid one f. Means unlike, by arrangement of the 
 
30 
 
 LOGIC FOR CHILDREN. 
 
 [art. 30. 
 
 ART. 30.] 
 
 LOGIC FOR CHILDREN. 
 
 31 
 
 bnt this arose from insufficient knowledge. It maybe convenient for 
 the teacher to have the old forms of syllogisms translated into the 
 present. They are therefore given here as Prof Bain lias presented 
 them, with the old barbarous Latin names having their vowel 
 quantities marked. The present letters A, E, I have been sub- 
 stituted for those of Prof. Bain. There are apparently 19 forms, 
 but 8 are merely verbal varieties of other forms, depending on 
 what is called figtire and mood, (which again depended on the 
 grammatical order of subject and predicate) distinctions rejected 
 by De Morgan, and needless to be considered now. The bracketed 
 numbers at the end refer to the three kinds of syllogisms and 
 their rules, as given above. Arts. 23, 25, 27. Observe that by "all 
 A IS E," Prof. Bain means, in the present case, " all words con- 
 taining A contain E," or fAe ; by " some A is E," he means " at 
 least one word containing A contains E," or JAE; by " some A 
 is not E," he means, " at least one word containing A does not 
 contain E," or JAe ; and by " no A is E" he means " not even 
 
 +t premiss. Conclusion, f extremes. Ex. ft AE • f^I .*. fAI. Use either 
 ft AE or its equivalent ffae, according to the phase of the mean in the 
 other premiss, for the sake of the rule. 16 cases. 
 
 11. One premiss ff, and %. Means unlike, by arrangement of ff, as in 
 No. 10. Conclusion, X extremes. Ex. ff AE • Jel .-. J AI. 16 cases. 
 
 12. One premiss tt and one f. Meana indifferent. Conclusion, X the 
 contrary of the extreme of the f premiss with the other extreme and also 
 with its contrary. Ex. tt AE • t EI .-. X{Ai, at). 8 cas«s. 
 
 13. One premiss ti, and one t Means indifferent. Conclusion.?. Ex. 
 JJAE-iEI.-.?|AI|. 8 cases. 
 
 14. Both premisses Xf- Means unlike. Conclusion, If extremes. Ex, 
 Xf AE • XfeL .-. tfAI. 8 cases. 
 
 15. Both premisses tf. Means like. Conclusion, J the contraries of the 
 extremes. Ex. tfAE-JfEI .-. +a». 8 cases. 
 
 ^ 16. One premiss tt, and one Jt. Means unlike, by arrangement of ft, as 
 m No. 10. Conclusion, Xf extremes. Ex. ft AE • f f^I .-. ff AI. 16 cases. 
 
 1 7. Both premisses ft- Means unlike, bv arrangement of ff, as in No. 10. 
 Conclusion, ft extremes. Ex. ffAE • ft^I .*. ftAI, which is the same as 
 ffae • ttEi . •. tt"'. 4 cases. 
 
 18. One premiss ff, and one Xt- Means indifferent. Conclusion. 11. 
 Ex. ttAE • ::EI .-. ttAI. 4 cases. ' ** 
 
 19. One premiss t|, and one Jf. Means indifferent. Conclusion, X the 
 contrary of the extreme of the tt premiss with the other extreme, and also 
 with its contrary. Ex. XX AE • JfEI .-. J (A», at). 8 cases. 
 
 20. Both premisses J|. Means indifferent. Conclusion.?. Ex. llAE-ltEI 
 .-. ?|AI(. lease. "*"*' ** 
 
 It will be an excellent exercise first to form all the cases of each and 
 apply the rule ; secondly, to form the restdtints of all, and show that even 
 when the conclusions are the same the resultants are different ; thirdly, to 
 show that the conclusions in every case can be deduced from a continual 
 application of the first five rules; and lastly, that everv one of the 676 
 syllogisms fiumishes a resultant and a conclusion consistent with one or 
 more of the last 49, forming the third class, and for any det4-rminHte case 
 of words selected must always be onn or othrr of tlu'S4; 49. Thtjre is 
 nothing in all this beyond the reach of iutcUigeut boys that can do a sum 
 in the rule of thrc«^ 
 
 I 
 
 * 
 
 one word contains both A and E," or fAE. Observe also that 
 the order in which the premisses are arranged differs from that 
 used in the preceding articles, but that this order is always in- 
 different, so far as the resultant and conclusion are concerned, 
 though made a matter of some weight in the old logic. The mode 
 of passing from the particular cases here worked, to the general 
 case of ordinary logic, is given hereafter. Arts. 4U to 42. Examples 
 of ordinary assertions will be found in Arts. 43 and 44. 
 
 First Figure. 
 
 1. Barbara. All E is I, all A is E, .*. all A is I; fEi- fAe, 
 
 .-. fAi, [1]. 
 
 2. Celdrent. No E is I, all A is E, .*. no A is I; fEI-fAe, 
 
 .-. fAI, [1]. 
 
 3. Darn. All E is I, some A is E, .*. some A is I; fEi'tAE. 
 
 .-. JAI, [3]. 
 
 4. FMo. No E is I, some A is E, .*. some A is not I; tEI * lAE, 
 
 .-. JAi, [3]. 
 
 Second Figure. 
 
 5. CesUr^. No I is R. all A is E, .-. no A is I ; fEI-fAc, 
 
 .'. fAI. [1]. This is the same as No. 2. 
 
 6. Cdinestres. All I is E, no A is E, .*. no A is I ; fel * fAB, 
 
 .-. fAI, [1]. 
 ■ 7. Festlnd. No I is E, some A is E, .*. some A is not I ; fEI • 
 JAE, .*. JAi, [3]. This is the same as No. 4. 
 
 8. Bdrokd. All I is E, some A is not E, .*. some A is not I : 
 
 fel • JA^, .-. lAi, [3]. 
 
 Third Figure. 
 
 9. Ddrapti. All E is I, all E is A, .*. some A is I : fEi * faE 
 
 .-. :AI, [2]. 
 
 10. DtsamU. Some E is I, all E is A, .*. some A is I ; tEI * faE. 
 
 /. lAI, [3]. 
 
 11. Ddtist All E is I, some E is A, .*. some A is I; fEi* JAB, 
 
 .*. JAI, [3]. This is the same as No. 3. 
 
 12. FSlajftton. No E is I, all E is A, .-. some A is not I: tEI' 
 
 faE, .-. JAi, [2]. 
 
 13. Bdkardo. Some E is not I, all E is A, .*. some A is not I ; 
 
 JEi-faE, .-. jAi, [3]. 
 
 14. F&risdn. No E is I, some E is A, .*. some A is not I; fEI* 
 
 JAE, .*. JAi, [3]. This is the same as No. 4. 
 
 Fourth Figure. 
 
 15. Brdmantip. All I is E, all E is A, .*. some A is I; fel* 
 
 t«E, .*. fal, [1], which is the full conclusion, and this 
 implies JAI (art. 14, vii), which is the old conclusion, 
 where JAI is used instead of fal, simply because the 
 old logic had no terms to express the latter assertion as 
 a conclusion, with A as the *' subject." Sir W. Hamilton 
 introduced the phrase : " some A is all I," for fal, (art 
 43, vi. and note,) and dismissed the old form BraniayUip^ 
 
 16. Camihiis. All I is E, no E is A, .*. no A is I; fel 'fAE 
 
 .*. fAI, [1]. This is the same as No. 6. 
 
t 
 
 32 LOGIC FOR CHILDREN. [aRTS. 30, 31. 
 
 17. Dtmans. Some I is E, all E is A, /. some A is I; JEI* faE 
 
 .*. JAI, [3]. This is the same as No. 10. 
 
 18. Fesdjpo. No I is E, all E is A, .'. some A is not I ; fEI • faE 
 
 .'. JA?*, [2]. This is the same as No. 12. 
 
 19. Frestson. No I is E, some E is A, .*. some A is not I : fEI • 
 
 JAE, .'. JAi, [3]. This is the same as No. 4. 
 
 The teacher who has read Archbishop Thomson's Laws of 
 Thought, will find 36 syllogisms given as Sir William Hamilton's, 
 in each of the three figures, of which the Archbishop retains 22 
 in the first figure, and 20 in each of the others. All of Sir 
 William Hamilton's, except 8, can be immediately solved by the 
 processes here given ; and of these 8, 4 belong to the class of 
 doubtful conclusions, and the other 4 are really not syllogisms, 
 because they implicitly contain a fourth name,* and, when solved] 
 lead also to doubtful conclusions. None of them have been ad- 
 mitted by other logicians, and hence need not be further consi- 
 dered. 
 
 These are merely notes to help the teacher to translate his old 
 books into these new symbols, and have nothing to do with the 
 children. All remnants of merely medieval formalism and its 
 consequences must be carefully kept from them. To resume : — 
 
 31. The three skeleton rules for syllogisms yielding certain con- 
 clusions, and the two rules for syllogisms yielding doubtful conclu- 
 sions, will enable the pupil to solve an immense number of ex- 
 amples without hesitation. Suppose, for instance, we take the ten 
 cases considered in the third kind of syllogism, art 26, (for example, 
 the one JfAe • ffEi, .'. JfAi,) we may solve each of them by a con- 
 tinual application of the rules. Thus JfAe means fAe ' JaE, and 
 tfEi means fEi • fel. Taking each of the first with each of the 
 second, we get cases to which the rules apply. Thus fAe • fEi, 
 .-. fAi, [1]; fAe -fel, .'. J«i, [2]; JaE-fEi, .-. Jal, [3]; JaE • 
 fel, .*. ?, [5] : the conclusion is then fAi (which implies J AI • Jat), 
 Jai, (already implied in the last,) and Jal, that is, on the whole, 
 JAI-fAi-Jal-Jai, or JfAi as already found. And so on for 
 every possible case that can occur. These rules therefore embrace 
 the whole ordinary theory of the syllogism, as enlarged by De 
 Morgan and Hamilton. 
 
 * Thus one of these syllogisms, reduced to a case of letters in words, 
 is : No words have E without I, at least one word containing A is dif- 
 ferent from at least one word containing E, .*. at least one word contain- 
 ing A is different from at least one word containing I. Now the assertion 
 that at least one word containing A is different from at least one word 
 containing E, means that the firat contains (or does not contain) some 
 other letter, say N, which the second does not (or does) contain. Then 
 we have JAN and +E». There are therefore three premisses fEt, JAN, 
 JE«. Then fEt • JE« .-. Jwl by [3], and J«I • JAN gives a doubtful 
 conclusion, by [5], but contains the expression of Sir W. Hamilton's con- 
 clusion, for we see that at least one word contains I without N, and 
 at least one word contains both A and N, so that these words must be 
 different. 
 
 ARTS. 32 — 35.] LOGIC FOR CHILDREN. 
 
 33 
 
 I 
 
 32. Again, the rules immediately enable the pupil to solve en- 
 thymemes ; that is, when the conclusion and one premiss is given, 
 the other can be found. Thus let the conclusion be f I, and one 
 premiss be faE, then the other must be fel, by [1], which shows 
 that the conclusion t«I can only belong to the first kind. Hence 
 also if JAE were given as a premiss, having fal as a conclusion, 
 we should see there must be an error. In fact, the only certain 
 syllogisms with JAE as a premiss, are JAE 'fEI, .'. JAi, and 
 JAE-fEi .-. JAI, (art. 27.) 
 
 33. Again, given a conclusion and a mean, we can find all the 
 syllogisms which will produce it. Thus : Let JAI be the conclu- 
 sion, and E or e the mean ; we see that t<^E * f Ei, •\ae ' f ei, faE * 
 JEI, t«e • Jel, JAE * fEi, JAfi ♦ fei, and no others, will give JAI, 
 the two first by [2] and the others by [3]. 
 
 34. We can also see whether any given syllogism is logically 
 bad. For, stating the premisses, we can immediately read off" the 
 correct conclusion, and compare it with the one given. Thus 
 fAE ' fEi .*. JAI is wrong, the conclusion is Jal, leaving PAI. 
 Thus logical fallacies are detected. The greater number of falla- 
 cies in reasoning depend upon assuming incorrect premisses. 
 One method of showing their error is to combine them logically 
 with correct premisses and obtain conclusions properly drawn, 
 which by their absurdity show the absurdity of the premiss which 
 involved them. 
 
 35. We do not disprove a conclusion, by showing that either one 
 or both of the premisses state precisely the contrary of what is 
 true ; for if we change J and f into f and J in either premiss sepa- 
 rately or in both together, we get a conclusion which is consistent 
 with the former. This should be showi>* for each kind of syllo- 
 gism, thus — 
 
 First kind. fAe • fEi .-. fA^, [1]. 
 But tAc • JEi .-. ?, [4] ; +Ae • fEi .'. .P[4] ; and JAe ' JEi .'. ? [5] ; and 
 a doubtful conclusion is consistent with any conclusion whatever. 
 
 Second kind. f^^E • fEi .*. JAI, [2]. 
 But JciE • fEi .-. Jal, [3] ; t«E; JEi .-. JA?:, [3], and +«E • JEJ .'. P [5]. 
 Now JAI is consistent with either or both of Jal and JAi, see Art. 
 14, i 
 
 Third kind. JAE-fEi.-.JAI, [3]. 
 But JAE • JEi .-. ?,[5] ; fAE • fEi.-. Jal,[2]; and fAE • +Ei .'. +ai,[3]. 
 And JAI is consistent with either or both of Jal, and Jai. 
 
 To disprove a conclusion, we must prove the correctness of an 
 assertion which is inconsistent with it. Thus to disprove fA?, 
 we must prove JA/, or fAI, or \ai, and the various ways of pro- 
 ving each of these with a given mean are obtained immediately 
 from the skeleton rules, as shown in art. 33. But if none of the pre- 
 misses required for this purpose are correct, we cannot disprove 
 the conclusion, even if we do not admit the premisses on which it 
 is based. Of course, it does not follow that the conclusion was 
 right, but merely that we have not found out the proper form of 
 mean for disproving it syllogistically ; and the choice of means 
 
a 
 
 34 
 
 LOGIC FOR CHILDREN. [aRTS. 35 — *67. 
 
 ARTS. 37 — 39.] LOGIC FOR CHILDREN. 
 
 35 
 
 t 
 
 is unlimited. More often, however, the concUision is disproved 
 by some of the inconsistent assertions being established by direct 
 observation or experiment, and not syllogistically. The favourite 
 method of disproving one or both of the premisses, merely shows 
 that a wrong line of argument has been followed, not that the con- 
 clusion is wrong. Thus, assuming the transition from words and 
 letters to ordinary instances, made in art. 40, let the argument be : 
 — "Pompey (A,) is a man (E); all men are mortal (I); hence 
 Pompey is mortal." RerePompey being an individual name is mark- 
 ed A.1, (art. 20,) so that the first premiss is J:A,E or tA,c, and the 
 second is fEi, the conclusion being fAr/, by [1], giving JA,I. Now 
 suppose we deny fAi^?, and declare JAiC, (Pompey is a dog, for ex- 
 ample), and again deny fEi and declare JEi, (at least Enoch and 
 Elijah were not mortal), then JAie • JEi.'. ?, and does not show 
 JAii (or that the dog Pompey is non-mortal !), that is, does not 
 disprove fAii (or that Pompey, be he man or dog, is mortal). 
 
 36. On the other hand, if we take one of the premisses and com- 
 bine it with an assertion which is inconsistent with the conclusion 
 taken as a second premiss, we shall find as a conclusion some- 
 thing inconsistent with the second premiss. This is also best 
 shown for each kind of syllogism separately. 
 First kind. fAe ' fEi .'. fAi, [1]. 
 
 Assertions inconsistent with fAe are J A'?, fAE, fiie 
 ft » „ fEt are JEi, fEI, fei 
 
 »» ♦, )) fAi are jAi, fAI, fai. 
 
 Now combine fAe with each of the three last, and we shall get 
 one of the three second in each case, thus : — 
 
 fAe ' tAl .-. JEi, [3] ; fAc • fAI .'. JEi, [2] ; f A^ • fai .'. fei, [1 ]. 
 
 Again combine fEt \^th each of the three last and we shall 
 get one of the three first in each case, thus — 
 
 fEI • lAi .-. :Ae, [3] ; fEr fAI .-. fAE, [1] ; fE^ • fai ,'. +A«, [2]. 
 Second kind. faE • fEi .-. JAI, [2] 
 
 Inconsistent with faE are JaE, fAE, fae 
 n n fEi ... JEi, fEI, fei 
 
 It „ JAI is fAI only. 
 
 Then faE • fAI .'. fEI, [1] ; fEi ' fAI .-. fAE, [1]. 
 Third kind. t AE • fEI .: JAI, [3] 
 
 Inconsistent with J AE is fAE only 
 ** „ fEi are JE/, fEI, fei 
 
 tf •• JAI is fAI only. 
 
 Then JAE • fAI .'. JE/, [3] ; fEi • fAI .'. fAE, [1]. 
 
 These syllogisms are sometimes called the opponents of those 
 from which they have been derived, but they have not been usually 
 treated with proper fulness. For example, only two ojiponents 
 have been allowed in each case, whereas it has just been shown 
 that the first kind of syllogism has six opponents. 
 
 37. Will, then, one of the premisses combined with the con- 
 elusion as a second premiss, give the other premiss as a new 
 conclusion ? Try for each kin 1, taking the three examples just 
 chosen. 
 
 « ^\ 
 
 n 
 
 First. tAc-fAi.-. JEI, which is notfE?:, but is consistent with 
 it ; and fEi • fAi .'. Jae which is also consistent with fAe. 
 
 Second. faE • JAI .-. ? [4], and fEi • JAI .-. ?, [4], which results 
 are of course consistent with fEi and faE respectively. 
 
 Third. JAE • JAI .-. ?, [4], fEi • JAI .'. ?, [4], which again are con- 
 sistent with fE^ and JAE respectively. 
 
 The new conclusion, therefore, in each case differs from, but is 
 consistent with the other premiss. 
 
 38. We may therefore believe that if any three assertions consis- 
 tent with the premisses and conclusion of a syllogism are assumed, 
 and any two of them are taken as premisses, the conclusion will 
 be consistent with the third. This belief will be raised to a cer- 
 tainty by considering that the premisses and conclusion are all 
 three of them mere portions of the resultant, which was produced 
 by the joint action of the premisses, so that if merely the resultant 
 were given, all three assertions, namely the two premisses and the 
 conclusion, could be deduced from it. This can readily be done 
 by the process in art. 19, hitherto followed in obtaining the con- 
 clusion from the resultant. The Teacher is recommended to set 
 up a resultant in counters, and obtain from it all the possible 
 results respecting A,E and E,I as well as A,I. 
 
 The pupil may also be exercised in deducing other assertions 
 respecting the three names A, E, I, from a given resultant. Thus, 
 in the resultant in art. 22, all words containing both A and E also 
 contain I. This is shown by JAKI and fAKi, which must neces- 
 Barily involve the fact that at least one word which does not con- 
 tain both A and E, that is, which is anon- AE- word, (and which is 
 therefore an Ae-, aB-, or ae-wovd,) does not contain I. Now the re- 
 sultant agrees with this by means of Jaei, although fAel, fAei 
 show that no words have A without bl, and ?ael shows that 
 although there is at least one word with neither A nor E (shown 
 by Xaei) yet, even if there are more such words, it is doubtful 
 whether even one of them contains I. 
 
 A gain ?aEl ' faEi show that though the existence of any words 
 having E without A is doubtful, yet if any such exist, that is, if 
 JaEI, they all will also contain I. The consequence of which is 
 that there must in that case be at least one word which does not 
 contain E without A, that is which is a non-aE-word, (and which 
 will therefore be an A E-, Ae-, or ae-word,) and also does not 
 contain I; that is, that J^(AEi Aei, aei); and we see, in fact, 
 that though fAEiand fAei, yet Xaei. 
 
 Judicious exercise of this kind, illustrated by examples in words 
 and drawing of diagrams, will materially strengthen the reasoning 
 powers, and at the same time may be made the source of much 
 interest by presenting unexpected problems. 
 
 39. The method here adopted is complete and systematic, but 
 the teacher may find it more interesting and instructive at first to 
 adopt an unsystematic method, and then come to system as a 
 means of classifying cases which have already occurred. The pre- 
 sent principles with the use of counters are eminently adapted for 
 such a course. Various cases of the same syllogism, arising from . 
 determining the ? compounds in the premisses as J or f , have 
 
■>-■ 
 
 .f 
 
 36 
 
 LOGIC FOR CHILDREN. 
 
 [art. 30. 
 
 also to be worked out and their results contrasted. This may be 
 done without recourse to the skeleton rules, which may be reserved 
 as convenient " skeleton keys" to pick future logical locks. It 
 should be remembered that the real thing to be obtained and 
 thoroughly understood is the resultant, which is altogether neg- 
 lected in ordinary logical treatises, and that the conclusion is only 
 the statement of jwr^ of the facts contained in the resultant, the 
 whole of which can alone completely determine what is affirmed, 
 denied, or left in doubt by the premisses, when taken jointly. 
 
 For these purposes the following modification of the forms in 
 which the investigations were conducted will be found convenient. 
 The table on this page shows how the ten definite forms of the 
 third kind of syllogism may be exhibited at one view upon a slate 
 or on the black board, and contrasted with the original indefinite 
 form (involving doubtful compounds). The letters are first writ- 
 ten down, and their columns for each case ruled. The letters for 
 the premisses occupy the upper, and those for the resultant the 
 
 Premisses. 
 
 ARTS. 39, 40.] LOGIC FOR CHILDREN. 
 
 
 AE 
 
 I 
 
 + 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 •f 
 + 
 
 8 
 
 9 
 
 + 
 
 10 
 
 + 
 
 + 
 
 + 
 + 
 
 + 
 
 ■f 
 
 + 
 + 
 
 -• 
 
 2, 
 
 Ae 
 
 + 
 + 
 
 t 
 
 + 
 
 t 
 
 4- 
 
 
 t 
 
 t 
 
 + 
 + 
 
 + 
 + 
 
 2. 
 
 aB 
 
 t 
 
 
 + 
 + 
 
 + 
 
 t 
 
 + 
 
 t 
 
 t 
 
 + 
 + 
 
 + 
 
 2.2, 
 
 + 
 
 ae 
 EI 
 
 + 
 ■f 
 
 X 
 
 + 
 + 
 
 t 
 
 + 
 
 + 
 + 
 
 i 
 
 t 
 
 + 
 + 
 
 ■t- 
 
 + 
 + 
 
 + 
 
 + 
 + 
 
 
 
 + 
 
 + 
 
 + 
 
 + 
 
 t 
 
 Et 
 
 t 
 
 t 
 
 t 
 
 t 
 
 t 
 
 t 
 
 t 
 
 t 
 
 t 
 
 t 
 
 n 
 
 f 
 
 el 
 
 + 
 + 
 
 + 
 
 + 
 + 
 
 t 
 
 t 
 
 t 
 
 + 
 ■f 
 
 t 
 
 + 
 + 
 
 t 
 
 + 
 JAE 
 
 ei 
 
 + 
 + 
 
 tt«E 
 
 + 
 
 + 
 
 + 
 
 ■f 
 + 
 
 + 
 
 + 
 
 + 
 + 
 
 ttA^ 
 
 + 
 + 
 
 + 
 + 
 
 ttA. 
 
 Jt^r^ 
 
 ttA. 
 
 n«E 
 
 Xfae 
 
 ttA^ 
 
 IXAE 
 
 UAE 
 
 m 
 
 
 ttE/ 
 
 ttE» 1 
 
 JtE» I 
 
 ttEt 
 
 ttE» 
 
 ttEt 
 
 ttEi 
 
 ttE» 
 
 ttE^• 
 
 ttEt 
 
 Resultants. 
 
 
 AEI 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 + 
 
 + 
 
 + 
 
 + 
 + 
 
 + 
 
 + 
 
 + 
 
 + 
 
 + 
 
 
 + 
 + 
 
 + 
 
 t 
 
 A Ki 
 
 t 
 
 t 
 
 t 
 
 t 
 
 t 
 
 t 
 
 t 
 
 t 
 
 t 
 
 t 
 
 ? 
 
 A^I 
 
 26 24 
 
 t 
 
 + 
 + 
 
 t 
 
 t 
 
 t 
 
 t 
 
 t 
 
 2b 2, 
 
 t 
 
 23 
 
 Act 
 
 2324 
 
 t 
 
 + 
 + 
 
 t 
 
 > 
 
 + 
 
 + 
 
 t 
 
 t 
 
 2324 
 
 + 
 
 3, 
 
 aFA 
 
 t 
 
 + 
 
 + 
 + 
 
 + 
 
 t 
 
 + 
 
 t 
 
 t 
 
 + 
 
 + 
 
 f 
 + 
 
 t 
 
 aEi 
 
 t 
 
 t 
 
 t 
 
 t 
 
 t 
 
 t 
 
 t 
 
 t 
 
 t 
 
 t 
 
 3, 
 
 ael 
 
 2«25 
 
 + 
 + 
 
 t 
 
 t 
 
 t 
 
 t 
 
 + 
 + 
 
 t 
 
 2«25 
 
 t 
 
 3,23 
 
 aei 
 
 2325 
 
 + 
 + 
 
 t 
 
 ■f 
 + 
 
 :tA« 
 
 + 
 
 t 
 
 ■♦■ 
 ttA» 
 
 + 
 ttA* 
 
 2325 
 
 + 
 
 JAI 
 
 :ai 
 
 ttA» 
 
 X\ai 
 
 XUi 
 
 Xf"i 
 
 tAI- 
 
 Xol 
 
 iJAi • 
 
 
 •i 
 
 37 
 
 I 
 
 under part. The symbols J, f, ?, for the indefinite form, stand on 
 the left; those for the various definite forms, on the right. Each pre- 
 miss is written at full length. The several columns are numbered 
 for convenience of reference. In the columns of the premisses appear 
 those variations of the premisses enumerated at the close of art. 26, 
 p. 27. The corresponding columns of the resultant are worked out 
 from them in the way there explained, and the limitate indices are 
 employed as there introduced. Under each column of the pre- 
 misses are written the short forms of the premisses, and under 
 each column of the resultant the short form of the conclusion. 
 The order of the cases is that adopted in art. 26. This mode of 
 workmg will be found convenient not merely for contrasting 
 several definite forms corresponding to one indefinite form of 
 syllogism, but for working out a number of unrelated syllogisms, 
 without writing the letter part over and over again. 
 
 40. We are now prepared to make the transition from the par- 
 ticular case of words containing letters, to the general case of 
 things possessing attributes distinguished by names, and thus 
 to transfer all the knowledge and mechanical faciUty already 
 gained to the propositions with which Logic ordinarily deals, — 
 propositions which from their abstract nature are generally ill- 
 adapted for elementary exercise. 
 
 The pupils will probably have been to a museum, or they may 
 be taken to one, or to any labelled collection of objects, for the 
 purpose of the present course.* Their attention must be drawn 
 to the fact that each object is labelled, and that each label contains 
 several names, and that these names have been given in order to 
 recall to the mind of observers certain thoughts or feelings or 
 sensations which they experienced when they fully examined the 
 objects, not only externally but internally, not only by sight, but 
 by touch, and occasionally the other senses, by breaking, dissect- 
 ing, weighing, burning, trying with acids, observing habits of life, 
 growth, and decay, if the object had been alive ; in short, by every 
 possible way that they could imagine which would lead them to 
 distinguish one from another. That any simple name when placed 
 upon an object implied a simple sensation experienced by the 
 observer, but that, in general, names were put for shortness which 
 implied a great many simple sensations, leaving it to dictionaries 
 or special treatises to explain what they were. Perhaps the teacher 
 may be able to go so far as to explain that there is presumed to be 
 m each object an antecedent of such sensations, and that this 
 antecedent is called an attribute, quality, or property of the object. 
 Ihis IS a difficult matter to make clear, and it must be approached 
 with caution, and always by drawing the thought from the pupil. 
 JNext make the pupil observe that these labels, and the words on 
 them, bear a wonderful resemblance to words and the letters they 
 contain ; that if the labels were taken off you could sort them first 
 
 * Most schools have museums of quite sufficient extent for this purpose ; 
 but, if necessary, the furniture, books, ink-pots, slates, &c. in a school- 
 room might be labelled for the occasion. Collections of shells, minerals, 
 flowers, or stuffed animals are the most convenient. 
 
 D 
 
38 
 
 LOGIC FOR CHILDREN. [aRTS. 40,41. 
 
 into those containing such a name and those without it ; then 
 could sort the first set into those containing another name and 
 those not containing it, and the second set- likewise ; and so on. 
 We miorht, in fact, suppose that the names were represented by 
 our letters A, E, I, &c., and that the labels not having the names 
 A E,I, had the letters a, e, i respectively, to show that the names 
 A E 1 had not been omitted by oversight, but that the other 
 obiects to which these names had not been affixed really had not 
 the names A, E, 1. This once done, the labels and our previous 
 slips of paper and counters become one and the same thing 15ut 
 now the slips or labels no longer mean words ; the capital letters 
 A E I no longer indicate that the words contain those vowels, 
 and the small letters a, e, i, that these words do not contain them. 
 The slips or labels or counters now mean objects m the museum. 
 The letters A, E, I are convenient abridgments of the names whicli 
 the obiects bore in the museum in order to certify tha. those 
 objects had certain corresponding properties. The Presence of 
 the letters a, e, i, indicate the absence of the names denoted by 
 A E T and hence the absence of the corresponding properties m 
 the objects. Also draw attention to the fact that many different 
 objects have some one name on their labels indicating the exist- 
 ence of the same property in all, and that almost all the objects 
 have several different names on each of their labels, mdicating 
 that one object has several different properties By this means the 
 pupil will readily appreciate the facts indicated m art. b. 
 
 41 Now make the transition to language in general. Show 
 how we have named things, (taking material objects first, then 
 their actions, &c.,) on account of some sensations they aroused in 
 ns how we have applied the same name, as man, to various objects 
 because many of the sensations they aroused were identical and 
 how, if each sensation had a different simple name, any name like 
 man would be in fact a compound name made up of the names ot 
 all the diff-erent sensations which must concur in order for us to 
 have the very compound sensation which the simple word man 
 now recalls. Turn to a bird; elicit (rather than show) many pro- 
 perties in common between a man and a bird as ^^^f fJ»Pj™ 
 both are animals and both two-legged. Then elicit their diffe- 
 rences, as feathers, wings, &c., and lead to the conclusion that 
 birds are non-men, and men are non-birds. Find other i^on-men 
 as quadrupeds, which are also non-birds, ^ow abbreviate the 
 ^ords Man into M, Bird into B, Two-legged into T, and Quad- 
 ruped into Q, when of course m, h, t, q will be names of non-men, 
 Zlrds, non-two-legged, non-quadrupeds. Ehcit he following 
 expressions of observations made on these object.^ Not only 
 tM^ giving as a logical consequence, art. 14, vi., JMT and jmi, 
 that' is, " not even one man is non-two-legged, every man is two- 
 legged, every thing which is not two-legged is not a man, but 
 
 also, as a new observation, u- j» „^+ « mon " 
 
 +wT ''at least one two-legged thmg (as a bird) is not a man, 
 
 Bo'thBt the logical ?mT is determined, and we have really 
 
 ++M< II tMT •mflmT-mt, in which every compound must be 
 
 thoroughly explained and exemplified by the pupils. Again, find 
 
 ARTS. 41 — 4o.] LOGIC FOR CHILDREN. 
 
 39 
 
 fMQ, giving as a logical consequence, art. 14, v., IMq and JwQ, 
 that is, " not one man is a quadruped, every man is a non-quad- 
 ruped, every quadruped is a non-man," and also, as a new obser- 
 vation, 
 
 Xmq " at least one non-man (as a bird) is a non-quadruped," so 
 that the logical ?mq is determined, and we have really 
 
 IfMQ II Wq-lKq-lmq-lmq, no term being left in doubt. 
 Similarly deal with fB^ and find J6T, so that really JfB^. Also 
 find fBQ and Ihq, so that JfBQ. 
 
 Next lead the pupil to observe that fM^ and fB^ show that what- 
 ever you can say of a Man or of a Bird, you really do say of a Two- 
 legged thing, so that there is no property possessed by a man or 
 a bird which is not possessed by at least one two-legged thing. 
 (This is an important general principle, see p. 41, note * ) But as 
 JmT and J&T, there is certainly at least one property possessed 
 by at least one two-legged thing which is not possessed by any 
 one man, and also at least one property, {not necessarily the same 
 as the last,) which is not possessed by any one bird. Show that if 
 it were otherwise, that is, if we had fmT and fhT, in addition 
 to fM^ and fB^ we should have ffMf, and ffB^, in which case 
 every single two-legged thing would be both a man and a 
 bird ! ! 
 
 42. Thus the assertions which we have lately considered respect- 
 ing the existence of certain letters in certain words, can all be read 
 as assertions respecting the attributes residing in certain things. 
 In fact all our thoughts may be considered as slips of paper or 
 counters, each containing some one distinctive mark or name 
 which "denotes" the particular slip or counter, and each con- 
 taining the marks or names of the particular attributes of those 
 thoughts which are " con-noted" by that first distinctive mark. 
 Once reduced to this form, all assertions whatev^er become pre- 
 cisely the same as assertions respecting letters in words, where 
 the word forms the "de-notation," and the several letters, being 
 the whole of the distinctive marks or attributes in the word, are 
 its " con-notation."* 
 
 43. A large number of assertions must now be analyzed. Ex- 
 amples may be collected from all elementary treatises on Logic. 
 
 * The order of the letters is important in a word, and must be con- 
 sidered as an attribute. But this may be here disregarded. Thus, so far 
 as possessing the letters I,P,T, and no others, pit and tip are the same. 
 But in writing down the letters we introduce a new attribute, order. In 
 thinking of things, this order is unimportant. "Whether we say a bird is 
 two-legged and feathered, or feathered and two-legged, is of no conse- 
 quence. This little point of difference should be borne in mind by the 
 teacher, but need not be mentioned unless he sees that a diflBiculty arises 
 in the pupil's mind ; and then the explanation must be elicited by 
 questioning. The distinction, however, between the two cases is that 
 mathematically important distinction between commutative and non-com- 
 mutative operations ; and Boole's mathematical theory of logic depends 
 mainly upon the commutative character of attributes, or iadiffcrouoe in 
 the order of mentioning them. 
 
 d2 
 
40 
 
 LOGIC FOR CHILDREN. 
 
 [art. 43. 
 
 The following specimens will show the relations of the present 
 strict notation to conimon language, and some of the difficulties 
 to be contended with * For convenience some of the words are 
 placed in ( ), and others in [ ] ; the letters A and E refer to these 
 words respectively, so that the expressions have the same form as 
 those considered in arts. 13 and 14. But in practice the letters 
 should be varied, and may generally be chosen so as to recall the 
 words, as in art. 41. 
 
 i. All (the righteous) are [happy], fAe, or "not one righteous man 
 is non-happy ;" implying JAE, or " at least one righteous man is 
 happy," and Jae, "at least one non-righteous man is non-happy,'* 
 and leaving absolutely doubtful, whether there is or is not even 
 one non-righteous man who is happy, or PaE. 
 
 ii. No (human virtues) are [perfect], fAE, implying JAe and 
 JrtE, but leaving ?ae. In putting these into language, remember 
 a is the name of a non-human, not of an in-human, virtue. The 
 forms fAE • JaE show that all perfect virtues are non-human. 
 If we were to write " all perfect virtues are not human," the 
 phrase would be ambiguous, and might also mean either JAE, 
 (" although all perfect virtues may not be human, some are"), which 
 is of course wrong. 
 
 iii. Some (possible cases) are [probable], JAE, or "at least one 
 possible case is a probable case." Observe that " some" in Logic 
 
 • The first six of these specimens are the six "judgments" of Arch- 
 bishop Thomson (" Laws of Thought, § 78, p. 135 of tenth edition, 1869), 
 which he SATubolizes, in order, by the capital letters A, E, I, O, U, Y. In 
 Sir W. Hamilton's formal language (i^., § 79), translated into the present 
 svmbols, thev art — 
 i. A. All A is some E || fAe 
 i'. E. No A is E || fAE 
 
 iii. I. Some A is some E || ^AE 
 
 iv. O. Some A is no E || JAtf 
 
 V. U. All A is all E || ttA<f 
 
 vi. Y. Some A is all E II t«E 
 
 Archbishop Thomson caUs these "at// the judgments," as he does not ad- 
 mit the two following of Sir W. Hamilton, which, however, he cites and 
 SATubolizes by the Greek letters ij, a> {ib.) : — 
 
 vii. 71. No A is some E || laH \ viii. a>. Some A is not some E I| J(AN,Ew). 
 
 Another form of this last assertion is somewhat enigmatically stated by 
 Sir W. Hamilton as : " some A is not some A." This reads like a cc»ntra- 
 diction in terms, but really means l (AN, A«), which is the same as art. 14, 
 note, i. 1) The assertion +(AN, E») states that there is at least one thing 
 called A which (possessing the attribute N) is different from at least <yne 
 thing called E (which does not possess the attribute N). 2) The assertion 
 X (AN, Kn) states that there is at least one thing called A which (possess- 
 ing the attribute N) is different from at least one (other) thing called A 
 (which does not possess the attribute N) ; that is, that there are at least 
 two things called A (one with and one without the attribute N). 3) The 
 assertion XKe shows that there is at least one thing called A which (not 
 possessing the attribute E) is different from every one of the things called 
 E. 4) The assertion XaY. shows that there is at least one thing called E 
 which (not possessing the attribute A) is different from every one of the 
 things called A. 5) And the assertion X (A^, flE), art. 14, note, vi., makes 
 both the third and fourth of these assertions at once, and consequently 
 implies the firet ; for even if there be only one thing called A (and not 
 
 ART. 43.] 
 
 LOGIC FOR CHILDREN. 
 
 41 
 
 means " at least one," or " at least a portion of," and though it 
 cannot be none, it may be all. Now this is different from its use in 
 common life. Hence "at least one," is preferable. To say "at 
 least one possible case is non-improbable," is merely putting 
 non-improhahle for probable, which will be correct if improbable 
 only means no7i-p^'obable, but not otherwise. 
 
 iv. Some (possible cases) are not [probable]. JAe, or " at least 
 one possible case is a non-probable case." This is the method in 
 which such sentences have to be understood. 
 
 V. (The just) are [the holy], ffAe, that is the same set of people 
 is called both just and holy, not one who is just is non-holy, not one 
 who is holy is non-just. Observe the extreme brevity of common 
 language and the consequent probability of its being misunder- 
 stood. It implies JAE • Jae ; and consequently that " all the just 
 are holy, all the holy just, all the non-just non-holy, and all the 
 non-holy non-just." Observe, however, that though in this case 
 the just persons are "identical" with the holy persons, within the 
 range of thought, the property of being just is quite different from 
 the property of being holy, and that, in fact, the use of the asser- 
 tion is not to declare the identity in the meaning of the words, but 
 only in the things to which they apply. (See art. 13, vii.) This 
 " identity" must be kept quite distinct from mathematical " iden- 
 tity."* The pupil must be taught due caution in the use of such 
 extremely hazardous words as "identical" and "same." 
 
 possessing the attribute E), and only one thing called E (and not possess- 
 ing the attribute A), these two things must, in this case, be different. 
 The difficulty usually experienced in understanding the assertions rf, w will 
 justify the insertion of this explanation for the use of teachers. Their 
 awkward wording by Sir W. Hamilton depends on his "quantification of 
 the predicate," which is further referred to in art. 44, note, p. 45. The 
 whole of this imperfect enumeration of assertions is superseded by the 
 complete enumeration of assertions respecting two elements in arts. 13 and 
 14, including the note. The assertion co in the form X (AN, An) belongs 
 to these, as we have seen. In the form X (AN, Ew), it in fact consists of 
 two premisses of a syllogism, as in art. 30, first note, No. 5, p. 29, which 
 gives the resultant — 
 
 2,AEN, 4^21 A^N, 43 «EN, 4443 a^N, 
 22AEW, 44 Acn, 4322 «E«, i^i^am, 
 where the limitates 2j and 2^ are due to |AN and JE« respectively [con- 
 ditioning + (A, E, N, «)], and the limitates 43 and 44 to Ja and Xe respec- 
 tively. Hence it is erroneously placed among assertions respecting two 
 elements, and consequently, when used as a premiss, leads to pseudo- 
 syllogisms, as explained in the second note to art. 30, p. 32. This analysis 
 of Su- W. Hamilton's assertion 00 will show how the method here proposed 
 for teaching children the most elementary notions of logic, is in fact a 
 powerful instrument for examining the laws of thought propounded by 
 some of our profoundest thinkers. 
 
 * Two mathematical operations of a distinctly different form are said to 
 be identical when they possess a common property — namely, that of both 
 producing a given result according to the mere laws of operation, without 
 any annexed condition. Two identical mathematical operations are con- 
 sequently merely members of the same class (art. 44, i.), which mat, and 
 
42 
 
 LOGIC FOR CHILDREN. 
 
 Fart. 43. 
 
 VI. Some (happy persons) are all [the righteous]. faE. 
 vii. All (the insincere) are [dishonest]. fAe. 
 yiii. Ko (unjust act) is [unpunished]. fAE. 
 ix. Some (unfair acts) are [unknown]. JAE. 
 X. Some (improbable cases) are not [impossible]. JAe. 
 xi. The (unlawful) is the only [inexpedient]. ffAe. 
 xii. Some (unhappy men) are all the [unrighteous]. faE. 
 xiii. (Fixed stars) are [luminous]. fAe. 
 xiv. (Life) every [man] (holds dear). fAe. 
 XV. Few not a few, many, a large number of, the maiority of 
 the minority of, most, not all, only some, not nearly all, tome at 
 least two or three, one or two, (men) are [honest]. XAE In or- 
 cSd ^^Thp^"^? ""^ these quantitative words ca^i be distin- 
 
 thil f^^V « f T^^.^ ^"^ ^J ^r^* supposed that they also agieed in 
 this, that " at least one A-thing is an e-thing," or tAe Persons 
 who use the words may often mean both :AE and J A., but on ex! 
 r^'T^r.!!' ^^",b^^"^d that the actuat assertion is only JAE 
 and hat the rest is left in doubt. The consideration of a definite 
 number of things, or a definite part of the whole number of the 
 things mentioned belongs to a peculiar branch of logic first 
 treated by Prof. I^ Morgan, with which children must not be 
 troubled, see art. 48. Observe that " not all (men) are [honest] " 
 is^^mbiguous, and might also mean, "no (man) is [hones?],' or 
 
 xvi. Every (mistake) is not [culpable]. This is ambiguous- 
 either "not one (mistake) is [culpable]," fAE. or "at lealt on« 
 (mistake) is non-[culpable]," JAe. ® 
 
 xvii. None but the (brave) [deserve the fair]. faE The state, 
 ment IS apparently entirely negative. Show thai it 'is equivalent 
 to all [men who deserve the fair] are (brave)," leaving ?Ae or 
 a doubt as to whether those who do not deserve the fafr include 
 even one brave man. luv^iuue 
 
 xviii. Nothing is (beautiful) but [truth]. fAe. 
 xix. All (planets) are not [self-luminous]. Ambiguous, either 
 no one planet is sell- uminous," fAE, or "at least one planet is 
 self-lummous", JAE, which would agree with ordinary lan^g^age 
 
 grncTally does mc ude many others. But members of the same class mav 
 always be substituted one ibr another, as long as merelv theh" clas" ^^ 
 peily IS concerned (see the general principle in art. 41, p^ 39) The X; 
 
 ;iis f Ld""Ev7r:r^ ''"".''^•' "^*^°^^*^'^^' ^° ^^p^^^^^ substitutions^ 
 
 lil .H ?. fi. '''' V''"^'^^.'^'' '^ annexed, it is supposed to be ful- 
 
 hlled for the purpose of carrying out the substitutionf and it is itself 
 
 ■~~ 1 o T D or X •K u. 
 
 ARTS. 43, 44.] 
 
 LOGIC FOR CHILDREN. 
 
 43 
 
 XX. (He) is [no fool]. fAje. Observe "he" is singular, and 
 hence Ai is used, see art. 20. Also observe that no-fool is much 
 more than a-non-fool, or not a fool, and really implies the pos- 
 session of a considerable amount of wisdom, or at least cleverness. 
 
 44. As examples of the great diversity of expression in use for 
 the same assertion, take the following, which will guide the teacher 
 iu checking interpretations given by the pupil, who must be made 
 to see in time, by a variety of concrete examples, and first by 
 means of words containing letters, that each assertion has pre- 
 cisely the one same definite meaning here assigned to it, and more 
 fully explained above as the fifth and second usual assertions, 
 art. 14. 
 
 i. fP'Z implying tPQ-tP^Z'Pi'Q-lM. 
 Not even one thing (or object of thought) exists within the 
 range of thought bearing the name P, (or possessing the 
 attribute P), without also bearing the name Q (or possess- 
 ing the attribute Q). 
 AllPisQ. AllPsareQs. 
 
 All P is some Q. Every one of the Ps is some one of the Qs. 
 Every P-thing is a Q-thing. 
 No P-thing is any non-Q-thing. 
 Everything is either Q or non-P, that is, either a Q-thing, or a 
 
 non-P-thing. 
 Whatever is not called Q, is not called P. 
 Nothing which is not Q, is P. 
 Every non-Q-thing is a non-P-thing. 
 All non-Q is non-P. All non-Q is some non-P. 
 The class Q contains the class P, or the class P is in the class Q. 
 The class P has the name Q as one of its class-marks. 
 The attribute Q belongs to the things in the class P. 
 The name Q is an essential* part of the name P. 
 The attribute Q is an essential part of the attribute P. 
 The names of the things P are to be sought only among the 
 
 names of the things Q. 
 The attribute P is only to be found among things possessing 
 
 the attribute Q. 
 The things P are a species of the things Q. 
 The things Q are a genus of the things P. 
 The name P is dependent on the name Q. 
 
 All the properties of the things Q are to be found among the 
 properties of the things P. 
 
 ii. IPQ implying JPQ • JHP^, pq) ' VipQ,, M-) 
 At least one thing (or object of thought) exists within the range 
 of thought bearing both the names P and Q, (or possess- 
 ing both the attributes P and Q). 
 
 ♦ The confusion attending the use of the word *' essence," is and will 
 probably remain enormous. But, at least in early teaching, the old philo- 
 sophical questions must not be raked up, and we must be content with 
 such a statement as this, which, if not the whole truth, in the opinion of 
 many thinkers, is at least all that comes into action in logic, the rest be- 
 longing to investigations into the " nature of things." 
 
u 
 
 LOGIC FOR CHILDREN. 
 
 [art. 44. 
 
 I^meols?- ^^™^P!«««^^Q. Some Ps are some Qs. 
 gome (,) IS F. Some Q is some P. Some Os arp anmo p! 
 
 ^"^IM^r* '""«^ '^"^ P is airsoSroneTtlf things 
 
 Some P is not any of the non-Qs. 
 
 The class non-Q does not contain the whole of the class P 
 
 The class non-P does not contain the whole of the cS O 
 
 All of the class P is not in the class non-Q. ^• 
 
 At east one of the class P is in the class Q. 
 
 At least one of the class Q is in the class P. 
 
 Ue cits'p ml" .? ^'M' '^^' °l!f """S in common. 
 
 The absence of the name Q is not essential to the name P 
 
 buteT "■' """''"*' Q " "" inessential Ttheattri- 
 
 "^^ W?L^ tr nairo' ""^' '°- °"'^ """"^ ^'''"^ "'"«'' 
 The attribute P does not exist only outside of the class O 
 The things P are not a specioi <rf iioo-Q. ^• 
 
 Ihe thmgs non-Q are not a geim, of P; 
 Alf Z"^ '* independent of ihc ak..,„c« of tb* name Q 
 
 ass?4ris^?5>'Q°7pt^i'::^ ';ie"ff;%r/.:s:.»'*-' 
 
 r ^rp- rj "n7nt'rtlJJ1^r^tr»S 
 a more advanced stage * ^ ImJ Wl for 
 
 ♦ The real intention of the (rnun^ll^ flmn. fc«^ -.wlj. *v i 
 terms subject, copula, and pr^^S^ ^^SSJ^^ ^"^^ 
 the relation fA,., in its most^c^Sm^ ^f^ ^\ M^i^ ^ ,TVT. 
 one thing caUed A, was also c^S^ ^thli oL^^ift^'' ^i*^*. ^ 
 were also called E, in En^rlish" A i. P^" U, ?!l2 T°?u *^/^*'^ -^ . 
 an^ndiv^dual in the set orthin6;''E tt ^SLS^^.^ 
 (which m many langmffes is oolv AsnaML^i^ Vk i J . S^*P"* 
 
 the same thing. But this T* ^ - "T^iit^. 1 il7 TT i^*^ *" 
 thingA,, and others also, for eaiir " ^5^ni „2!I!i^J?V2' 
 E- Then. u«tead of sp^ng^JT. X til ll' ESS? j^il^^^ 
 .poken of as " the E thi^" Ld whm jT^ ^/SST^'AT 
 C,. D,. and other things possessed Ok alMtlT. bvEt I*"^ "•* A,. B,, 
 
 !^>=t <?<» the class F i^ prSt^t^fc^ -•» rF^'oTtT?/"^ 
 ceeding further in knowWiTA A ^.Vi » ^^ >. *' * '^ <*r JtR/: p^o* 
 
 st&d^^h^'r^e.Sr<!Sr%S^-^^^ 
 
 art. 45.] 
 
 LOGIC FOR CHILDREN. 
 
 45 
 
 45. Definitions all depend upon the complete assertion ffFa 
 showmg that the two names P and Q apply to the same things 
 and no others, (Art. 13, No. vii). But care has to be taken that 
 definition and connotation are not confused. Supposing that all 
 P-things are M-things, or fPm, and that all P-things are also 
 N-things, or fPw, and also that all things which are at once 
 M-things and N-things are also P-things, or fi^MN, then it is 
 clear that if we have once ascertained that a thing has both the 
 attributes M and N, we have found that it is P, and if we find that 
 It has the attribute P we shall know that it has both the attribut^js 
 M and N. But P need not mean MN ; that is, MN need not be the 
 connotation of P. Thus in a certain room we may know that all 
 Enghshmen wear blue coats and buff waistcoats. This becomes 
 
 language), the first relation XfEf had also to be quanffjied, as " all E is 
 F," and, as now the Xe¥ wiis immaterial, this phrase came to mean (in 
 logic at legist, thoui,-h ordinairy speech is ambiguous) only fE/. and not 
 necessarily ttE/. But there v«i still % diffic«hy. Uun ^ttw^ |o dth 
 ^.«" "^ ^>^,^ ^ ^' •' H ynu ckar tisit to my - aU hUihUxfrt m 
 F4lung». Of t>^, Mn4 «bo " Mue ioilividiMl £.(hiiig»a«v m.n.l' .«hW" 
 <c : V. wvMiM Ic loeoMiiictit I^ ibenMomiy. ^^mmt E^hinKK «n?G. 
 ihm^ 01 :t.Cr, «kI -»n K^ihiL^ tn wn-G-thlng«," or tEG,«x)ukl 
 abc l^ inti»iM»Uni, But It <lil ik* ikqut to wmktn «f UnffwiM to fatlk 
 ora^.F-<hing«, or loo-U-thingv, Thc^T c.:*ti»i>coi1inff tanMiiini not m> 
 |tfc•i^ and oOm go Viyood the mtak. Hicoe the diiKc^r vw rc< ov» 
 tt <«o eti.. \^; aJtflrijjp the «|.als **rawE m im< (J." or 1>V. w»i •" ih» 
 ol^bv nH/A/;;t„^ tJto •iilmct, -H* E b O." cir fEG. llui Kn«it cod. 
 tapn then ftR«iabavi XBO and fEil, fwit wasiooit fuinkl that vtt ^Mbi 
 »<« tkj other ^y Ortqr, «ad my with inputl tnitlj, '^Mcnd O mlL** 
 wrj^ G U K" Uiuf maida^ tbemtimtf i»to a »uhjf<4, which «m a 
 oooMMit |i<m«rvicin of tJbo oi%iiBil iK4ion. Ilitioi xntt nil tb* kiriad 
 
 but tko form :K^, t]^, oould mat. Why f Wo m^ht tappoM th*tit 
 
 IW, niT. E4hm«K" Bnl it «m^ la iin]iirwQ& kid not «a»civ«d 
 Jl?^i*",""^ ** *■ «cist<nt nrtilitr, but oolr m gi nMitMa of Q-tlite*. 
 ThitdAyiiltj- <Iuii^ to all b^c down to Uh time of PMf. Ut, Mofsn. Hi 
 SLTT!!''^ prchaAi Hurt c«iTc«lon hv which tho rndtetoWmne • »ub. 
 jfcUMaMut the old <VMouhcattiin of tbefiib|ttfl, led 8irWBIiam Hamiitcs 
 *« iBi^ctimUo eMo«|4ion of th« «Miintiflc)Mio& of tho pwdic^tc;* 
 HI!! ^M^ti^li"''!?^^?^*^ W abi.*e of UaiTB^^ kto which 
 
 . i » , tt***^ t« 9«i«uo it hum, Tho hopolMt omfttskitt iht^y 
 cratol by th^ Mtem^t to bcttdjomdur tonui iatoiciaitUlc Initi— di. 
 ip« l»wi the mAio diffic«Hj in bficftl tn(«4tm; The old inimrf<«t ihmbm 
 of the iant m^kfytn of kngni^pr, whkh /civc tlM UittinctMo of wS&a 
 tjBd pn^f ^ with tho tUewDt to tMrr it thmiiKh» nistod nu^ n^hnt 
 A^Ui<h .^ tLri impfrfbctthoary oS utfltt$Um tad Msdmm^m Utk<4m^ 
 •mm ti nuSU\ au tho fovMktna of lof^kal tho««ghL All this is oUMy 
 ^^f^f^yX ^^'^ «»<*l»«» hcif expUiaoi, and it wxiold be c«t/u«« to % 
 t\xxU\ » mmd to Morpln it with thn nH«»h« of Mch Uxnx% liTwbioct. 
 oopiila. ^ mfioilc^ coDOfpt m i:rMnmB4icAl ■olMot. In tlBO tb» oim*. 
 ■oUcui^i pupil ^ «0me toaak th.> t«t«ha-of kgic about it; Md fcr 
 Ujit vrouu^.ty this not*, hm l.*.n wHttfl^ to Ao«r the tctcher the fo- 
 ntKo* which ore railly xu^nX to be nr[irmol 
 
46 
 
 LOGIC FOR CHILDREN. [aRTS. 45 — 47. 
 
 for the raoraent our definition of an Englishman, the marks by 
 which we find him out. But it is not the connotation or meaning 
 of the word EngHshman. Bearing this in mind, the teacher will 
 have no difficulty with the cases treated in other books, and with 
 the distinctions of essence (which belongs to the connotation) and 
 accident (which does not belong to, but is not inconsistent with, 
 the connotation). Observe that in the last case fPm, fPn, and 
 fpMN, gives, by the general condition, JPMN * t(PMyi, PmN, Pmn, 
 J9MN) • l^ipMii, j77wN, pnm) ' J'(i>wN, pmri) ' l\pMn, pmn) ; that is, 
 in the instance given, there will be in the room at least one Eng- 
 lishman with a blue coat and buff waistcoat, and at least some one 
 who is not an Englishman, that will either not have a blue coat, 
 or not a buff* waistcoat, or have neither one nor the other ; and 
 there will be also at least one person without a blue coat, who is 
 certainly not an Englishman, but may or may not have a buff 
 waistcoat ; and lastly, there will be at least one person without a 
 buff waistcoat, who is certainly not an Englishman, but may or 
 may not have a blue coat. 
 
 46. Disjunctive assertions can now be understood. Thus " all 
 A is either B or C," nieans simply fAhc, or the name A is never 
 found without at least one or other of the names B and C. But the 
 assertion is ambiguously worded, for it may or may not imply that 
 A can be both B and C. If A cannot be both, we have fABC in 
 addition. Hence the assertion " all A is either B or C" means, at 
 full, either fAhc'ViABC, ABc, AtC), or t(ABC, Afec)-J»(ABc, 
 AfcC), the remaining compounds being doubtful. Whenever an 
 ambiguous assertion is made, both its meanings should be dis- 
 covered, and in all trains of reasoning, each of the meanings should 
 be separately considered, and then the various conclusions should 
 be separately exhibited, as if each had been originally derived 
 from a single unambiguous assertion. This is the only certain 
 way to avoid en'or. Other ambiguities are pointed out in art. 43. 
 All such ambiguities should be avoided, if possible, and the use of 
 the present symbols in place of or as an aid to ordinary language, 
 will render their avoidance always possible. When the second 
 kind of assertions are dealt with, similar ambiguities will also 
 arise, as will be seen later on, (art. 59, i, j. 66, 67, 68). 
 
 47. The impossibility of referring to any text-book to explain 
 the method of teaching logic to children, which I ani here en- 
 deavouring to indicate, must be my apology for entering at so 
 much length into minute details. What has been here advanced 
 will however suffice, I hop^, to enable any teacher who has him- 
 self studied some elementary treatise on logic,* to apply the 
 
 ARTS. 47, 18.] 
 
 LOGIC FOR CHILDREN. 
 
 47 
 
 * JF. Stanley Jevons, Elementarj' Lessons in Logic, deductive and induc- 
 tive, with copious questions and examples, and a vocabulary of logical 
 terms, London, Macmillan and Co., 1870, small 8vo., pp. 340, contains 
 almost all that is wanted. Lesson 23 contains Prof. Jevons's own arrange- 
 ment of Dr. Boole's system, which is the foundation of that here adopted, 
 and it is my duty to mention that it was through Prof. Jevons's Substi- 
 tution of Similars, 1869, and Pure Logic, 1864, that T was induced to study 
 
 method I advocate to every case which presents itself, and to solve 
 every example furnished so far forth as the simple assertion and 
 syllogism are concerned. 
 
 48. But the simple assertion and the syllogism are the mere 
 beginnings of logical studies. Far more complicated assertions 
 involving the composition of any number of names, and combined 
 in any variety of ways, have to be considered by the advanced 
 student, but of course cannot be presented to a child, as much 
 more highly developed mental powers, combined with a much 
 greater range of general knowledge, are required to comprehend 
 the mere intention of the investigations. Much might be done 
 by extending the examples of words containing or not containing 
 certain letters ; but the more advanced student would feel this to 
 be a childish game (as in fact it is, and is meant to be,) unless he 
 could leel that it had a real application to the nature which sur- 
 rounds him. Hence I do not recommend proceeding beyond the 
 point already reached, with assertions of the nature hitherto con- 
 sidered. Numerically definite assertions, although extremely im- 
 portant, and really easily exemplified by definite collections of 
 words containing or not containing certain letters, are also scarcely 
 adapted for children, even when presented in the simple manner 
 here advocated. The utmost that can be done in this way, is to 
 make clear the very simple case, that if most of the words in a 
 given collection contain A, and also most of them contain E, then 
 at least one of them contains both A and E. This is easily illus- 
 trated by a case like this. 
 
 AAAAAAAaaaaa 
 e e e e e EEEEEEE. 
 
 ( 
 
 Dr. Boole's abstruse mathematical system of logic. As my plan bears in 
 its foundations a great resemblance to Prof. Jevons's, they might appear 
 identical on a superficial examination. It is necessary therefore to state, 
 that the resemblance is entirely superficial, and that my views are not only 
 more extensive than Prof. Jevons's, but aim at accomphshing all that is 
 contained in Dr. George Boole's Investigation of the Laws of Thought, on 
 which are founded the mathematical theories of Logic and Probabilities, 
 (London, 1854, 8vo., pp. 424,) without his mathematical expression, and 
 without those doubtful points of theory on which his mathematical views 
 are based, and also at including the whole of Prof. Augustus Be Morgan's 
 Formal Logic, or the Calctilus of Inference, Necessary and Probable, (Lon- 
 don, 1847, 8vo., pp.336), and his more recent Syllabus of a proposed 
 System of Logic (London, 1860, 8vo., pp. 72), and of furnishing a com- 
 plete explanation of the fundamental theories of Sir William Hamilton's 
 New Analytic, as presented in ArchbisJiop William Thomson's Outline of 
 the necessary Laws of Tl).ought, a treatise on pure and applied Logic, 
 (London, 10th thousand, 1869, 8vo., pp. 304). Some account of these 
 three latter systems is given in Prof. Alexander Bain's Logic, Part I., 
 Deduction, (London, 1870, 8vo., pp. 279), to which, and to Mr. Thomas 
 Fowler's Elements of Deductive Logic (3rd ed., Oxford, 1869, pp. 176), the 
 teacher who shrinks from studying Archbishop Thomson, or the great 
 works of Dr. Boole and Prof. De Morgan, is referred for further infor- 
 mation. Br. F. Ueberweg's System of Logic and History of Logical Doc- 
 trines, translated by T. M. Lindsay, (London, Longman, 1871, pp. 690, 
 8vo,) gives a full historical account of the old logic. 
 
48 
 
 LOGIC FOR CHILDREN. [aRTS. 48, 49. 
 
 ARTS. 49 — 51.] LOGIC FOR CHILDREN. 
 
 49 
 
 Here are 12 words of which only the A's and E's are written, of 
 which 7 contain A, 7 contain E, 5 do not contain A, and 5 do not 
 contain h. By arranging them as above, it is clear that at least 2 
 must contain both A and E ; but as they could be arranged thus— 
 
 AAAAAAAaaaaa 
 EEEEEEEeeeee, 
 
 we might have as many as 7 words containing both A and E, and 
 may have any intermediate number between 2 and 7. Particular 
 cases of this simple nature may be readily solved by elementary 
 arithmetic : but the general statement involves algebraical con- 
 oiuerations. 
 
 Passing over these, and assertions respecting pure combi- 
 nations, we come to what really form the most frequent and most 
 important case,— assertions respecting the consistency of other 
 assertions The greater part of all actual ratiocination turns upon 
 the test of consistency, and the whole doctrine of probabilities can 
 be naade to rest upon its proper development. Yet in ordinary 
 treatises on Logic this principal part of the subject is usually dis- 
 missed in a few pages, with one or two almost self-evident 
 examples, under the head of Complex or Hypothetical Propo- 
 sitions and Syllogisms. At the end of a lecture already far too 
 long. It would be out of place for me to attempt giving this class 
 of assertions their proper position; and, fortunately for my present 
 purpose, the real development of their theory is so far beyond any 
 child s mental powers, that it would be useless to make the attempt 
 in these hints for teaching logic to children. But I must endeavour 
 to give some account of the elementary notions, not only because 
 they are usually insufficiently explained, but because some of the 
 most ordinary processes of reasoning in common life, and in trea- 
 tises on geometry, turn upon their application. 
 
 49. We are already familiar with cases of consistent and incon- 
 Bistent assertions (art. 15). In the syllogism we have learned 
 that the premisses and conclusion are all consistent with the 
 resultant and with one another, and that although assertions 
 wfthX nn' T'^ *hf Premif es are not necessarily inconsistent 
 with the conclusion, they will necessarily be inconsistent with the 
 resultant (arts 31 to 39). In this consideration of consistency, we 
 did not trouble ourselves with the truth or co.Tectness, th^t is 
 the consistency with actual fact, of the assertions themselves 
 i3ut It will be more convenient in future to speak of this truth, not 
 ?!''. f{ importance m itself, but as being consistent or not With 
 ^e truth of other assertions. " Supposing an assertion to be true, 
 will other assertions be true or false ?" This is the question which 
 InL *^7r^ ofainnvestigations into the consistency of Isser- 
 mZ'tion'^'lT.l ^"?''^ T'*^^^ of proposing it came the deno- 
 mination hypothetical." Again, the results are frequently stated 
 
 namP^T' ^^^^^V-^I^ 9a«^«^ that will be true," and hence the 
 n^me disjimchve, which implies that the two cases are incon- 
 
 ftsefft ^""^ WK "'^*'''^- \"«ther form in which this case presents 
 Itself IS : Whenever or wherever this happens, that will happen " 
 Ihis 18 a purer form of stating the question of consistency. What- 
 
 <l 
 
 ever form however be taken, consistency is at the root of the inves- 
 tigation. It is enough to state this, in order to show its radical 
 importance to all thinkers, especially the least advanced, who are 
 most exposed to the besetting sin of all speakers, (not to say 
 arguers,) consistent inconsistency. 
 
 50. Now we cannot advance far without a notation less cum- 
 brous than that of ordinary language. Letters must be employed, 
 which must, as before, be considered as labels, or titles used in 
 place of the whole expression of the matters, sometimes very 
 lengthy, with which we have to deal. Let any capital letter, as X, 
 signify the truth of a certain assertion, which for similarity may 
 be written X', (read X.-dash or assertion-X), concerning a certain 
 event, which, also for similarity, maybe written X" iresLdX-douhle- 
 dash or event X).* The pupil will therefore learn to associate the 
 conception of truth, assertion, and event, with -jmaccented, once 
 accented, and twice accented letters, such as X, X', X". It is totally 
 indifferent what the assertion, or event may be. But when several 
 assertions are made respecting different events, different letters 
 must be employed, distinguished by the same accents. Thus, if 
 the assertion be " a house has fallen," and we call this assertion 
 X', then the event of the fall of the house will be X", and the truth 
 that the house has really fallen will be X simply. This will be a 
 difficult thing for a child to understand, and great care must be 
 taken not to hurry over this step. Pupils must themselves make 
 and note different assertions in this way, till they become familiar 
 with these abbreviations, for they are nothing more. 
 
 51. Now, if the assertion X' is not true, if the event X" has not 
 happened or will not happen, there will be some other assertion 
 which is true, some other event which does happen, and this asser- 
 tion and this event will have been incompatible or iiiconsi«tent with 
 X', X". Call this assertion x', and this event x", and the truth of 
 jr' call X, reading x as non-X. The truths of X' and x' are incon- 
 sistent ; that is, X and x are inconsistent. We cannot have both^ 
 X and X, but we must have one or the other. The assertion X' 
 must be true or false. If it is true, we have X ; if it is false, we 
 have X. This must also be well felt, be quite familiar, before the 
 pupil advances at all. Draw his attention to the fact that x' may 
 represent any one of several assertions. Thus, if X' be fAE, any 
 
 * In place of X', X", I have used oX, eX, in my Contributions to Formal 
 Logic. These forms might be used, if preferred, the Greek o, f being only 
 considered as peculiar ways of writing a, e. Otherwise as. X, ev. X, or 
 even assertion X, event X, might be used. The teacher must suit himself 
 to the powers of his class. In my Contrihutiom, kc, I also use tA for the 
 attribute A, and OK for the thing possessing the attribute A, and hence 
 called by the name A. The use of vK for the number of the things 6 A 
 under consideration, and i/X for the number of times that the event iX 
 occurs, is also convenient in the investigation of numerically definite as- 
 sertions and statistical conditions, of probabilities ; but these matters re- 
 quire a knowledge of algebra, and he Tar beyond anything here attempted. 
 
^^ LOGIC FOR CHILDREN. [aRTS. 51— 53. 
 
 one of the assertions J AE, fA^ faE will be inconsistent with it 
 and hence any one of these will be represented by x\ because the 
 truth of any one of these implies the falsehood of X'. Hence 
 although X may be single, x' may be ambiguous. Truth we 
 know is one. error hydra-headed. If X" does not happen, any one 
 of several events x ' may have prevented it. " There is nmmi a 
 slip twixt the cup and the lip." But this is of no matter ; we are 
 only interested in knowing that if X" occurs, x" does not occur, 
 and if a occurs, X'' does not occur. 
 
 52. This occurrence and non'Occnrrencc of X" or x'' may be con- 
 sidered as the presence or absence of X and x, and hence' we may 
 use our old signs of black and red lett<;rs on counters,and. in writing 
 our old signs J and f. We do not require a doubtful sign be- 
 cause o)u^ or othf^r must occur, there is no doubt about it. Then 
 +X (or black side of the counter X) indicates the jrresence of the 
 ti-uth of the assertion X' concerning the event X'\ and conse- 
 quently implies fx (shown by the red side of counter x) or tixe 
 absence oj the truth of the assertion x* coiiccoiiog the event s^ • 
 one of the assertions X', x' must and imly one cwiU irue. th:4t in* 
 one of the two events X'', y', must and only one c«ii hnppc*. In 
 writing, we use J, (X, x) to indicate tlii», whct^ J, re«d " pwimi 
 only one of," is the sign introduced in iirt. 2 ». Wit4 coiint<>w it m 
 sufficient to place them on their sides, iImb [^ . H | . AllcovntcrM 
 8opW will be subject to this law. or form a gnmn. culled » 
 unicate or Yfhich one must be present, and wir one am be rir^- 
 sent, although the conditions assigned hare not boon anOkaml lo 
 determine which one that will be. 
 
 63. If two events X", Y" are compatihlf, or eampfmO^, the two 
 assertions X', Y' are consistent, and liirncv tbo two IriHK* X Y 
 may co-exiM The two events may tluT„ bi« wid to fona n iymvfe^ 
 event, (X.Y)', [where the full stop or dut (.) » n*d •*r«m/'l the 
 two assertions form a complex ascrikin (X.Y)'; nmi the two 
 truths form a complex truth, or simplv » fowpUr X.Y. Here be 
 careful to distinguish a complex X.Y,' <two tniUi8, Xaiid Y.cw. 
 sidered as coexistent), from a compouN^t AK (Cho nftino of » thing 
 to which each of the names A and K apply siintly). and draw 
 attention to the presence and absence of the full *ton Or dot ( . ), M 
 characterising the two symbols, X.Y nod AE. AU dbtinniijih 
 this dot on the line (.), used for forming complcxot, frum the dot 
 above the line (), used for connecting nsejwrtioei*. iS«e art. 12. oo 
 p. 13, 1. 6.) Observe that the expression X.Y (tho CQontcr on it« 
 side) only states that X and Y may conift. Tlmt llier do CJcifit in 
 expressed by JX.Y Uhe counters erect. blftHt «iao to the front), 
 read *' present X cum Y"; and that tbcv do n^4 condU u ox- 
 pressed by fX.Y (the counters erect, rtd' «iii<t to thv frootK rttA 
 " absent X cum Y." The nature of tbo Cruib*^ X and * iJi Uion 
 such that fXjr, at all times and under all oircttuitftAnom. TliU 
 may be compared with tA.i, wkich cxpro«M the (nci^MtrnXtd wi 
 p. », I. o rrom bottom. T\te (fiftreooe bcUc^jn iXa aud fAa 
 moflt be tborMghlj* aeixod. 
 
 ARTS. 54 — 57.] LOGIC FOR CHILDREN. 
 
 51 
 
 ( 
 
 \ 
 
 54. Observe that with X. there must always be either Y or y. No 
 event occurs by itself. When it does occur, it is always accom- 
 panied by some other event, which is compatible with it, and that 
 other event is again either compatible or not compatible with 
 some other event. Hence if X" occurs, either Y" or y" must 
 occur, but both cannot occur. Hence if tX, then either JX. Y, or 
 |X.y. Similarly, if J*. then either J.p.Y, or IX.y. But either 
 Jx, or Jaj. Hence, under all circumstances, one of the four com- 
 plexes X.Y, X.^, a;.Y, x.y, must be present, and only one can 
 be present. Why only one ? Because if we could have X.Y and 
 X.y present together, we should have Y and y present together, 
 which we know to be impossible. Hence, whatever the two events 
 X" and Y" may be, we have always Ji (X.Y, X.y, x.Y, x.y), ex- 
 pressed with the counters by ranging them thus (compare art. 21), 
 
 >-• 
 
 • 
 
 X 
 
 5ri 
 >< 
 
 • 
 
 
 Those foar €oinpl«<3Dt6 Wkay bo readily rcoMmbered fi«<m ihb fbttr 
 OMnpoondis AE, Ae, iiR, 4i«, of wbicti the moaning ntu) Iuwh are, 
 a* we know, totAlly different; for wc licvc Icumcd tbnt till iho90 
 <o«npoonds may bo i)rf)!iei«t> axmI that at Vtani t%so of thetn m^rM 
 be preicnt^ to fulfil tiiO fcwnrral coaditkinii (arta. 12. l;^ 1 1>. It iti 
 knpoflrfeant t^t thiM re^^imbUiieo (but not identity.) in form aud 
 dilferanoo in mo&ning should bo tliorongbly woU uudontood. 
 
 55. r vc in the lamc way, we »«yi thai if JX.Y, then 
 
 eithier ;.X.V.yi, or JX-Y.r, and w on for cm<+» <if Iha (Xirnplnxca 
 of two tmt!i«. ITcficc, OB Tvcurdn cumi)lw»>K of ibreo trutlift. wo 
 mu^itt havi) *.(X-Y.Zw X.V.:, X,y,2tt ^-y-Sp *.Y.Z, x.Y.2, 
 ir^.Z, r,tf,T), And no o« for complexee of a«iy «ninb«r of tnaths). 
 tbo modo of fodrnuUion bcinir ixrwiwly tbo Kuoe a* for oonipoiind^ 
 of any niimlKr of namcft. The t^wThor, in dnvnlofiin^ thi» «*t of 
 eom^mutf^ wilU of course, iinx^tvd with oountcnrs in a manner 
 prec«w1y (nmilar to tlua exp^uvd in rvforeDCD to oocnpo«nd« in 
 vt. 17. 
 
 t 
 
 Ml Kow obaenrc that. «t OQlllf one in each of thejc »c<t« of 
 plexoft c*n bo pretM(nt> if wo lewiw thut o«c i# porc«cntv we know 
 that all tbo rem ax^ abe««U Th«K, if the tixcntn X^ and Y" nre 
 focnpettible, that i«. if both th^ aAf^rtionsi X' and Y' arc coa5i«t<9it> 
 UicnJX.Y. and tX.y, ar,Y. x.jf), or. th# annrtioii X' ia inoon- 
 •bUynt w ith //. and htAh 7^ and / ar<» idu^ether fal:*©, Thn^ if it b 
 \vm Ihiit Join tnKtA oi» KUhard <X*% *nd dbud toiih hiw <Y^^ 
 it wooli) U inoonnstcnt to say J(^hn ndhd <m Biekard (X'*). and 
 did not dint ipiih him (/0» or that JWiin did HiU mI? oh Sithard 
 (x^)* even if nocbio^ ia BBfortAKl abomtdintn^ or that Jchn did not 
 dine wiM Hirkard ^>. ereo if noiUikig is aascrted abo«t calling. 
 JlliiBtfatfi thi4 cnae by n yan*.*(y of flinikr ample cxnmplM of 
 erorj-dny life. 
 
 07. If wc know tbai one or mare of scycnil mdi oomplesM iM 
 nr«*ont, wo know for a c<^rtiiinty that oil tbo o4Ii6t» are ftb^^t* 
 ling, if wo know J.lX.V, ^.y)^ that is, thaX either both tbt 
 
52 
 
 LOGIC FOR CHILDREN. [arTS. 57—59. 
 
 theftf know ^HY?^\*C^''J"''''- °^ "^"''«'- "^ ^^^"^ happen. 
 hapDen sinX w!'^' f •^'' '^' '*■• "'^» "^'t^er of them can 
 
 tenTt^^alUrrr; rse17th™7tl"*" "*f '^''Pr "'^«- 
 neither of them happen sb^y. ^ '"'*° '"^ ""'^ '''" 
 
 a set weTn^r*! *^* ""^ T^ 7 ""'•^ complexes are absent from 
 
 aTd"didU^earon'hr^ ^Glneral/J'thf^'^h'''^'^'^-^?' ^■'"'»^^' 
 ^^hich involves :. (X.^.-Yr/tr'tl^^^^^^^^^^^^ 
 +Y, then Xx. Or, as it is frequently put, if X" hacDen? V ' Arl 
 not happen, or if Y" happens, X' does not hanr^n^TL ^ 
 e^P^ess.on for either of Xse'^.^o^J^JLtSs iftheSe 
 
 (a.) fA.i implies ti(X ?/ »• V n/\ "V j v/ 
 &^"x'l^'^''V^Y'•^>on^^^ of both' I' Tdt'tr: 
 
 ^% is^r^'e!T- Vs'T''. if ^Y-'S'' I' 1^^ ^^ 
 ^^n^be eoneluded from the supposition Thl't I' l^ '^!,, J:^^-$. 
 
 "ifVutl^X' irl-^-^""^' 'i'-^'' "■' ^ -^ *™-. Y'is true," 
 
 suDiisidon h^t Y' <?i ^- ^ ""'iV*' "*" ''«' concluded from the 
 supposition that A is false, or that Y' is true 
 
 faUe " ^"/y' k'jli^! I-'*^;^' ,H.^'-2'); "'f X' is false, Y' is 
 
 the s'uDDosilon thTV^ ^^T""" .?°"'l"« ^'"' ^ concluded from 
 me supposition that A is true, or that Y' is false. 
 
 (d.) fa-.y implies t,(X.Y. X?/ a- Y ^ "T'or.^ v x u .1 
 
 false" "X' Qlr^rio ^,. v/ 1 • ^' '^ ^ ^"^ ^ are not both 
 
 car. be concluded from the suppoLionSt I /s true!" or th^t y"5 
 
 ARTS. 59 — ^\^^ LOGIC FOR CHILDREN. 
 
 53 
 
 fi 
 
 (e.) t(X.Y, X.?y) implies +i(f.2/, x.y)\ "X' is false." 
 (f.) t(X.Y, aj.Y) implies Ji(X.]/, «^7/) ; " Y' is false." 
 (g.) t(.K.Y, a?.?/) implies t.(X.Y, X.^/); "X' is true." 
 (h.) t(X.2/, a;.?/) implies Ji(X.Y, a;.Y); "Y' is true." 
 (i.) t(X.Y, a;.?/) implies JiCX.i/, a;.Y); " X' and Y' cannot be 
 both true or both false," " either X' alone is true, or Y' alone is true," 
 *'if X' be true or false, then Y' is false or true respectively; and if 
 Y' be true or false, then X' is false or true respectively." Compare 
 this with (a ; and with (d), and remark the differences. Observe 
 that it is common to say, " either X' or Y' is true," which is am- 
 biguous and might mean either (a) or (i) ; or else to say, " either 
 X' or Y' is false," which is also ambiguous and may mean (d) or 
 (j). Remember that different eminent logicians interpret this 
 ambiguous phrase differently, and hence, when it occurs, work out 
 both cases separately. The results will ahvays be very different. 
 
 (j.) tvX.7/, x.X) implies tJX.Y, x.y), " X' and Y' are both true 
 or both false together," "X' and Y' stand or fall together," 
 " neither X' nor Y' can be true or false separately ;" " if X' is true, 
 Y' is true, and if X' is false, Y' is false, and if Y' is true, X' is 
 true, and if Y' is false, X' is false." 
 
 (k.) t(X.]/, xX, x.y) implies +X.Y; «X' and Y' are true 
 together." 
 
 (1.) t(X.Y, x.Yy x.y) implies JX.?/; "X' is true and Y' is 
 false." 
 
 (m.) t(X.Y, X.y, x.y) implies t^.Y ; " X' is false and Y' is 
 true." 
 
 (n.) t(X.Y, X.7/, X.Y) implies Ix.y; " both X' and Y' are 
 false." 
 
 60. From these we see how two or three assertions are some- 
 times necessary to arrive at a conclusion. Take the ex ahsurdo 
 proof Suppose it has been proved that " if X' is true, Y' is true," 
 this is (b). From this nothing can be inferred respecting Y. But 
 if we are able also to show that to suppose Y' to be true when X' 
 is false, leads to an absurdity, we can assert that " if X' is false, 
 Y' is false," or (c). Now (c) has been shown to be the same as " if 
 Y' is true, X' is true," and this is the converse of (b), which was 
 required. But having both (b) and (c), we have (j ), and hence all 
 the results there given, and this is much more than is generally 
 mentioned in an ex ahsurdo proof. 
 
 61. Next apply the same sort of analysis to a series of com- 
 plexes of 3 truths, X.Y.Z, X.Y.z, X.t/.Z, X.y.z, aj.Y.Z, x.Y.z, 
 x.y.Z, x.y.z. It will be an excellent exercise to suppose any 
 2, 3, 4, 5, 6, or 7 of these absent, and determine the meaning of the 
 result. Real assertions should in all cases be invented to suit the 
 abstract cases. These exercises are most conveniently worked 
 with counters. The following are examples. 
 
 (a'.) tX.Y..^ implies +, (X.Y.Z, X.v/.Z, X.y.z, aj.Y.Z, x.Y.z 
 x.y.z, x.y.z) : Hence, " if X' and Y' are both true, Z is true," 
 
54 
 
 LOGIC FOR CHILDREN. [aRTS. 61, 62. 
 
 ARTS. 62 — 65.] LOGIC FOR CHILDREN. 
 
 55 
 
 " if X' is true and Z' false, Y' is false," " if Y' is true and 
 7J false, X' is false." The pupil should be made to see how this 
 agrees with the theory of opponent syllogisms already given, taking 
 X', Y' as the premisses and Z' the conclusion of a syllogism (art. 
 36). But also " if X' and Y' are both false, or one false and the 
 other true, nothing can be concluded respecting Z'," and this 
 agrees with what was shown respecting the insufficiency of deny- 
 ing the premisses for the purpose of denying the conclusion 
 <art. 35). 
 
 (b'.) t(X.Y.2, X.y.z, X.Y.Z) implies X,{X.Y.Z, X.t/.Z, a^.Y.Z, 
 se.y.Z, x.y.z) ; " if or whenever X' or Y are separately or jointly 
 true, Z' is true," "if or whenever X' and Y are separately 
 but noi jointly false, Z' is true." Nothing can be concluded 
 respecting the truth of Z' when X' and Y are jointly false. Noth- 
 ing can be concluded respecting the truth of X! and Y when Z' is 
 true, but " if Z' is false, both X' and Y' are false." 
 
 (c'.) t(X.Y.Z, x.y.Z) implies :,(X.Y.2, X.v/.Z, X.y.z, a;.Y.Z, 
 aj.Y.z, x.y.z) ; " if or whenever Z' is true, either X' alone is 
 true or Y' alone is true ;" " if X' and Y' are both true, Z' is 
 false." Nothing can be concluded of the truth of X' and Y' when 
 Z' is false, but " if X' and Y' are both false, Z' is false." 
 
 (d'.) f x.y.Z implies J. (X.Y.Z, X.Y.z;, X.y.Z, X.y.z, iu.Y.Z, 
 X.Y.Z, x.y.z), "if or whenever Z' is true, X and Y' are not 
 both false," "if X and Y' are both false, Z' is false." Nothing 
 can be concluded respecting the truth of X' and Y' from the false- 
 ness of Z'. 
 
 (e'.) The same process can be extended to any complexes. 
 Thus t(X.|/.Z.W, X.y.zM, .c.Y.Z.W, x.X.z.w) means, "if or 
 whenever X' and Y' are not both together true or both together 
 false, then Z' and W are also not both together true or both 
 together false," and " if Z' and W are both together true or false, 
 then X' and Y' are also both together true and false." 
 
 (P.) -fiX.y.Z.w, X.y.z.W, x.Y.Z.w, x.Y.z.W) means "if or 
 whenever X' alone or Y' alone is true, then Z' and "VY are either 
 both true or both false," and " if Z' alone or W alone is true, then 
 X' and Y' are either both true or both false." 
 
 62. When the pupil thoroughly understands the meaning of 
 such symbols, he will be ready to solve any of the usual ques- 
 tions of this kind, and, in fact, be in a position to undertake prob- 
 lems of much greater complexity. The process he has to pursue 
 is, first to determine how many assertions are made, and to re- 
 present them by letters, as X', Y', Z', W. Then to write out the 
 corresponding list of complexes, for the same number of truths, 
 X, Y, 2, W. Then from the given relations, which will always 
 be expressed in one of the ways just considered, to determine 
 what complexes are absent. Each given relation will determine 
 one or more absences, and all the absences must be marked in 
 turn. Then, of the remaining complexes, one, and only one, must 
 be present. It may happen that some simple truth, or some 
 smaller complex, may be present in, or absent from, every one of 
 
 the remaining complexes ; if so, the corresponding simple or com- 
 plex assertion is certainly true or false. It is this partial result 
 alone which is contemplated in ordinary logic. It is usual to 
 state these in the form of a syllogism ; but this is an unnecessary 
 complexity. 
 
 Examples are usually furnished in the singularly inappropriate 
 form, " if A is B, C is D." This is very misleading. It would 
 seem to exclude such cases as "if all the things called A are also 
 called B, then at least one of the things called C is called D ;" or " if 
 John comes home to dinner, we will have the sirloin roasted." 
 Again, any such statement as " if X' is true, then Y' is true," is 
 itself an assertion P', which may often, with great advantage, be 
 used as an abbreviation. Thus, take two instances from Fowler's 
 Deductive Logic, p. 114, where they are called dilemmas, but 
 they really present nothing peculiar when the above device is 
 adopted. 
 
 63. For the first example, let "A is B" be X', and let " C is D 
 and E is F" be Y'. Then any assertion inconsistent with Y', 
 such as " either C is not D, or E is not F," will be y'. Then it is 
 asserted, "if X' is true, Y' is true," or ^X.y, see art. 59, (b). It is also 
 asserted secondly, " Y' is false" or fY, and consequently t(^-Y, 
 jt'.Y), see art. 59, (f ). But (b) and (f ) together make up (n), and leave 
 J^.7/, or both X' and Y' are false. This is immediately obtained 
 by forming the series of complexes, and seeing that only x.y is 
 left. If instead of simply asserting Y' is not true, we said " neither 
 C is D, nor E is F," or " either C is D, or E is F, but not both," 
 we should still be saying Xy', or some assertion inconsistent with 
 Y' is true. The conclusion will of course be the same. But if the 
 second assertion were simply Y' is true, or JY, or Ji(X.Y, x.Y) see 
 (h), this would dispense altogether with the first assertion ^X.y, 
 (which it implies, art. 67,) and give no information respecting X. 
 
 64. In the second example, let " A is B, or E is F, but not both" 
 be X', and let " C is D" be Y'. Then the assertions are, "if X' 
 is true, Y' is true," or ■fX.y; and " X'is true" or JX or ti(X.Y, 
 X.y), in which X.y cannot be J owing to the first assertion, so 
 that the result is JX. Y, or both X and Y are true. If the second 
 assertion had been " C is not D" or " Y' is false," that is fY, or 
 t(X.Y, X.Y), this, in conjunction with the first gives Ja?.^/, or both 
 X' and Y' are false. 
 
 65. The next two examples are also from Fowler, p. 115, and 
 also called dilemmas, according to his statement. I have, however, 
 at once put them into the present notation, and will for an example 
 pursue the method of developing the complete series of complexes. 
 For brevity only one series of complexes is used for both examples. 
 When the sign f is placed on the left it refers to the first example, 
 and when placed on the right to the second. The sign f must, of 
 course, be considered as omitted till the work begins. The reader 
 should copy out these complexes, and add the sign f ^s it is re- 
 quired in the process of work. 
 
56 
 
 LOGIC FOR CHILDREN. [aRTS. 65 — G8, 
 
 X.Y.Z.Wt 
 tX.Y.Z.iyf 
 
 X.Y.z.Wt 
 
 X.Y.Z.W 
 
 fX.y.Z.Wf 
 
 fX.y.Z.wf 
 
 fX.y.z.Wf 
 
 fX.y.z.wf 
 
 aj.Y.Z.Wf 
 fx.Y.Z.wf 
 fx.Y.z.Wf 
 \x.Y.z.w 
 
 x.y.Z.W 
 fx.y.Z.wf 
 fx. y.z.W 
 fx.y.z.w 
 
 66. First, " If X' is true, Y' is true," or fX.?/, 
 
 " if Z' is true, W is true," or fZ.w, 
 " X' and Z' are not both false," or fx.z. 
 
 Work. — Take the marks to the left only of the complexes in art. 65. 
 First fX.y, makes the second column f; 
 next fZ.iv makes the second line f ; 
 next fx.z makes the two last complexes in the 
 two last lines absent. 
 
 There remain 5 complexes, none of which contain y.w, hence the 
 conclusion fy.ic, or "Y' and W are not both false." Observe 
 that Fowler gives the third assertion and the conclusion in the 
 forms, " either X' is true or Z' is true," and " either Y' is true or 
 W is true." These are ambiguous forms, see art. 59 (i), but accord^ 
 ing to Fowler's definition, p. 113, he means by them f{X.Z, x.z) and 
 t( Y.W, y.vy). Now it is readily seen by this example that the addi- 
 tion of fX.Z would only make the first complex f also, and would 
 leave the complexes X.Y.z.W, and ar.Y.Z.W, so that Y' and 
 W' would be true together whenever X' and Z' were true sepa- 
 rately, and therefore Fowler's conclusion is faulty. 
 
 67. Secondly," if X' is true, Y' is true" or fX, yl^ , . ^ 
 
 " if Z' is true, W' is true" or fZ, w S ^^ ^^^°^^ 
 " Y' and W are not both true," or fY.W. 
 Work. — Take the marks to the right only of the conplexes in art. 65^ 
 •f"X.^ and fZ.w make the second column and second line absent 
 as before. And fY.W make the first and third complexes in the 
 first and third columns absent. There remain 5 complexes in none 
 of X.Z occurs. The conclusion is, that fXZ or " X' and Z' cannot 
 both be true." Observe that as before. Fowler makes his second 
 assertion t(Y.W, y.w) and his conclusion t(X.Z, a-.z), but the 
 addition of fy.w will only make ifx.y.z.w in addition,, and leave 
 two complexes ar.Y.^.w, ;r.?/.z.W, so that " X' and Z' will both 
 be false when Y' alone or W' alone is true." Fowler's conclusion 
 is consequently again faulty. 
 
 68. These remarks serve to show the importance of attending 
 to ambiguities of language even in formally stated assertions. 
 When thev are not stated formally, the utmost caution is neces- 
 sary. It IS quite common to assume t(X^-Y, x,y) when in fact 
 this assertion depends upon some other, or when further considera- 
 tion will show from the nature of the assertions made that either 
 X.Y or x.y may be present. " John is either a knave or a fool," 
 is either " John is a knave" (X') or " John is a fool" (Y'), and is 
 ambiguous, meaning far.]/, or t(^-Y, x.y\ and the latter is usually 
 assumed, although knavery and foolishness (even insanity) are 
 compatible. " John must either win or lose," is either *' John will 
 win" (X') or " John will lose" (Y'), and is usually taken as t(X.Y,. 
 
 ART. 68.] 
 
 LOGIC FOR CHILDREN. 
 
 0/ 
 
 X y) But in the first place John may not play, and in the second 
 place the game may admit of being drawn, so that, unless some 
 other condition is annexed, X.7/ is possible. ,, ^ . -^^ 
 
 These remarks apply to the old story cited by ±owler, p. lib, 
 which is amusing and instructive enough to reproduce and com- 
 plete — " Protagoras, the sophist, is said to have engaged with his 
 pupil Euathlus, that half the fee for instruction should be paid 
 down' at once, and the other half remain due till Euathlus should 
 win his first cause. Euathlus deferred his appearance as an advo- 
 cate, till Protagoras became impatient, and brought him into 
 court. The sophist then addressed his pupil as follows : ' Most 
 foolish young man, whatever be the decision, you must pay your 
 money if the judges decide in my favour, I gain my fee by the 
 decision of the court, if in yours by our bargain.' This dileinma 
 Euathlus rebutted by the following : * Most sapient master, what- 
 ever be the decision, you must lose your fee ; if the judges decide 
 in my favour, you lose by the decision of the court ; if in yours, by 
 our bargain, for I shall not have gained my cause.' " l^ow as this 
 is clearly a logical exercise of the old school, and not a fact, it is 
 lawful to invent a conclusion, as follows : " Whereupon the Chief 
 Judge frowned and said : ' Why waste my time ? Protagoras had 
 no ground for action till Euathlus had gained a cause. I enter a 
 non-suit. There has been no action. Neither wins and neither 
 loses a cause. But both pay costs.' " . • ^.t 
 
 The full statement of every assertion here made is rather 
 lengthy, but the ground of the Judge's decision is easily made out. 
 
 Let the event, 
 
 " Euathlus pays Protagoras be X , 
 
 " Euathlus gains a cause" be Y", 
 
 " Euathlus loses a cause" be Z". 
 Then " if Y" happens, X" happens," or f-^^Y, 
 
 " if Z" happens, X'' does not happen," or fX.Z, 
 
 " if T' does not happen, X" does not happen," or fX.y; 
 while " it is not possible that Y" and Z'' should happen together,'* 
 or fY.Z, although ** it is possible that neither Y" nor Z" should 
 happen,'' either by Euathlus not appearing in court at all, or only 
 appearing in a non-suit. Developing the series of complexes, and 
 placin<^ the t as indicated by these assertions, we have 
 tX.Y.Z ■fX.y.Z fa^Y.Z x.y.Z 
 
 X.Y.Z f^-y-z f^.Y.z xy.z. 
 
 Conclusion Ji (X.Y.^, x.y.Z, x.y.z), one of three things must 
 happen : Euathlus gains and does not lose a cause, and pays Pro- 
 tagoras, or Euathlus does not gain a cause (either by losing, or not 
 losing, as in a nonsuit) and does not pay Protagoras, and, as the 
 Judge declared that ly and Jz, (or that Euathlus neither lost nor 
 gamed,) there remains only tx.y.z, so that Protagoras was un- 
 paid.* 
 
 * Since the lecture was delivered, I have read the original of the story 
 in Aulus Gellius, 5, 10, and am sorry to report that his judges were not 
 quite 80 sensible as my chief judge; we read: "Turn judices dubiosum 
 hoc inexplicabileque esse, quod utrimque dicebatur, rati, ne sententia sua, 
 
I-OGIC FOE CHILDREN. [arts. 68—70. 
 
 a treaty is not Decessarily nor eenerallv an « or,^ u? ^ 
 
 ment" (for examnl*. ih^i- K^? generally an amicable arrange- 
 .077. ^^°'\^^^™P'f' t'lat between Germany and France 26fh irik 
 
 Eschew "negotiations by understandino- '' o^? t H^ ?^ ^^- 
 
 . so far a« in language lie/avotr^^^.^. ShTXhs. feo* 
 
 69. Very often the complete develonment of all fK^ i 
 
 P'. Q', E', such that :.(X Y tuP^o'm^T:^°''''3^'^^'' 
 fZ.r. Show that t^.P fvO tl R ^W ' ^^^.P*' tY.g, and 
 the complete series wo^ldlo^iain 2 2 2 2 "2 fe^^ «'■;"*«, 
 which would be troublesome to write out ' But tT,;l ^"fP'e^es. 
 immediately seen to be satisfied only by Lx,?* Par xY°°'n'''' 
 '^■y.Z.p.q.n). For a complex contai^in» P^ ''^ I '•Y-2.i>.Q.r, 
 or R, in virtue of + 7p ^n ^""'^'""g ^ ^^aonot contain Q 
 9 and r; but by tY « andS?'' ??<? ^ence can only contaiS 
 
 i't can only contlfn J an'd ^and not Y andT^'^J'^i^ ^T^ '' 
 virtue of ^ (X V 7\ ;f v ii ^^^ '^' ^^'^^ ^9» b). Now in 
 
 Jn l^i^J'g^ef 'h'^ tfrco,;:l'°"'"J" ^ ^^^l ^ it must Ton" 
 siderations givfthe others and «d'rfR''^''''^'y ^^^ ^«« <=on- 
 
 utramcumque in partem dicta esset, ipsa sese res<-;T.rt»,^t"T ^~^~7 
 
 rchquerunt, causamque in diem lonm^^m.^ ^ .■ ^ ' "*" ">judicatam 
 c&tulenmt." This'was a cowar^rpr^riT 'bufn?r^^^ 'T^."J 
 been given, we have ty and t-, and L^cT^.^.^ tfo« '"''" '^""^ 
 
 They^'s^Xte^^SfTt^S^lr^lS^^^^^^^ 
 
 pcnd upon views of nature have been siTr^H^ i, *v°^'= ^^'^^ de- 
 more accurate observations and LwrimenK ^J''' ^¥ J^'^its ot the 
 deductive logic we havp tn »..:. ^ . , """e™ mduction. For 
 "clearly" anf" rtinctTy" det^^n' J'^t f* ^'^"f'^ "''J'*' '« ^ 
 word is by its visible leVtJrs iC^.^^- '^ 1°^" "«"*«'«. as a written 
 altogether ideal, and nev?; eSts i^ anv ."*' <^»f ''?«»»" thus formed is 
 Hence the distribution oruchTdhriduTlJoZ^r '' ""'"*"'' '■'^ ^^- 8»)- 
 or ultimate classes, is al^pZwi^l¥^^T"^°'t^'''f'^°"<><' 
 arc merely collections of rude^nceitf "'L h» 1i °*"" ^ "^ Aristotle 
 and not on an accurate I^ttTgS'r dStio'ns ""Zr .i''"?"" ««' 
 
 J?/ietfT;,:rsr.^!;,sr^^^^^^ 
 
 -in..,)mu.tbc...cgaidcda'trtK'c':X.tr^^^^^^^^^^^ 
 
 ART. 70.] LOGIC FOR CHILDREN. 59 
 
 but I felt that a mere oral exposition of the method proposed 
 would have been barely comprehended when heard, and would 
 have been forgotten immediately. To have a chance of doing any 
 good therefore, I was obliged to furnish something approaching 
 to a Teacher's Guide— to be carefully concealed fro m the pupil , 
 
 are'nTv^deally or ah^lutely, but merely relatively, « clear' ' (sufficient to 
 distinguish their objects from all others) and ''distinct" (sufficient to dis- 
 tinffui^'sh all the parts of any one from all the parts of any other concep- 
 tion). An attribute becomes simply a mark (as in the old nota, reKti-npiou), 
 in the sense of art. 40. ^ ^, v ^ aa 
 
 Let both the conceptions A and B be of the class C ; then, by art. 44, 
 i p. 43, tAc and tBt", and on developing the resultant we have, smce 
 J(AC, «;, BC, *.), ^^^^^^^^ 2.A5C, 2,«BC, F«^C, 
 
 fABc, fAbc, faBc, Xabc, 
 
 (compare art. 24). This gives the diagram 1, where all the different am- 
 ^ ^ ' biguities are clearly shown by the 
 
 blank spaces. This most general 
 
 case, where none of the limitates or 
 
 doubtful compounds are determined, 
 
 80 that it is not certain whether there are or are not any conceptions 
 besides A or besides B which are mcluded in the class C, has not been 
 distinctly contemplated by the old logic. The sub-vaneties JfAc with 
 tB., or tA(j with ;tB^ have also been omitted The case reaUy first 
 ' ' ' taken into consideration was Xt Ac with 
 
 2. 
 
 A- 
 
 C- 
 B 
 
 JfBc, giving diagram 2 ; compare the 
 observations on the predicate, p. 44, 
 note. In this case the conceptions A 
 
 and B were said to be co-ordinated, and each of them to hesub-ordmated to 
 the conception C, which was in turn said to be super-ordmated to them. 
 The resultant is ^^^^^ 2,2,A/.C, 232,«BC, 2,2^bC, 
 
 fABp, fAic, t«Bc, Xabc, 
 
 which is more determinate than before on account of J(«C, 6C) ; but there 
 are four compounds stiU left in doubt, concerning which we can make 
 
 '^Let^^n^^^^^^^ addition to tfA. and tfB., so that not one of the 
 conceptions A is subordinate to any one of the conceptions B, and con- 
 vSnot one of the conceptions B is subordinate to any one of the con- 
 ceptions A. Observe that, as Xabe gives %ab, we could not assume t AB 
 SpTv, atthat would leave ?.^ or assume ffAB, as that would give faj ; 
 80 that it AB was the only assumption involving f AB that could possibly 
 be made The effect of this is to make fABC, and consequently, through 
 The'^itates 2. and 2„ to make jA^C and >BC, and then again t^^^^^^^^ 
 
 the hmitates 23 and 24, to leave iab^y 
 
 fA as in diagram 3, which is formed from 
 
 3. < c the last by filling it up to meet this 
 
 Lb case. The conceptions A and B are 
 
 now entirely separate or incompatible ; but they remain co-ordi^iate^^ suh- 
 ordinate to the conception C. This case is not ^^^^^^^''^^i:^^'^'^^^ ^! 
 old logic, but A and B may be caUed incompatibly co-ordinate, ihe re- 
 sultant is .^ J. T.r>. o in 
 
 fABC, jAiC, t«BC, ?«*C, 
 fABc, ^Abc, t«B(?, Jaif', 
 
60 
 
 LOGIC FOR CHILDREN. 
 
 I \l 
 
 [art. 70. 
 
 whom It could only worry and confuse. I think that if any 
 teacher will have the counige to read my lecture in its extended 
 form, and then work through some treatise by its aid, he will find 
 much clear which was formerly dark, and at least emerge from 
 Cimmerian gloom into a kind of twilight.* 
 
 so that there 18 still one doubtful compound, ?al>C, to be determined. If 
 we take JrtiC, there will be part of the conception C which is not covered 
 by any part of either A or B, and then the conceptions A and B are said 
 to be (hynnct, bemg incompatible and subordinate to C, but not filling up 
 the whole of C (see diagram 4). If, however, we take iabC, then A and 
 B 3,re contrary opposites, being incompatible, but filling up the whole of C 
 (see diagram o). 
 
 4. 
 
 { 
 
 Disjunct. 
 
 A- 
 C- 
 B. 
 
 '( 
 
 Contrary opposites. 
 
 A- 
 
 C- 
 B. 
 
 It should be mentioned that the diagram for contrary opposites, which is 
 given in Lindsay's Ueberweg, is quite fallacious. 
 
 Without reference to co-ordination, we have +tAc for the sub-ordination 
 of A to C, and super-ordination of C to A ; ff Ac for the equipoUence of the 
 conceptions A and C, answering to the identity of objects (p. 41, v.) ; 
 a AB ioT contradictory opposition ; t(A, B), that is, J AB • +A^ • +«B • ?/?A 
 (see p. 17, note, vii.), for intersection, where each conception has some 
 part m common, and also some part not in common with the other ; JfAB 
 for incompatibility, as distinguished from contradictory opposition. These 
 are, however, very imperfect representations of the twenty-sLx assertions 
 m arts. 13, 14, and note, applied to conceptions. 
 
 In order that two conceptions A and B should be disparate, it is neces- 
 sarj' that they should not be both subordinate to any conception except 
 ^t of an object of thought. Let C be any conception with this exception. 
 The possibihty of having either of the conceptions A or B equipollent with 
 C (or non-Cs or of both A and B beina: subordinate to C (or non-C), is 
 excluded by excluding the pair iAc'\)ic (or the pair tACfBC, respec- 
 tively) ; and these exclusions, whatever be C, are the conditions of dis- 
 parity. Hence the only possible disparate combinations are the twenty- 
 three other pairs, 
 
 •made up of one of J+AC, XfaC, Xfac, fAC, \ke, 
 and one of ++BC, Jf/.C, If be, fBC, fBc. 
 
 Of these, the two tAC-fB^" and f Ac -fBC give fAB, by art. 23, [1], im- 
 plying complete separation ; while any one of the other twenty-one pairs 
 will give either ^:AC-+BC or else XKc'X^c, imphing intersection in 
 either C or c. 
 
 For all these cases, which, though difficult to understand in the abstract 
 language of the old logicians, are simple enough when reduced to the pre- 
 ceding form, see Lindsay's Ueberweg, pp. 130—135. 
 
 • Instead of reading this lectiure, a mere oral explanation of the nature 
 of the processes was given, and one or two syllogisms, with the example 
 m art. 68, were worked out with the counters. The whole lecture, endino- 
 with art. 70, was printed in the Educational Times for June, July, and 
 August, 1872. Several paragraphs and foot-notes have been added in 
 this reprint, and the whole has been carefully revised. 
 
 ARTS. 71, 72.] LOGIC FOR CHILDREN. 
 
 61 
 
 Pajrt II. Inductive Logic. 
 
 Til .V ? /. '"^"^ ?'' press the preceding explanation of a 
 Method of teaching Logic to Children, by means of words and 
 counters, it became clear to me that I ought not to content myself 
 with tlie passing allusion to induction there made (art. 6), but 
 that i should endeavour to show how induction could be tauffht 
 to children, at least so far as their limited knowledge and capa, 
 cities allow.* In attempting now to do so, my language must be, 
 as before, addressed to tea<;hers ; and, from the nature of the sub' 
 ject, rather consist in an exposition of the general principles which 
 they will have to inculcate, than a j)recise indication of any method 
 which thev can pursue. Hints will be given as I proceed, but de- 
 tails would require a course of instruction rather than a single 
 lecture. » 
 
 72. In deductive logic we dealt with heaven-sent assertions. 
 Ihese were assumed to be always correct, or rather all inquiry 
 mto their correctness was tabooed. All we had to do was to as- 
 certain precisely what they affirmed, denied, and lefL in doubt, 
 separately and jointly. With regard to separate assertions, attend 
 tion was confined to the most elementary and most frequently 
 treated, illustrated by assertions respecting the occurrence of 
 certain letters m certain groups of words (arts. 7 to 20) The 
 combined action of assertions was illustrated by the common syl- 
 logism where the premises also related at first to letters in words 
 (arts. 21 to S^) but were afterwards made more general (arts. 40 to 
 11 1 -^^i m, ^^ ^^ inquiry into the consistency of assertions of 
 all kinds. The very large class of numerically definite assertions, 
 whether of the first kind (art. 48). or of a statistical nature, were 
 passed over as too difficult for children,— the latter were indeed 
 not even mentioned.f The class of assertions respecting mere 
 combinations or juxtapositions of objects was also passed over 
 (art. 4»). And yet numerically definite assertions, statistical con- 
 aitions, and juxtapositions play a great part in all strict induc- 
 tions. Moreover another class of a'^sertions relating to order or 
 succession, m inference, in space, in time, in action, was not 
 even alluded to ; and such assertions have been, on the whole, so 
 little treated by logicians, who for the most part were not already, 
 and were not qualified by their previous studies to become, iuduc- 
 tionists, that even an elementary treatise upon them would be an 
 eaort ol human genius. Yet induction depends greatly upon a 
 
 The sources of inductive philosophy are the treatises of Bacon and 
 i^esc^rtes, and the prmcipal modem works are by Corate, Whewell, Mill. 
 Md bpencer. The teacher wiU find useful compendiunis in Fowler's and 
 Barn's Inductive Logic, 
 t Except in the note to p. 49. 
 
62 
 
 LOGIC FOR CH1LT)KET^. [ARTS. 72, 73. 
 
 ARTS. 73, 74.] LOGIC FOR CHILDREN. 
 
 63 
 
 proper method of handlmg snch assertions* Again, the absten- 
 tion from all inquiry into the correctness of assertions conditioned 
 an abstention from any inquiry into the probability of their correct- 
 ness Yet in the greater part of induction, even m the mam tacts 
 on which induction and the verification of induction depend, every 
 assertion has only a greater or less degree of probability of which 
 even the approximate evaluation presents mathematical ditfacul- 
 ties not yet in all cases satisfactorily overcome. Moreover, even 
 it we assume that the main facts can be stated, and the conclu- 
 sions verified with absolute correctness, the method of reaching 
 those conclusions occasionally involves deductions of a nature so 
 much more complicated than any which T have ventured to ad- 
 duce, that their mere statement would be utterly unintelligible ex- 
 cept to persons who had already undergone special and laborious 
 trainincT. It will suffice to say that in order to reduce to a con- 
 dition of verifiability the most complete and general induction 
 with which we are acquainted, the law of gravitation , it was neces- 
 sary to invent an entirelv new branch of mathematical analysis.t 
 and for its subsequent verification so far as that has hitherto pro- 
 ceeded, other and more delicate processes were initiated by 
 Laplace, and carried out by men of the highest mathematical 
 genius. But inquirers even now find themselves stopped by in- 
 superable difficulties in many special domains, as for example, in 
 hvdraulics. In other branches of natural science we are much 
 less capable of arriving at exact verification; and it has been the 
 criory of such a man as G. S. Ohm, who has only lately died, to 
 indicate, and of such a man as Helmholtz, who still lives, to 
 verify approximatively, inductions respecting such extremely 
 elementary and long considered subjects as the sounds of music, 
 on which Pythagoras himself is reported to have dogmatized. 
 
 73 With our tew and easily stated assertions m that portion of 
 deductive logic which I previously considered— and I warned you 
 that I was only scraping the soil of a great subject (art. 3)— at was 
 very easy to arrive at clear, precise, and certain results. Ihis is 
 entirely changed in induction. All our data are more or less 
 hazy • and hence our conclusions must share the same fate. Ihe 
 word' induction runs so glibly off* the tongues of most speakers, 
 that I may seem to have been raising giants for the purpose ot 
 slaying them. But unfortunately I found the giants already there : 
 and I am quite unable to slay them. The only comfort for our 
 present purpose is, that even if a steam mitrmlleuse noethod of 
 slaughtering these giants wholesale had been perfected, school 
 children would be quite incapable to manage it, and that if we are 
 content merely to show the nature of the method by rough in- 
 stances which a child can manipulate, we can skirt the fastnesses 
 of these ogres, and make a real advance into their country, on a 
 road suited for infant feet. In fact, as we shall see, much may be 
 
 • The (imperfect) method in which such assertions of succession are 
 reduced to assertions of mconsistency is illustrated below (art. 81). 
 
 t Newton's "Prime and Ultimate Ratios," subsequently kno^m as 
 '•'^t^uxions," and now generally replaced by some form of "Differential and 
 Integral Calculus." 
 
 done without any systematic knowledge of even deductive logic 
 (art. 84). Wha,t I am now anxious to do is to show how teachers may 
 put young minds upon what to present science appears the right 
 track, thus saving them from dogmatism upon insufficient data on 
 the one hand, and from scepticism through despair of all know- 
 ledge on the other; and leading them to feel that when the truth or 
 real law evades us, we can and must put up with a contrivance, a 
 theory, which we are convinced is not the reality, but which we 
 
 must treat as such, till the means are at hand for correcting it, 
 
 they probably never will be at hand for perfecting it,— valuing it 
 as the most trusty attainable weapon, but ready to reject it at 
 once if it snaps with any blow, however unexpected ; or, at least, 
 to limit its use to less adamantine obstacles, and leave the others 
 unassailed.* We have, indeed, constantly to be content with a 
 statement of laws which we are convinced are not correct, and to 
 argue from them as if they ivere correct, in order to discover the 
 limits within which they may be used with safety. This is, in fact, 
 our position with regard to almost all physical, biological, social, 
 and moral science. Our knowledge is all on its trial, but is not 
 the less valuable ; for wherever we can predict results with certainty, 
 our theonj, so far, if not true, is at least as good as true. Some 
 metaphysicians are inclined to think that man has never got, and 
 can never get, beyond this. Whether they are right or wrong is of 
 very little consequence. It is quite enough to have got so far. 
 
 74. We scarcely ever speak without making an assertion. De- 
 duction, as we have seen, dealt with the meaning of assertions al- 
 ready made. Liduction deals with the method of making verifiable 
 assertio7is. It is easy to make assertions right and left, if we are never 
 called to account for them. Theodore Hook described the pleasure 
 felt by a barrister who had turned parson, that, when he preached. 
 
 * We shall thus escape the error of those " who set up their own concep- 
 tions of the orderly sequence which they discern in the phenomena of na- 
 ture as fixed and determinate laws, by which those phenomena not only 
 are, within all human experience, hut alwa\ s have been, and always must 
 he, invariably governed," stigmatised by Dr.'Carpenter in his Presidential 
 Address to the British Association at Brighton, 14th August, 4872. This 
 very widely circulated essay contains many illustrated allusions to the 
 matters here treated, in a form having especial reference to children in 
 science as well as adults— the Associates as well as the Members of the 
 British Association. Attention will consequently be frequently drawn to 
 it in futiue notes. The present lecture had been completed a fortnight 
 before Dr. Carpenter's address was deHvered ; and nothing could have been 
 more unexpected than such an address on such an occasion. But the neces- 
 sity for something like scientific instruction in schools was well illustrated 
 on this occasion, for the Lord Lieutenant of the coimty, the chairman of 
 an educational society in high repute, in proposing the usual vote of thanks, 
 seemed to take it as a matter of course (according to the report in the 
 Brighton Daily News for 15th August, 1872, p. 10, col. 1) that outsiders 
 should know nothing about such elementary matters, saying, " I should 
 like to suggest to many of the friends here present, who perhaps, like myself, 
 have not had a very scientific training, that the address is one which will 
 amply repay a second or a third perusal. A good deal of it, I must confess, I 
 did not at first quite understand." 
 
 
6i 
 
 LOGIC FOR CIIILDREX. 
 
 [art. 74. 
 
 ART. 74.] 
 
 LOGIC FOR CHILDREN. 
 
 65 
 
 there was no longer any one to get up on the other side. But the 
 man of science is his own opponent. He is not satisfied with his 
 own assertions till he has run them to earth, deduced from them 
 as many of their consequences as he can (that is, as we now 
 know, discovered their meaning separately and jointly so far as 
 possible), and contrived means of contrasting these consequences 
 with fact. To him vn'ifinhle hnnwhcIgG is the only IcnovHedge. 
 He is very well aware, however, from sad experience— the veri- 
 fication requires no coaxing— that many of his theories are non- 
 verifiable, often as to bare possibility ; but he at least takes care 
 never to call them knowledge. In tliis respect he differs widely 
 from the metai)hysician, the intuitional philosopher,* who fre- 
 quently contends that he is as certain as he is of his own existence 
 that some un verifiable theory re])iesents a iact. Such theories are 
 not without a cei-tain value, and sometimes a vcrj' high value ; but 
 they are never knowledge; they are not inductions, and they do 
 not belong to our present subject, except so far as it is necessary 
 to warn young inductionists that even induction is not everything, 
 and that our world would be a poor world indeed if it only dealt 
 with what happens at the present moment to be verifiable, and 
 that, as the admission of verification into the arena of science is 
 scarcely more than two or three hundt ed years old, and as our 
 nieans of verification have largely and rapidly increased as its ne- 
 cessity has been more and more felt, we ought not to be too hasty 
 in determining the limits of future verifiability.f But this is a 
 question for the master, not the learner, of his art, and we must 
 strictly limit our school inductions to what is apparently verifiable. 
 
 * Dr. Carpenter says, " By tho iutu'tionalists it is asserted that the ten- 
 dency to form these priraarj- beliefs [« which constitute the groundwork of all 
 Bcientifie reasoning,' as p^e^^«^usly defined] is inborn in man— an original 
 part of his mental organisation ; so that they j,t^w up spontaneously m his 
 mind as its faculties are gradually unfolded and developed, requiring no 
 other experience for their genesis than that which suffices to call these 
 faculties into exercise. . . . The wteUectunlintuitions of one generatian are 
 
 the embodied experience of the pririous race It appears to me there has 
 
 "been a progressive improvement in the thinkiytg potrer of man. ... As there 
 can le no doubt of the hereditary- transmission in man of acquired consti- 
 tutional peculiarities, which manifest themselves alike in tendencies to 
 bodily and to m-ntal disease, so it seems equally certam that acqmred 
 mental habitudes often impress themselves upon his organisation with suffi- 
 cient force and permanence to occasion their transmission to ofltspring as 
 tendeneiefi to similar modes ofthoitqhtr This doctrine, as Dr. C. mentions, 
 was " fii-st txpUcitly put forth by IMr. Herbert Spencer." Miss F. Power 
 Cobbe, in her essay called Barnimsm in Morals, combats Mr. Spencer s 
 
 statement. ^ ,r j 
 
 + At the Br ghton meeting of tho British Association, on Monday even- 
 in<-, 19th Augi 6t, 1872, Prof. W. K. Cliffunl delivered a lecture on "The 
 Aims and Insln ments of Scientific Thought," which has not obtained so 
 wide a circulaticn as the President's address, but is of extreme value to 
 any persons wl o \n ish to obtain a correct notion of scientific thought. It 
 h-^i boen printed in Macmillan's Magazine lor October, 1872, from which 
 all the citations will be made. " When we are told that the infinite extent 
 of space, for example," savs Prof. Clifford, and the same appUes to any 
 other existing iuia c rifiabilitv, "is something which we cannot conceive at 
 present, we mav i eplv that this is only natural, since our eipenence has 
 
 Observe that unverifiable assertions may still be made, and deduc- 
 tive rules applied to them ; but the conclusions, as we know can 
 never have anymore weight than the assertions themselves on which 
 they depend. This limitation at once strikes off all supernaturalism. 
 all suppositions of "occult causes," all assumptions of what may be 
 because it cannot be shown not to be, all hypotheses concerning 
 the existence of fluids, ethers, spirits, atoms and what not. made 
 with any other view than colligating results, and obtaining veri- 
 fiable laws * This remarkably simplifies the questions to be con- 
 sidered. Many persons will be induced to conclude that with 
 these limitations " thinking is but an idle waste of thought." but 
 the field will soon be found far larger than man at present can 
 plough, and sow, and harvest. And as far as school instruction 
 IS concerned, there is the utmost value in drawing the line thus 
 sharply. It is not quite that between secular and reliorious in- 
 struction, for it cuts off much which is purely secularf but the 
 wide debateable land thus left between induction proper and 
 religion, will serve not ouly to render the distinction clearer— 
 
 never yet supphed us with the moans of conceiving such thing's. But then 
 we cannot be sure that the facts will not make us learn to conceive them • 
 in which case they will cease to be inconceivable," (p. 511, col. 2) And 
 speaking of another supposition, he says, "The knowledge of that fact 
 would be different from any of our present knowledge, but we have no 
 right to say that it is impossible." {lb., col. 1.) 
 
 * "Electric fluid" is a word of constant occurrence in newspapers. Its 
 existence is now entirely discredited, yet two electric * currents,' a positive 
 and negative, are familiarly spoken of by men of science. The * luminous 
 ether, supposed to pervade all space, and to generate, by its various * modes 
 ot motion, heat and Ught, if not other physical phenomena, must be re- 
 garded as a mere vehicle for obtaining and colligating laws. Its existence 
 has never been verified. The mathematical laws might be the same if 
 derived from other sources. Nothing is more delusive in this rejpect than 
 a mathematical formula. Thus, xy = a\ results from a consideration of a 
 bemg a mean proportional between a: and y, from x and y being len"-ths of 
 the sides of a varying rectangle with a constant area, from .rand y beinc 
 the coordmates of a point in an hyperbola, or the distances of two point! 
 from the centre of an involution, &c. "Spirit," in Newton, is an ether, 
 conditioning physical forces. See the last paragraph of the Frincipia. 
 Adjicere jam liceret nonnulla de spiritu quodam subtUissimo corpora 
 crassa pervadente, et in iisdem latente ; cujus vi et actionibus particulae 
 corporum ad mimmas distantias se mutuo attrahunt, et contiguae factae 
 cohasrent; et corpora electrica agunt ad distantias majores, tam repellendo 
 quam attrahendo corpuscula vicina ; et lux emittitur.reflectitur.refringitur, 
 intiectitur, et corpora calefacit ; et sensatio omnis excitatur, et membra 
 ammalia ad voluntatem moventur, vibrationibus scilicet hujus spiritus 
 per soUda nervorum capillamenta ab externis sensuum organis ad cere- 
 brum et a cerebro in musculos propagatis." Such ethereal and spiritual 
 vibrations are only another name for periodically recurring states, and that 
 IS all which 18 expressed by the resulting mathematical formulae. The 
 old "animal spirits" are quite dead. The "spirits of the dead" do not 
 yet belong to science. " Atoms" are familiarly spoken of by chemists, who 
 have formed from them a powerful theory for the investigation of the rela- 
 tions of substances, which we cannot at present replace, but which can only 
 be regarded as provisional ; for though the resulting laws can be controlled, 
 tho existence of atoms is quite unverifiable. 
 
66 
 
 LOGIC FOE CHILDREN. [aRTS. 74, 75. 
 
 because distinctions always become confused near the limits*— 
 but to impress very strongly on the pupils* minds that induc' 
 Han is not the sole sanction to thou{fht recoffnized among men, and 
 hence, to save them from a scientific, wliich is quite as hurtful as 
 a metaphysical or supernaturalistic pedantry. There are several 
 classes of researches where verification is not possible ; as, for 
 example, in theories of the origin and treatment of disease, m 
 geological and historical investigations, where we cannot put back 
 the c'ock of time, and contrast the actual past with its hypo- 
 thetical aspect. Such studies are admitted as partially positive, 
 because the different results are such as contemporary men might 
 have verified, so that, though unverifiable to the present genera- 
 tion, they are not so to humanity at large. But, inasmuch as they 
 are not absolutely verifiable,t they do not possess the same amount 
 of evidence, and require special and very delicate treatment to be 
 admitted as scientific facts at all. In all cases the power of veri- 
 fying must not be limited to the inductionist's own powers, but 
 must be extended to all investigators in all time, the main point 
 being that future verification should not exceed the bounds of 
 human power. But until verification has taken place, we have, at 
 most, reached probability, and real knowledge, science proper, 
 
 does not exist. „-,.■, ^- • ^ 
 
 75. The intention of all the processes called inductive is, /rom 
 ihe Icnotvn present and known past, to discover the unknmcn present 
 and past, and to predict the unknown future. This discovei7 is, 
 at any rate, scientifically, never designed to gratify idle curiosity, 
 but rather to obtain a basis for further prediction. And this pre- 
 diction must not be looked upon as a gipsy peep into futurity, but 
 as the only means we have of preparing lor contingencies, and 
 advancing in any way, material or moral. It is often said, that if 
 man knew what was to happen, he would be the most miserable of 
 beings. Yet man is always striving to know what will or may 
 happen, and to provide for the rainy as well as the sunny day. 
 If, in winter, we could not look forward to seed-time ; if, when 
 sowing, we could not look forward to harvest; our existence would 
 be exhausted in a hand-to-hand fight with death. But we now 
 know how the seed-time can be best utilized by proper prepara- 
 
 • " The animal and vegetable kingdom have a debateahle ground be- 
 tween them, occupied by beings thiit have the character of both, and yet 
 belong distinctly to neither. Classes and orders shade into one another 
 all along their common boundarv. Specific differences turn out to bo the 
 work of time. The Unc dividing organic matter from inorganic, if drawn 
 to-day, must be moved to-morrow to another place ; and the chemist wiU 
 tell you that the distinction has now no place in his science, except in a 
 technical sense, for the convenience of studj-ing carbon compounds by 
 themselves." (Prof. Clifford, p. 505, col. 2.) . w ^ • 
 
 t The geologic induction that there will bo a seam of coal tound ma 
 certain locuhty is of course verifiable by digging down. But the geologic 
 theory which accounts for the formation of that seam of coal is not veri- 
 fiable, as the formation of coal toik j^kv in jct-hi^tfinc «««•. 17,* i^t"-*! 
 and dynamical theories of geolog>- *n UArfly UitftttfU HI Um vv*knco «a 
 which they rest. The text especially ftppUoi to Um 1«tl«r. 
 
 ARTS. 75, 76.] LOGIC FOR CHILDREN. 
 
 6? 
 
 tion of the ground, and how, by commercial intercourse, involving 
 predictions innumerable, we may even supply the dearth of one 
 country's harvest by the plenty of another. The teacher must 
 always be prepared for the old old question. What's the use ? But he 
 should resolutely refuse giving a detailed answer. It is enough 
 to know generallv that, without a means of predicting, however 
 roughly, life would be an abject, objectless chaos, and that what- 
 ever contributes to more accurate prediction, no matter how slight 
 the apparent connection between the discovery and any individual's 
 special needs, is advancing so far the whole human race. We gain 
 knowledge in order to predict, and we predict in order to pr<vid(\* 
 not by or for ourselves individually, but by and for the whole 
 human race at large. And in thus acting for a remote or uncer- 
 tain future, we must remember that an incalculably greater part 
 of all the knowledge and all the comforts therefrom arising whirh 
 the present generation enjoys, is due to a remote and uncertain 
 past. It is a text on which the teacher may often be able to 
 preach with advantage. 
 
 76. The inductive faculty shows itself in earliest youth, and is 
 certainly developed in the lower animals. Hence, when a child 
 comes to school, he is fully competent to feel the processes through 
 which he may be put, though it would be absurd to lay down the 
 laws in a dull, dry, abstract manner. We must get him to perforin 
 from mere habit certain processes with which he has become more 
 or less familiar, but of the meaning of which he is far from being 
 conscious ; and then make him feel what he has done, so that he 
 may aftenvards apply those processes intelligently. The ability 
 to educe tliese powei-s marks the educator. One of the most fre- 
 quent processes of tiiought the child goes through, when he sees 
 or hears any tiling, is to make a guess (English), a supposition 
 <Latin), or an hypothesis (Greek). They are all the same thing. 
 He sees one tiling, and iramodiately looks forward to another, or 
 assumes that some other has gone before. You may call this in- 
 stinctive, if you will. It is certainly part of our constitution, 
 and it is the foundation of all inductive reasoning, which is 
 ffierely regulated gues sing.f The child guesses from certainly a 
 
 ♦ According to Comte's maxim : " Savoir pour prevoir, afin de poiuvoir." 
 t This is, in fact, the "common sense" spoken of in Dr. Carpenter's 
 address ; " the ralue of its results dependhig in each case upon the quali- 
 fications of the individual for arriviug at correct results the trust- 
 worthiness of their common sense decision arises from its dependence, not 
 on any one set of experiences, but upoa our unconscious co-ordination oftlie 
 whole aggregate of our <?jrjw<;nV«<r<"«— not on the conclusiveness of any one 
 train of i-easoning, but on the convergence of all our lines of thought towards 
 this one centre.'' These words must not bo pressed home. True, our judg- 
 ment at any time is the habitual outcome of all our exi>erience and pre- 
 vious judgments, but that experience and those judgments are not gene- 
 rally present to our mind when making the new judgment. So frequently 
 are important links forgotten, and judgments first foi-med have to be sub- 
 sequently modified, that "second tlioughts are best," has past into a pro- 
 verb. It is only so iw as this " uncas^sniouA co-ordination" becooMM 
 **ccaMsMu** tkst tniftw«cthiiati»« rwulU,*T/i thcnodl>* (OMugh nut nl- 
 ^y*!' whca -the «1u>lo ^fgitgBt.^ ofoir «xp«ri«i»(«*' kw bcca Uigo, 
 •s ut thiioM* of "oDciU.'^ Tfho r«»l wbols Mgronto to ht conaulM 
 is tkf rcdot^cd oMpakntt of koMniiy. 
 

 s 
 
 
 I 
 
 68 
 
 LOGIC FOR CHILDREN. [aRTS. 76, 77. 
 
 Tery narrow field of observation and experiment, with, most 
 likely, a very imperfect appreciation of the circnmstances pre- 
 sented to his mind, and also generally in a very awkward and 
 round-about manner.* But the fundamental principle of all rea- 
 soning is merely to form the simplest supposition which is con- 
 sistent with the WHOLE of the circumstances to he represented ; and 
 what is this but the child's process clarified ? All the so-called 
 canons of induction are but evolutions of this one principle 
 (art. 82). The supposition must be the simplest possible, and it 
 must be consistent with every one of the circumstances involved, 
 considered not merely isolatedly, but more especially as a whole, 
 as parts which mutually affect each other. The intentional sup- 
 pression of some circumstances is scientifically unthinkable, ex- 
 cept they be recognized as immaterial, but the accidental suppres- 
 sion of some material circumstances through ignorance is frequent, 
 and hence the necessity for means of detecting them, by observa- 
 tion and, when possible, experiment. Occasions will very fre- 
 quently arise in the course of instruction or in school life in which 
 the teacher will have to suggest material circumstances which 
 have escaped the observation of the pupil. This is constantly the 
 case where the young reasoner sees no difficulty at all. Without 
 a knowledge of any circumstances, no supposition at all can be 
 made. Hence, so-called a priori assertions are misnamed; they 
 are not assertions made without any knowledge, but assertions 
 made with extremely imperfect knowledge, occasioning the em- 
 ployment of inappropriate analogies, and, as we now know, leading 
 generally to worthless results. The light of nature^ which is a 
 very favourite source of inspiration for quick children, is also power- 
 less in presence of ignorance of circumstances, though it will often 
 help towards the solution of a troublesome problem in a more or 
 less awkward manner, for the light of nature is a lamp which re- 
 • quires trimming with the scissors of experience and plenishing 
 with the oil of observation. It burns however with a very different 
 flame in different minds, and the danger is that children will come 
 to rely upon it without troubling themselves with learning, igno- 
 rant that, as has been well said, genius is above all things a capa- 
 city for hard ivorJc. 
 
 77. Now the basis for making any supposition concerning cir- 
 cumstances which are presented to the mind, is that the supposer 
 has seen similar circumstances concur, and observed results 
 similar to those which he consequently anticipates. There is no 
 doubt of this fact as regards the supposer, but the question is, 
 what justifies him in this conclusion ? And the only answer as 
 as yet given is, the event. A feeUng has grown up among men, 
 which is stronger the more opportunity there has been for accu- 
 rate examination, that there exist invariable and unconditional re* 
 
 ♦ This last circumstance does not mark the child only. Every inves- 
 tigfator is aware that the way in which he is first led to some new concep- 
 tion is generally very obscure, and that the processes he consequently 
 adopts to work it out are subsequently discovered to be most unnecessarily 
 circuitous. Simphcity results from much knowledge and many trials. 
 Success is the outcome of numerous failures. 
 
 ART. 77.] 
 
 LOGIC FOR CHILDREN. 
 
 69 
 
 lations respecting the succession and co-existence of all circum' 
 stances* It is felt that it is only our ignorance which stands in 
 the way of our predicting with certainty what events will follow 
 after or concur with any given event. But the mode of express- 
 ing this feeling is yery diverse, and is often accompanied with 
 strange limitations, due to previous ignorance and present pre- 
 judice. On these I refrain from entering. If any child start 
 some question of necessity and freewill,— and he may do so, for he 
 may bring his school knowledge home, and his crude statements 
 of the teacher's doctrines may alarm some sensitive parent into a 
 dread lest the child may be taught some '* unsound" notions of 
 divine government, — the teacher has only to say : "As you learn 
 more, you will understand these things better. You must learn 
 very much to understand them well, but you know now that if 
 you strike a ball, the ball will move from you, and that you need 
 not strike the ball unless you like. The motion of the ball de- 
 pends upon the kind of blow you give it. Your liking to strike it 
 
 * " The step from past experience to new circimistances must be mad© 
 in accordance with an observed uniformity in the order of events. This 
 uniformity has held good in the past in certain places ; if it should also 
 hold good in the future and in other places, then, being combined with our 
 experience of the past, it enables us to predict the future, and to know 
 what is going on elsewhere ; so that we are able to regulate our conduct in 
 accoixlance with this knowledge. The aim of scientific thought, then, is to 
 apply past experience to new circumstances ; the instrument is an observed 
 uniformity in the course o'' events. By the use of this instrument it gives 
 us information transcending our experience ; it enables us to infer things 
 which we have not seen from things that we have seen ; and the evidence 
 for the truth of that information depends on our supposing that the imi- 
 formity holds beyond our experience." (Prof. Clifford, p. 502.) " We say 
 that the uniformity which we observe in the course of events is exact and 
 universal ; we mean no more than this, that we are able to state general 
 rules which are far more exact than direct oxperiment, and which apply to 
 all cases that we are at present hkely to come across." (Id., p. 506, col. 1.) 
 ** It is possible that by-and-by, when psychology has made enormous ad- 
 vances and become an exact science, we may be able to give to testimony 
 the sort of weight which we give to the inferences of physical science. It 
 will then be possible to conceive a case which will show how completely 
 the whole process of inference depends on our assumption of imiformity. 
 Suppose that testimony, having reached the ideal force I have imagined, 
 were to assert that a certain river runs up hill. You could infer nothing 
 at all. The arm of inference would be paralyzed, and the sword of truth 
 broken in its grasp, and reason cou'd only sit down and wait until recovery 
 restored her limbs, and further experience gave ler new weapons." (76., 
 p. 506, col. 2.) The principle of the uniformity of nature is the postulate 
 of mduction, an assumption without which any inductive reasoning would 
 be iinpossible. We have not yet seen our way to any means of demon- 
 strating its truth a priori. Every coincidence of a result with a pre- 
 diction founded upon it, helps to verify it. But any absence of such 
 coincidence that could not be traced to mere errors of deductive reasoning 
 of observation or of record, would be destructive of its value, would, as 
 Prof. CHfford remarks in the passage just quoted, " paralyze the arm of 
 inference," and leave us ignorant of everything which we are now sup- 
 posed to know. 
 
70 
 
 LOGIC FOR CHILDREN. 
 
 [arts. 77, 78. 
 
 ARTS. 78, 79, i.] LOGIC FOR CHILDREN. 
 
 71 
 
 or not to strike it, depends upon something else. Thus you may 
 not like to strike it, because you are told to do so, or because you 
 are afraid you cannot make it go far enough, or strike it in the 
 best possible way, or feel ill, or lazy, or * don't see the use,' and so 
 on. The diflerence between the ball's motion and your liking to 
 strike it, is so far a difference of simplicity. Lot us try to under- 
 stand the motion of the ball, which you will find useful at fives 
 and at cricket, and which is quite difficult enough for neither you 
 nor me to understand thoroughly, and leave the liking for another 
 time, knowing that at best we shall always understand it very 
 imperfectly." 
 
 78. It is necessary that occasion should be taken very early in- 
 deed to show that this absolute invariability and unconditionality 
 can never be thoroughly appreciated in the things we see or feel, or 
 deal with, but only in certain imaginary things which we come to 
 conceive by leaving out of consideration all those little peculiari- 
 ties that are too complicated to allow for.* Thus, in determining 
 the inotion of the ball, we should first suppose it quite smooth, and 
 this is not the case in the best cricket ball, much less in a common 
 fives ball ; and also that it is perfectly spherical, which we know 
 is never the case ; and also that it is equally elastic in all places, 
 which those who have tried to make a fives ball for themselves 
 will know they could never manage. All these things are better 
 obtained in a billiard ball or a glass globe, but these would pro- 
 bably shiver to pieces with a blow of the bat. We must however 
 consider that the ball will never break, nor the bat either, hit we 
 ever so hard. We hit an abstract ball with an abstract bat, and 
 then we can tell tolerably well how the ball would fly, provided 
 there were no air ! But we have a real ball, a real bat, and plenty 
 of air, already in motion generally, in the shape of wind, hence 
 the ball never goes exactly as we foretold, but always nearly so, 
 and the more nearly the nearer the realities approach to the ab- 
 strations. That is, there are invariable unconditional relations, 
 but these hold in all their perfection, solely for ahsir actions, ab- 
 stract things and abstract events. For realities we have always 
 to make certain allowances.f Why not go to the realities at once ? 
 
 • Science, therefore, deals with an abstract, ideal world. It is never 
 more than an approximation to reality. This must be carefully home in 
 mind in weighing? the api^irently unqualified assertions of men of science, 
 who are so familiar with the fact that they forbear to mention it, as even 
 the ecclesiastical wTiter neglects putting " D. V." alter every future tense, 
 notwithstandiDg Jas. iv. 13 — 15. 
 
 t The tlu-ee angles of a triangle, says EucHd, together make up two right 
 angles. Probably no one doubts the fact, although all geometricians know 
 the extreme difficulty occasioned by one of the assumptions (in this case a 
 pure assumption) on which it is based. But try to put it to the test. What 
 man ever drew on paper a triangle which strictly satisfied the condition ? 
 Nay more, what surveyor, in making his triangulations, either with his 
 measuring chain, or with all appliances of scientific apparatus, as in na- 
 tional trigonometrical operations, ever found such a triangle? In the 
 latter case, indeed, where the sides of the triangle are many miles in length, 
 the observer always finds the sum of his three angles greater than two 
 
 %^i 
 
 Simply because man*s mind is unequal to the task, the abstrac- 
 tions giving him already more than he is able to accomplish. All 
 we can do is to make the realities as near the abstractions as pos" 
 sible, and alluwfor the differences. And in this precept is involved 
 another extremely important principle, setting forth the bounds 
 of human power. We cannot alter the invariable unconditional 
 relations which determine the order of events , but w& can alter the 
 intensity with which the circumstances enter into the events whose 
 order is thus determined. We cannot prevent the ball starting 
 from the bat when struck, but we can regulate the strength and 
 direction of the blow, and we can abstain from giving the blow 
 altogether. We cannot prevent ignited gunpowder from explod- 
 ing,°but we can keep the heat to so low a degree that ignition will 
 never take place. It really does not require a very great step 
 further to reconcile necessity and free-will, and the teacher will 
 do well to show over and over again, by simple instances, when- 
 ever they occur, that re Zah'otis are fixed, conditions variable. The 
 knowledge of these fixed relations, and of the effect of that vari- 
 ability of condition, is, as Bacon long ago expressed it, coin- 
 cident with human power.* 
 
 79. There are a few words which arise out of what has been 
 said, and from being in constant use, and therefore having vague 
 popular senses attached to them, require careful consideration. 
 
 i. The first of these is law. A sciejitifie, or, as it is often called, a 
 natural law, has only the one determinate meaning, of an invari- 
 able unconditional relation of succession or coexistence. Any other 
 meaning is unscientific. The clear and definite statement of such 
 a relation bears a vague resemblance to the written law of the 
 realm, which however can be altered, and can be evaded, and 
 which has been settled by the will of certain legislators. This law 
 of the realm lays down penalties in case it should be disobeyed, 
 but the appointed executive is often powerless to enforce them, 
 and even to detect the disobedience. These laws are therefore in 
 no respect invariable or unconditional. Owing to our power of 
 modifying the intensity of circumstances, we are able to bring 
 things or events within the conditions which render the results 
 of scientific laws appreciable or otherwise ; but having brought 
 them within this influence, there is no choice of obedience or dis- 
 obedience, the invariable relation invariably asserts itself. What 
 then can be meant by saying that a person by over-eating " broke" 
 the natural laws of health, and suffered the " penalty" of disease ? 
 The whole sentence is one of confused analogy, which is however 
 very widely employed. The real meaning is", that the person has 
 
 right angles, the difference (known as " the spherical excess," because due 
 to the nearly spherical form of the globe) havmg to be carefully allowed 
 for. If, then, we only knew our geometry by concrete instances, we should 
 be in a state of muddle. It was the happy exercise of man's power of ab- 
 straction by the ancient geometers which allowed us to see the invariable 
 and unconditional abstract law through the tangled net of concrete ckcum- 
 
 stances. . . 
 
 • "Scientia et potentia humana in idem coincidunt, qma ignoratio 
 causae destituit eflfectum." — Novum Orffanim, Aphorism 3. ^ • 
 
72 
 
 LOGIC FOR CHILDREN. 
 
 [art. 79, 
 
 ART. 79, i. li.] LOGIC FOR CHILDREN. 
 
 73 
 
 brought himself under the action of that invariable relation be- 
 tween food and vital functions which results in disease ; whereas, 
 if he had not brought himself within that action, this disease 
 would not have occurred. It does not follow that some other 
 disease might not have occurred, for the person might have come 
 within the conditions for the occurrence of that second disease, 
 even if he had avoided the first. There is, however, a little more 
 meant, namely, that the over-eater has put himself beyond the re- 
 lations which result in health; and it is this voluntary part which 
 apparently gave rise to the whole conception, most widely dis- 
 seminated by George Combe's Constitution of Man, a work which 
 in its day produced a profound impression among a very numerous 
 class of readers * This should be illustrated by such a simple 
 case as a boy putting his finger too near to any flame. The dis- 
 tance at which he holds it is generally voluntary; but he may fall, 
 or be thrown against the source of heat. It could hardly be said 
 that a law had been enacted against his putting his finger 
 too close to the flame, and that the destruction of the skin, and 
 
 * In the introduction to his work on '* the Relation between Science 
 and Religion," of which the 4th enlarged edition (here cited) appeared in 
 1857, a year before his death, Mr. Combe says (p. xxx.) " To prevent mis- 
 understanding, I beg here to explain the meaning which is attached in the 
 following work to the expressions * Laws of Nature' and * Natural 
 Laws.' Every object and being in nature has received a definite consti- 
 tution, and also the power of acting on other objects and beings. The 
 action of the forces is so regular, that we describe them as operating under 
 laws imposed on them by God ; but these words indicate merely our per- 
 ception of the regularity of the action. It is impossible for man to alter 
 or break a natural law, when understood in this sense ; for the action of 
 forces, and the effects they produce, are placed beyond his control. But 
 the observation of the action of the forces leads man to draw rules from it 
 for the regulation of his own conduct, and these rules are called natrn^l 
 laws, because Natm-e dictates or prescribes them as guides to conduct. If 
 we fail to attend to the operations of the natural forces, we may unknow- 
 ingly act in opposition to them ; but as the action is inherent in the things, 
 and does not vary with our state of knowledge, we must suffer from our 
 ignorance and inattention. Or we may know the forces and the consequences 
 which their action inevitably produces ; but from ignorance, that through 
 them God is dictating to ua rules of conduct ; or from mistaken notions 
 of duty, from passion, self-conceit, or other causes, we may disregard 
 them, and act in opposition to them : but the consequences will not be 
 alteiei to suit ignorant en-ors or humours; we must obey or suffer." 
 There if, therefore, a constant confusion of the two meanings attributed to 
 natural laws, (1) invariable, unconditional relations, and (2) rules for apply- 
 ing conditions so as to effect desired ends in accordance with those rela- 
 tions. These latter state, ** If you do A, you will obtain the desired B, if 
 you do not, you will not obtain B, and if you do X, or Y, or Z, you will 
 obtain the undesired C." And it is very j^robable that there will be moro 
 or less error in the statement of condit ons and result, because the gene- 
 rality of the terms in which the conditions are stattd leaves out of con- 
 sideration a vast number of peculiar circumstances which seriously affect 
 the result in particular cases. " To break the law," means '* not to use the 
 condition A," or more frequently, " to use one of the conditions X, Y, Z." 
 Upon the " coercive" character of law, see the last foot note to No. iv. 
 
 consequent pain arising from too near an approach, were the 
 penalty attached to breaking this law. It becomes evident that 
 there was only a certain definite relation between the amount of 
 heat and its effect on the tissue, and something of the same kind 
 was the case in over-feeding. 
 
 ii. This leads to another great difference between a country's 
 laws and scientific laws. The former are carefully promulgated, 
 so that the judge is always entitled to disregard the plea of ignor- 
 ance of the law. The latter have to be discovered. They are not 
 only in great part unknown still, but are, more correctly speak- 
 ing, only known to a very small extent indeed, and it is the busi- 
 ness of men of science to discover them. What becomes of the 
 notion of penalty attached to breaking an unknown law ? But 
 here steps in a very important word, which will have to be well 
 understood. Chance applies to all those events whose Law is un- 
 known. The feeling that law is universal is now so generally en- 
 tertained, that men of science refuse to believe in the absence of 
 law, much less to erect the absence of law into a disposing goddess 
 co-equal with fate. Hence, when they say that a thuig happens 
 accident' illy, or hy chance, they simply mean that its laws are un- 
 known.* And they point to a remarkable confirmation of this 
 view in the so-called law of avn-ages, which shows that when a 
 large number of events of the same kind is taken into account, 
 although we are unable to predict what will be the result for any 
 one in particular, they will all be grouped round one central 
 phenomenon. This should be illustrated. A boy strikes a ball, 
 say in the game of 'trap, bat, and ball.' The number of times which 
 he strikes it, and the distances at which the ball comes to rest, 
 should be noted. (The problem would be a little too complicated 
 if the exact places of rest, instead of merely the distances, were ob- 
 served.) A distance found by adding all the observed and mea- 
 sured distances together, and dividing by the number of blows 
 
 * Dr. Carpenter {if/id.), using the flint implements found at Abbeville 
 and Amiens as an illustration, says : " The evidence of design to which, 
 after an examination of one or two such specimens, we should only be 
 justified in attaching a probable value, derives an irresistible cogency 
 from accumulation. On the other hand, the /«/probabihty that these 
 flints acquired their peculiar shape by accident'" — where the phrase only 
 means by some unknown, and even nnguessed relation, different from that 
 previously assigned by ourselves, of having been formed by man as instru- 
 ments — "becomes, to'our minds, greater and greater as more and more 
 such specimens are found ; until at last this hypothesis, although it can- 
 not be directly disi)roved"— that is, although we cannot show directly that 
 our own hypothesis is correct, or cannot verify it — "is felt to l>e almost 
 inconceivable "—that is, incapable of representing to our own minds the 
 whole of the circtunstances knowTi to ?/«— " except by minds previously 
 *l>08sc8sed' by the 'dominant' idea of the modern origin of man." This 
 " dominant idea " means a conception formed from what Dr. Carpenter 
 considers insufficient consideration, or on grounds which he rejects, but 
 which tlie holder maintains to be one of the conditions which he cannot 
 ignore, so that to concludo that the flints had been humanly formed would 
 not represent the whole of the circumstances which he regards as known to 
 him. 
 
74 
 
 LOGIC FOR CHILDREN. 
 
 [art. 79, ii. 
 
 ART. 79, ii.iii.] logic for chh^dren. 
 
 75 
 
 struck, would give the "average" distance. It would be found 
 that if this average were determined from a large number of trials, 
 say 100, it would be almost precisely the same for another hun- 
 dred, provided the same boy made the trial. Also^ if 100 boys 
 were to try each three times, the average distance for each set of 
 100 blows would be found nearly the same. Make trials with toss- 
 ing a penny;* with picking balls from bags of white and black 
 balls; with walking to a place or touching squares on a chessboard 
 blindfold : with the number of words a boy can read intelligibly, 
 or write legibly in a minute, the number of words in a line of 
 writing or print, and so on. Show the application of this in the 
 average number of persons who travel by different classes in rail- 
 ways or over different lines, by which the officials have to regulate 
 their accommodation ; by the average number of things sold in 
 the market, and the average supply brought in, and the average 
 prices at which they are sold. In all examples, point out the 
 large number of influences that there must be at work, but which 
 we are unable to evaluate, and hence the great importance of the 
 
 * Tossinj^ a penny is an admirable illustration, and it is worth while 
 spending a quarter of an hour over it, the whole class experimenting and 
 registering the results as follows. Let each place the penny head upper- 
 most each time, toss it a little height only, and receive it on a crumpled 
 handkerchief, to prevent rebounding. Number 32 lines on a slate, write 
 h for head and t for tail, and write in each line the result, till you get to a 
 h. There will then be 32 trials. The following is the result of 32 actual 
 trials, the commas dividing the lines : k. A, M, //, tth, A, A, th,, A, thy /i, th, 
 /<, /<, A, h, A, A, tith, th, h, h, h, ttttth, tttth, th, th, tttth, h, tth, h, h. Now 
 as there is an " even chance" of throwing h each time, we should expect h 
 in IG of the 32 trials ; there are actually 19. As in two throws we might 
 have hh, ht, th, or tt, it is 3 to 1 against having th, and hence we should ex- 
 pect th in 1 case out of 4, that is in 8 out of 32, we really find it 7 times 
 in the 32 trials. Again, in three throws we might have hhh, hht, hth, hit, 
 thh, tht, tth, ttt, so that it is 7 to 1 against tth or '1th as we may write ; 
 we should therefore expect 2th once in 8 trials, or 4 times in 32, we really 
 find it twice. Similarly for Zth, we should expect it once in 16 times, or 
 twice in 32 times, we really find it once. Again, Ath we should only ex- 
 pect once in 32, we find it twice. But bth should be only half a time in 
 32, that is once in 64 trials, we really find it once in the 32. Let the 
 pupils add the results, in a table thus : — 
 
 Now this is a very small experiment, 
 expected, found, and the results are really much closer 
 
 Trials 
 
 32; 
 
 th, 
 1th, 
 Zth, 
 4th, 
 6th, 
 
 Total trials 
 
 16, 
 8, 
 4, 
 2, 
 1, 
 1, 
 
 32 
 
 19 
 7 
 2 
 1 
 2 
 1 
 
 to the theoretical amounts than we 
 have any right to think would often 
 happen. But if all the results in the 
 whole class are added after registra- 
 tion, wo should probably come much 
 nearer. The following table repre- 
 sents the restUts for h and th actually 
 found for a very large number of trials, 
 the first 2,048 having been made by 
 the celebrated naturalist Buffon, and are calculated from a table given in 
 Prof. De Morgan's Budget of Paradoxes, p. 170 (London, 1872). The first 
 column givos the actual number of trials, the second the numbers of h or 
 th theoretically expected, the third the numbers of h or th actually found, 
 the fourth what this would give in 100,000 trials at the same rate, and the 
 
 32 
 
 !'l 
 
 1/ V A 
 
 I t< 
 
 i 
 
 ' 
 
 { 
 
 law of averages, and of the observations on which they are based, 
 such as the decennial census. 
 
 iii. Draw attention to the fact that this law of averages applies 
 to volnntarij actions (leading a little further to the solution of the 
 question of necessity and free-will); for notwithstanding the 
 general controlling average, there is no sense of restraint in any 
 actor. This leads to a more accurate notion of restraint, which, in 
 voluntary agents, amounts to the presentation of a choice, coupled 
 with the withholding of a choice usually open. We have no 
 
 fifth the theoretical amount in 100,000 trials, and the sixth the difference 
 between the two last amounts : — 
 
 trials 
 h 2048 
 4096 
 6144 
 8192 
 
 th 
 
 1th 
 3th 
 4th 
 bth 
 
 eth 
 
 7th 
 
 Sth 
 
 9th 
 
 loth 
 
 nth 
 
 12th 
 Idth 
 nth 
 loth 
 
 2048 
 4096 
 6144 
 8192 
 8192 
 
 It 
 » 
 
 >» 
 »» 
 
 » 
 ft 
 
 expected 
 1024 
 2048 
 3072 
 4096 
 
 512 
 
 1024 
 
 1536 
 
 2048 
 
 1024 
 
 512 
 
 256 
 
 128 
 
 64 
 
 32 
 
 16 
 
 8 
 
 4 
 
 2 
 
 1 
 
 i 
 
 i 
 
 4 
 
 found or in 100,000 instead of difference 
 
 1061 
 2191 
 3126 
 4165 
 
 494 
 
 1001 
 
 1548 
 
 2028 
 
 982 
 
 480 
 
 266 
 
 132 
 
 71 
 
 36 
 
 17 
 
 9 
 
 2 
 
 1 
 
 
 
 1 
 
 
 
 2 
 
 51806 
 51489 
 50879 
 50842 
 
 24121 
 
 24438 
 
 25212 
 
 24755 
 
 11987 
 
 5859 
 
 3248 
 
 1611 
 
 867 
 
 439 
 
 208 
 
 110 
 
 25 
 
 12 
 
 
 
 12 
 
 
 
 25 
 
 50000 
 
 »» 
 
 ti 
 »» 
 
 25000 
 
 99 
 
 12500 
 
 6250 
 
 3125 
 
 1563 
 
 781 
 
 390 
 
 195 
 
 97 
 
 49 
 
 24 
 
 12 
 
 6 
 
 3 
 
 2 
 
 + 1806 
 
 + 1489 
 
 + 879 
 
 + 842 
 
 - 879 
 
 - 562 
 + 212 
 
 - 246 
 
 - 513 
 
 - 391 
 + 123 
 + 48 
 + 86 
 + 49 
 + 13 
 + 13 
 
 - 24 
 
 - 12 
 
 - 12 
 + 6 
 
 - 3 
 + 23 
 
 It is thus seen that the observed corresponds nearer and nearer with the 
 calculated average as the number of trials increases for h, and for the three 
 first for th, but for th it is evident that the numher of trials was not nearly 
 enough. After th the discrepancies become still greater. The table is 
 then reduced to the case of 8192 trials, which gives the sum of four different 
 sets of 2048 each. It is remarkable that in two of the sets 10th occurred 
 once, and in one of the sets llth, 12th, IZth each occuiTed once, while in 
 two of the sets 15^A occurred once. The table is arranged by heads, each 
 trial ending with an A. But the tosses observed might have been arranged 
 by /. Then the actual 32 trials at the beginning of this note would ap- 
 pear as hht, hht, t, hhht, hht, hht, hhhhhhht, t, t, ht, hhhht, t, t, t, t, ht, t, t, t, 
 ht, ht, ht, t, t, t, hht, t, hhh ... giving only 27 trials ending in t and part of 
 another trial, and the trials would have had to be continued. It is best 
 to use such a number of trials as 32, 64, 128, &c., because these numbers 
 divide so often by 2 exactly. Hence 2048 = 2", 8192 = 2'3, were selected 
 for the trials in the table. A few examples of this kind actually worked out 
 from a child's own experience will do more to convince him of the reahty 
 of a law of averages than a month's talk or a year's reading. Thejluctim- 
 tims '* found" above and below the " expected" average will also show him 
 the meaning of " leaving a margin." 
 
76 
 
 LOGIC FOR CHILDREN. [aRT. 79, iii. i\'. 
 
 ART. 79, iv. v.] 
 
 LOGIC FOR CHILDREN. 
 
 77 
 
 choice with regard to coming into existence or ceasinfr to exist- 
 we have no choice as to living by airandfood; we have no choice as 
 to breathing air or water, flying or walking, having two legs or 
 tour, and so on ; but we never feel restraint respecting these mat- 
 ters. A prisoner who submits to be taken to prison feels restraint 
 because he has the choice of going or of being afflicted for not 
 going, and he has all further choice withheld, and so on. In in- 
 voluntary agents, restraint consists in the introduction of unusual 
 conditions, which bring the event under the action of additional 
 laws. Thus a ball, when thrown, describes a certain path under 
 the action of gravitation. But if a string had been fastened to 
 It, the motion would have been greatly changed. The strin'v in- 
 troduces a new set of laws. Properiy speaking, restraint always 
 consists in coming within the influence of unusual laws. 
 
 IV. Cause and effect are two extremely common and very loosely 
 employed terms. Scientifically they are used for two sets of events 
 which occur in the order fixed by scientific law. But even scienti- 
 hcally the whole of each set of events is rarely thought of- many 
 being so much a matter of course, that their absence rather than 
 presence would have to be noted. But popularly the one seen and 
 observed event which, like the spark applied to a train of gunpow- 
 der, IS immediately followed by the change, is called the cause, and 
 the change itself the effect.* But, in this very instance, it would 
 be absurd to say that a spark was the cause of the ruin of a fort 
 ^ot only the gunpowder, but a certain collocation of the gimpow^ 
 der with respect to the fort, and a certain constitution and dryness 
 ot the powder, were also required, and formed part of the scientific 
 cause Hence cause and effect are words rather to be avoided f 
 and the teacher must be ca reful that they are not used vaguely. 
 
 ♦ See Method of Differences, Art. 82. 
 
 t Prof. Clifford (p. 509), referring to the common phrase, "every effect 
 has a cause asks, "What do we mean by this ?" and proceeds to say. 
 In askmg this question we have entered upon an appalling task. The word 
 represented by cause has sixty-four moanin-s in Plato, and forty-ei-ht in 
 Anstotle. These were men who liked to know as near as might be what 
 they meant; but how many meanin-s it has had in the writings of the 
 myriads of people who have not tried to know what they meant by it will 
 1 hope, never be counted. ... I shall evade the difficultv by tolling you Mr' 
 Urote s opinion You come to a scarecrow, and ask;\\1iat is the cause 
 ot this .-* lou find that a man had made it to fric^hten the birds. You so 
 away, and say to yourself, * Everj'thing resembles this scarecrow : eveiT- 
 thing has a piin)Ose.' And from that day the word canse for vou means 
 what Anstotle meant hy^fij^al ca,w. Or you go into a hairdresser's shoi), 
 and wonder what turns tlu- wheel to which the rotarv brush is attached. 
 On investigating other i)aits of the premises, vou find a man working away 
 at a handle. Then you go away and say, * Everything is like that wheel. 
 It 1 investigated enough, I should alwavs find a man at a handle ' And 
 the man at the handle, or whatever corresponds to him, is from henceforth 
 known to you as ' cause,' and so generally. When you have made out 
 any seipience of events to your enth-e satisfaction, so that you know all 
 about it, the laws involved being so familiar that you seem to see how the 
 beginning must have been followed by the end ; then you applv that as a 
 simile to any other events what<3ver, and your idea of cause is determined 
 Dy it. Only when a case arises, as it alwavs must, to which the simile 
 
 The invariable unconditional order of events is a much more im- 
 portant conception, and one to which the former should be always 
 reduced. The invariability and the unconditionality are, however, 
 often difficult to establish, and we are not entitled, when we see a 
 certain order always existing in all the observations we have made 
 on two events, to suppose that that order is invariable and uncon- 
 ditional. The order of day and night is one of the most invariable 
 observed ; we know, however, that it is conditional on the posi- 
 tions of the sun and earth, and the rotation of the earth. This is 
 the distinction of post hoc (after this) and propter hoc (because of 
 this).* We cannot declare the latter unless we feel that there is 
 sufficient evidence of invariability and unconditionality ; this it is 
 the province of scientific induction to investigate ; the mere suc- 
 cession is patent from a single observation. 
 
 V. When the invariability and unconditionality cannot be tho- 
 roughly established, they may be found to hold within certain 
 Innits, so that, the limiting condition being observed, others may 
 be left out of consideration. This gives rise to experimental or 
 empirical laws, which are of the utmost value as provisional rules. 
 
 will not apply, you do not confess to yourself that it was only a simile, 
 and need not apply to everything, but you say, * The cause of that event 
 is a mystery which must remain for ever unknown to me.' On equally 
 just grounds the nervous system of my umbrella is a mystery which must 
 for ever remain unknown to me. INIy umbrella has no nervous system ; 
 and the event to which your simile did not apply, has no cause, in your 
 sense of the word. When we say that every effect has a cause, we mean 
 that every event is connected with something in a way that might make 
 somebody call that the cause of it. But I, at least, have never yet seen 
 any single meaning of the word that could be fairly applied to the whole 
 order of nature." 
 
 * INIany writers are of a different opinion ; but so far as science is con- 
 cerned this difference is inoperative. Dr. Carpenter {ibid.) says :— ** As Sir 
 John Hcrschel most truly remarked, the universal consciousness of man- 
 kind is as much in accord in regard to the existence of a real and intimato 
 connection between cause and effect, as it is to the existence of an external 
 world ; and that consciousness arises to every one out of his own sense of 
 personal exertion in the origination of changes 'by his own individual 
 agency." That sense of personal exertion arises, we maybe said to know, 
 from the corresponding i)hysioiogical changes within him. It therefore 
 only removes the invariable unconditional relations one step further off. 
 But let us go back to the antecedent of these changes, to the causa causa- 
 rum, and we at once transgress the hmits of verifiable science, and hence 
 sail beyond our present purpose. It is widely different to say that the 
 invariable unconditional verifiable relation is the only one to be considered, 
 and to say that that relation is not itself a consequent of an unverifiable (or 
 even possibly at some future period verifiable) antecedent. The latter 
 statement is thoroughly unscientific. Dr. Carpenter considers force, as 
 deduced from considerations of personal exertion, to be the invariable an- 
 tecedent of causation, and says, " whilst no * law' which is simply a gene- 
 ralization of phenomena can be considered as having any coercive action'' — 
 ' law ' is merely a statement of invariability and unconditionality with- 
 out reference to coercion, — " we may assign that value," viz., the posses- 
 sion of coercive action, " to laws which express the universal conditions of 
 the action of a force whose existence we learn from the testimony of our 
 
 Q 
 
78 
 
 LOGIC FOR CHILDKEN. 
 
 [art. 79, V. vi. 
 
 ART. 79, vi. vii.] LOGIC FOR CHILDREN. 
 
 79 
 
 and as grouping phenomena for subsequent inductions.* Thus 
 the etlect of friction is taken to be exactly proportional to the pres- 
 sure, but this strict proportion ceases when the weight is large. 
 Part of a beam of sunlight will be reflected from an unsilvered 
 glass surface, according to the usual law of reflection, and part 
 will pass through. Attempt to reflect both portions from another 
 piece of unsilvered glass. The amount of the second reflections 
 will differ materially according to the angles and planes of both 
 reflections, and it is possible so to arrange the positions of the 
 glasses that either the light reflected or transmitted by the first glass 
 will either not be transmitted or else not reflected respectively by 
 the second. The law of reflection and transmission therefore be- 
 comes conditional. The investigation of this condition produced 
 within the present century the w hole theory of polarized light. 
 Such an experiment requires next to no apparatus, and will amuse 
 as well as instruct the children. But the teacher must have care- 
 fully tried the experiment himself before attempting to show it or 
 to lead a child to make it. Those teachers who think they can 
 perform an experiment in public, which they have not easily suc- 
 ceeded in performing in private, have seriously misconceived 
 their vocation. 
 
 vi. The reason icliy is constantly asked, and has two meanings 
 requiring discrimination. As respects the acts of a known volun- 
 tary agent, we suppose an acquaintance with laws of nature, and 
 with the method of putting himself or his actions within the influ- 
 ence of certain laws; when therefore he does so, we presume that 
 he desired to bring about the consequent effect. The answer to. 
 Why did you do so ? is consequently expected to be. Because I 
 wished to get so-and-so. That is, the reason ivhy expresses a mo- 
 tive, a desire, an intention, a purpose, an object. A large induc- 
 tion shows that human actions are regulated by such motives, and 
 
 consciousness" — that is, from ourselves coming consciously under the rela- 
 tion it expresses. " The assurance we feel that tho attraction of gravita- 
 tion niffnt act imder all circumstances according to those simple laws which 
 aiise immediately out of our dynamical conception of it, is of a very dif- 
 ferent order from that wtich we have in regard (for example) to the laws 
 of chemical attraction, which are as yet only generalizations of pheno- 
 mena." The distinction is chiefly one of degree. It is the infinite num- 
 ber of verifications to which science has accurately, and common experience 
 roughly, subjected gravitation, as compared with those to which chemical 
 theories (const.intly changing since Triestley's discovery of oxygen i can 
 be subjected, which gives the real balance in favour of the foi-mer. Force 
 is op/t/ measurable by motion : as Dr. Carpenter says {i>'jiJ.) ** the mecha- 
 nical philosophy of the present day tends more and more to express itself 
 in terms of motion rather than in terms of force." In English books on 
 mechanics tho " force of gravity" is always expressed by the number 32-2, 
 representing the number of feet which a body would describe horizontally 
 on a frictionless surface in vacuo during one mean second of time, if it were 
 to move uniformly with the velocity it had acquired after falling vertically 
 iM vacuo fur one mean second of time previously. Any other mode of mea- 
 siu"ing this force is almost " inconceivable " at the present day ; but how 
 could it have ever been discovered or used " from the testimony of our 
 conscioiLsnesnii " ? 
 
 [4 
 
 bonce we naturally inquire into the motives of any human action 
 Frequently, however, we find that the reason why a human being 
 produced a certain result was nob that he intended it, but that 
 intending something else, and ignorant of all the conditions necesl 
 sary for attaining it, he ended in doing what he did not in the 
 least mtend. When the conditions were such as he could not 
 anticipate, their occurrence is given as an explanation of his failure 
 Thus boys questioned on some school offence, allege such and 
 such unexpected conditions as an excuse. " Why were you late ?" 
 "I fell as I was running to school." •* Why did you not write 
 the exercise ? " "I hurt my hand." 
 
 vii. But when we get beyond human or living actions, it is not 
 correct to assume intentions. Such an assumption has however 
 often been made supernaturalistically, and leads to iinal causes, 
 the intention of a supernatural being, or of an abstract beino-,' 
 nature herself. But these are excluded from induction proper l>y 
 the total absence of verifiability (art. 74). Suppose the question 
 were, " Why did such a boy fall ill and die ?" No such answer as, 
 " Because it was the will of God," can be admitted in inductioni 
 for it is quite evident that it cannot be verified. The only an- 
 swers would be such as a physician would give, doscribincr t1 e 
 constitution of the boy, the conditions to which he was exposed, 
 the growth and treatment of the disease, and so on. Even these 
 answers are only partially verifiable by a registration of similar 
 cases, and the whole facts on which repose the science of life. 
 How can the reaso7i why be given scientifically? Bi/ showing the 
 known laws wJiich iticlude the jiarticular case. Nothino- more 
 should be attempted, and very frequently even this cannot be 
 done.f It is also evident that we must soon come to the end of 
 
 * Properly speaking, all the laws that we discover are empirical, limited 
 by our powers of verification. But relatively to us they are perfect when 
 they give results which are as exact as, or are more exact than, our means 
 of verification. Prof. Clifford shows that such considerations apply even 
 to the laws of geometry, as, for example, to the conclusion that the three 
 angles of a plane triangle are exactly equal to two right au'jles. (Id , n. 
 604.) \ ^ i 
 
 t Prof. Cliflford, alluding to an illustration, says, " The explanation de- 
 scribes the unknown and unfamiliar as made up of tho lcnoN\Ti and familiar, 
 and this, it seems to me, is tho true meaning of explanation" (p. o08, col. 1 ). 
 "By known and familiar, I mean that which we know how to deal with, 
 either by action in the ordinary sense, or by active thought" (p. 508, col. 2). 
 "That a process may be capahle of explanation, it 'must break up into- 
 simpler constituents which are already familiar to us. Now, first, this 
 process may itself be simple, and not break up ; and secondly, it may 
 break up into elements which are as unfiimiliar and impracticable as the 
 original process It is no explanation to say that a body falls be- 
 cause of gi-avitation. This attraction of two particles must alwaj^s, I 
 
 think, be less familiar than the original falling body, however early the 
 children of the future begin to read their Newton. Can the attraction 
 itself be explained ? . . . . The attraction may be an ultimate simple fact ; 
 or it may be made up of simpler facts utterly unlike anything that we 
 know at present ; and in either of these cases there is no explanation. 
 We have no right to conclude, then, that the order of events is alwavs 
 
 q2 
 
80 
 
 LOGIC FOR CHILDREN. 
 
 [art. 79, vii. 
 
 iRT. 79, viii.] 
 
 LOGIC FOR CHILDREN. 
 
 81 
 
 onr chain, and get to laws which we are unable to include in any- 
 higher, and then all " reasons why'* are at an end." " Why was 
 this plate smashed?" "A stone fell upon it." "Why did the 
 stone fall ?" " The wind blew it from the top of the wall." " Why 
 did the wind blow it?" "There was a sudden powerful gust in 
 such a direction." " But why sudden, why sufficiently powerful, 
 why in this direction ?" " I don't know." — " Why was that stone 
 on the top of the wall ?" " The builders placed it there as a 
 coping." *' Why was it not tight ?" " The mortar had ceased to 
 hold it ?" " Why ?'* " 1 don't know."—" Why did the stone fall 
 on the plate, and not beside it ?'* " Because the plate was placed 
 in a certain position, and the wind blew with a ceilain force.'* 
 •' Why ?" " 1 don't know." — " Would a little stone have smashed 
 the plate?" " Probably not." "Why?" " Because it would not 
 have acquired sufficient force in falling." " Why ?" " This can 
 be calculated from its weight, the height from which it fell, the 
 law of gravitation, and the cohesion of the parts of the plate.'* 
 " Why was the weight or the height such ? Why does the force 
 of gravitation exist? Why do the parts of a plate so cohere?" 
 " I don't know." — And so on. It is often useful to follow up such 
 a train of questions (here purposely much curtailed) and then 
 finally to show that all the reason given was the inclusion of facts 
 within laws, and these within more general laws, so that the re- 
 sult was simply showing the manner, not the motive or purpose, 
 and that, using whij in the human sense, we may sum up the re- 
 sult by saying, inductive science seeks to know tht how, and not the 
 why.* 
 
 capable of being explained" (p. 509, c. 1). At the same time, we have no 
 right to conclude that any given order of events will remain for ever in- 
 explicable. As Prof. Chtford says in reference to another question, " It 
 seems to me that we do not know, and that the recognition of our ignor- 
 ance is the 8iu:est way to get rid of it" {ib. p. 505, c. 1). 
 
 * In the words of Comte, inductive science " ecarte comme radicalement 
 inaccessible et profondement oiseuse, toute recherche sur les causes propre- 
 ment dites, premieres ou finales, des evenements quelconques. Dans sea 
 conceptions theoriques, elle explique toujours comment et jamais pourquoi. 
 Mais, quandelle indique les moyensde dinger notre activite, elle fait, au con- 
 traire, prevaloir constamment la consideration du but." Cath. Fos. p. 13. 
 Dr. Carpenter concludes his presidential addiess thus : — " The science of 
 modem times .... fixing its attention exclusively on the order of 
 Nature," that is, of phenomena — "has separated itself wholly from theo- 
 logy, whose function it is to seek after its cause. In this, science is fully 
 justified ahke by the entire independence of its objects, and by the his- 
 torical fact that it has been continually hampered and impeded in its 
 search for the truth as it is in nature" — for the discovery of invariable im- 
 conditional relations, giving the power of prediction and consequent verifi- 
 cation — "by the restraints which theologians have attempted to impose 
 on its inquiries. But when science, passing beyond its own limits, pre- 
 sumes to take the place of theology, and sets up its own conception of the 
 order of nature as a sufficient account of its cause" — I do not know any 
 man of ' science' who has done so ; the words seem aimed at Comte, to 
 whom the preceding quotation shows that they are inapplicable ; the 
 whole stattmtut should have been worded hypothetically,—" it is invading 
 
 < 
 
 viii. One other word, nature, remains to be considered. It has 
 been greatly abused, and is often used euphemistically for a 
 supernatural power. At other times it means the universe itself, 
 or its constitution, or the laws which any object obeys; sometimes 
 it is even used for all the universe except man, or except animals. 
 It must consequently be employed with caution. If we talk of a 
 " natural" spontaneous action, we mean an action according to the 
 laws which we usually see aflfecting the object, so that there is no 
 restraint, in the sense already explained. When we " naturally'* 
 conclude that so and so is the case, we draw our conclusions from 
 such knowledge as we recollect at the moment, without investiga- 
 tion. Policemen see a man stagger from a public-house, and fall 
 down. They " naturally" conclude that he is drunk, and lock him 
 up. But in several instances of the kind, the man has been found 
 dead the next morning, having fallen in a fit. " Naturally" often 
 means " hastily, negligently, ignorantly.'* 
 
 a province of thought to which it has no claim, and not unreasonably pro- 
 vokes the hostility of those who ought to be its best friends;" and woidd 
 be so if they had had sufl&cient education in those merest elements of sci- 
 ence to understand what science does seek to attain, and not to attribute 
 to science aims utterly alien to any possible conception of science admis- 
 sible in these days. " For whilst the deep-seated instincts of humanity, 
 and the profoundest researches of philosophy, alike point to mind as the 
 one and only source of power" — in man, to assume the same;, or anything 
 approaching to the same superhumanly, is to emulate Phaethon — trans- 
 nature is degraded by reducing it to eis-nature — "it is the high prerogative 
 of science to demonstrate the uuitt/ of the power which is operating through 
 the Hmitless extent and variety of the universe," — ^how if the power itself 
 is beyond scientific ken ? really, science strives to demonstrate the harmo- 
 nious character of the invariable unconditional relations ; it can't do more, 
 and it can only at present forefeel, scarcely state, far from demonstrate, 
 even this — " and to trace its continuity" — continuance of the invariability 
 — "through the vast series of ages that have been occupied in its evolu- 
 tion." The last words appear to be a mistake, for " in the evolution of the 
 universe," see the conclusion of this note. It is curious that Dr. Carpenter 
 should have been betrayed into thus committing the very mistake that he 
 deprecates, — transplanting religion into science. That he has done so, 
 and that it was one of the very objects of his address to do so, Dr. Car- 
 penter admits in a letter subsequently published in the Echo (the date of 
 which I am unable to give), where he says : "I expressed the opinion that 
 science points to (though at present I should be far from saying that it 
 demonstrates) the origination of all power in mind ; and this is the only 
 point in my whole address which has any direct theological bearing. 
 When metaphysicians, shaking oflf the bugbear of materialism," — which 
 is generally the appUcation of lower laws to higher phenomena, and is fl^ 
 best a doubtful analogy, — " will honestly and courageously study the phe- 
 nomena of the mind of man in their relation to those of his body, I believe 
 that they will find in that relation their best arguments for the presence 
 of Infinite mind in universal Nature." This lies beyond the purpose of a 
 purely scientific investigation. It is important that intellectual verifiable 
 Bcience, and emotional unverifiable religion, eihouldbe sharply distinguished. 
 The following quotations from the same letter throw further light on Dr. 
 Carpenter's notion of * coercive' laws. " My object" — in the Brighton ad- 
 dress — "was to show — (1) That what we call *laws of nature' are simply 
 
82 
 
 LOGIC FOR CHILDREN. 
 
 [art. 80. 
 
 ART. 80.] 
 
 LOGIC FOR CHILDREN. 
 
 83 
 
 80. Tlio province of scientific induction then is to discover lairs, 
 which take tlie form of genenil assertions respecting such matters 
 as are appreciable by the senses, and are hence called appparancea 
 (Latin), or phenomena (Greek). These appearances are, however, 
 generally supposed to refer to some underlying he!n</s (English) 
 or substances (Latin) which merely exist in our thowjhf, and are 
 hence called noumena (Greek). This distinction is, of course, not 
 one which can be proposed to children in terms, but in the course 
 of object lessons it can be easily shown that all we know of thinqi 
 are their appearances to our senses; that in fact we apply the 
 names as merely a short way ot'ty!nrf together (English) or colUgat' 
 ing (Latin) these appearances, and that we know nothing about 
 the things themselves {die Binge an sich, as the German philo- 
 sophers word it). The process of such observation gives first an 
 impression on our senses, which is regarded as an appearance, and 
 then we have the recollection of this appearance, often called an 
 idea (Greek) or notion (Latin). It is with these ideas or notions 
 that we think, but these internal images are always le^s prt^ris". and 
 d'finite than' the appearances as derived from external impressions, 
 
 oui- own expressions of tho orderly se [ucnce which wo discern in the j.lio- 
 nomona of the uiil verse; and tliat us all the history of science shows how 
 erroneous these have been in the past, so we have no right to assume uur 
 present conceptions of that sequence to bo either universidly or necessarily 
 true. (2) That these so-called ' laws' are of two kinds— one set being 
 merely generalisiitions of phenomena, of which Kepler's * Laws of riane- 
 tary Motion,' or tho * Laws of Chemical Combination,' are examples ; 
 wlule another set express the conditions of the action of a force, of which tlie 
 existence is, or may be, made known to us by the direct and immediate evi- 
 dence of our own conscience— our cognition of matter being indirectly 
 iormed through the medium offeree. (3) That * laws' of tho first kind 
 (\N Inch we may for convenience term phenomenal) do not really explain or 
 account for anything whatever. Nothing is more common than to hear 
 8<-:entific men speaking of such laws as 'governing phenomena,' or of a 
 phenomenon being * explained' when it is found to be consistent with some 
 ont! of such laws; though the fact is, that the law is a law, merely be- 
 cause it IS a generahsed expression of a certain group of phenomena ; and 
 t«) say that any new phenomenon is ' explauied' by its being shown to be 
 in conformity to a 'law,' is merely to say (as Professor Clifford well i)ut in 
 h.s lecture) that a thing previously unknown is * explained' by showing it 
 to be like somethin*? previously known." This is hardly a proper inter- 
 prrtation of Prof. Cliflbid's words, givi-n in the preceding note. " (4) I'hat 
 <.n the other hand, every 'law' of the sec(md kind (which we may distin- 
 guish as dynamical) is based on the fundamental conception of a force or 
 power ; so that if tlie existence of such a force (as tliat of gravity or elec- 
 tricity) be admitted, and the conditions of its action can be accurately 
 stated, then the law which expresses them may be said to * govern' tho 
 phenomenon ; and any phenomenon which can be shown to be necessarily 
 tieiucible from it may be said to be 'explained' so far as science can explain 
 it. But the utmost that science can positively do, as I statt^d towarrls the 
 conclusion of my address, is to demonstrate the unity of the power of 
 which the phenomena of nature are the diversified manifestations, and to 
 trice tho continuity of its action through the vast series of ages that have 
 been occupied in the evolution of the universe." The text and preceding 
 remarks will enable the reader to appreciate these explanations. 
 
 k 
 
 and hence they require to be constantly refreshed by recurrence 
 to these actual appearances, that is, to ohservation, either on the 
 events which occur to us without any planning on our part (natural 
 ohservation), or on events artifically arranged (e^dperimental oh- 
 servation). Scientifically we must be considered to have no ideas 
 or notions which have not been thus derived. Conceptions are 
 properly internal constructions of ideas thus derived, and must 
 hence be always subordinated to external materials. When they 
 are not, we say, in very evident cases, that a thinker is mad, or 
 subject to illusions. In other cases, we soften the term, and talk 
 of prejudice, or eccentricity, or bias. In all cases we acknowledge 
 untrustworthiness.* Whom are we then to trust ? It is clear to 
 anyone Avho has attempted to make any inductions, that his own 
 individual range of thought is very limited. When he looks upon 
 the lives, the many generations of lives, exhausted over obtaining 
 a few results, he sees also the absolute necessity of trusting, and 
 the difficulty of trusting even himself. The test of any persofi's 
 authority is simply the degree of verijication which his results have 
 received. But even this amount of verification we have constantly 
 to take on evidence. Our trust, our absolute faith — for it comes 
 to that — ^an only be founded on our own personal knowledge of 
 the kind of processes by which results are obtained and verified. 
 Hence the importance of an early familiarisation of the minds of 
 children with the great processes of induction in matters which 
 they can appreciate. But we must recollect that to talk to them 
 of things they have not seen, or felt, or sensuously appreciated, is 
 only to confuse their minds and waste our own strength. All in- 
 duction is founded on facts derived from natui'al or experimental 
 observation, and one business of the teacher in all he teaches is to 
 make that observation as thorough and precise as possible. 
 
 * As Dr. Carpenter (?i«VA) says : "The scientific conception of nature . . . 
 is a representation framed by the mind itself ont of the materials supplied 
 by the impressions which external objects make upon the senses ; so that 
 to each man of science nature is what he individually believes her to it?." 
 Whence the importance of the verification of scientific theories by others 
 than their proposer. " It may be said," he adds, " is not this view of the 
 material universe open to the imputation that it is * evolved out of tho 
 depths of our own consciousness,' — a projection of our own intellect into what 
 surrounds us — an ideal rather than a real world ? If all we know of matter 
 bo an ' intellectual conception,' how are we to distinguish this from such as 
 we form in our dreams ? . . . Here our ' common sense' comes to the rescue. . . . 
 Every healthy mind is conscious of the difference between its waking and 
 its dreaming experiences .... and . . . finding its own experiences of its 
 waking state not only self-consistent, but consistent with the experiences 
 of others, accepts them as the basis of its beliefs, in preference even to the 
 most vivid recollection of its dreams. Tho lunatic .... is so ' possessed' 
 by the conception fi-amed by a disordered intellect, that he does project it 
 out of himself into his surroundings ; his refusal to admit the corrective 
 teaching of common sense, being the very essence of his malady. And 
 there are not a few persons abroad in the world, who equally resist the 
 teachings of common sense, whenever they run counter to their own pre- 
 conceptions ; and who may bo regarded as — in so far — iiffocted with what 
 I once heard Mr. Carlyle pithdy characterize as a 'diluted insanity.'" 
 
84 
 
 LOGIC FOR CHILDREN. 
 
 [art. 80. 
 
 Hence the importance of an early introduction of some science 
 of natural observation, as hotany, which, at least in country 
 places, and partly even in towns, can be made such a valuable and 
 interesting study, as has been shown in several recent works * 
 And this, or some similar science of observation, becomes the host 
 vehicle for the introduction of strict views respecting scientific 
 names, terminology, and classification, to which considerable ex- 
 tension can be given when pupils have gone through a course of 
 deductive logic, but which can be brought home to the feelings of 
 pupils who have never even heard of logic. The great principle 
 of natural class ificat ion f by the increasing or decreasing generality 
 of the appearances exhibited, or the laws obeyed, by the objects or 
 events classed, can readily be shown even without taking up a sys- 
 tematic study. Children are by natural observation sufficiently 
 
 ♦ La 3Iaout'8 Le(jons Elementaires de Botanique (16 francs) recom- 
 mended by Mr. J. M. Wilson of Kugby, obtainable at Haeh« tie's, King 
 William Street, Strand. Miss Youmans' First Book of Botany (Kini^, 65 
 Comhill). Miss Youmans' Culture of Observing Powers of Children, edited 
 by Mr. J. Payne (King, 2s. 6d.), is published separately. Dr. Master's 
 Botany for Beginners (3«. 6(1.), 100 iUustrations (Bradbury and Evans). 
 
 t The classification of deductive logic is artificial, proceeding by tlie pre- 
 sence and absence of immistakable marks, as the presence or absence of 
 certain letters in a word (arts. 7 to 11). But in natural classification this 
 sharp determination of marks no longer exists. Concretely the marks re- 
 lied upon shade by degrees into each other, till it becomes* difficult to de- 
 termine whether a given object can be rightly said to have or not to have 
 the assigned mark (quality or property), the Linnsean {artificial) and 
 the Jussieuan {natural) classifications of plants are very useful in familiar- 
 ising the mind with this difference. But even in broader distinctions, as 
 between plants and animals, it will be as well to take an opportunity of 
 showing how difficult it is to say whether a given organism (for cxami>le, 
 a Diatant) is a plant or an animal, and hence to lay down unmistakable 
 cliaracteinstics which should determine that a man and a zoophyte are ani- 
 mals, and a Sensitive Plant or a Veuus's Fly-trap {Diotiaea muscipnla) are 
 not animals. Much harm has been done by confusing artificial and natural 
 distinctions, and by forgetting that language is founded upon loose obser- 
 v;ition of natural differences which are treated as artificial and strict. One 
 ot the great sources of difficidty in judging of the permanence of species, 
 or natural character of genem, arises from this very looseness. Hence, 
 again, in all cases, the danger of pui-hing home mere verbal distinctions. 
 ( observe how loosely we use such words as hot, cold, black, white, full, 
 « mjtty. One would think these must be absolute terms. But we have 
 hotter and colder, blacker and whiter, fuller and emptier ! To dr?iw the 
 line between hot and cold, black and white, full and empty, becomes really 
 difficult. Don't blame a boy, as I have heard boys blamed, for saying that 
 one cup of tea is "fuller" than another, or that a third is "almost quite 
 empty." They are merely endeavouring to give a precise impress to words 
 long w'om smooth by use. But the teacher may well " imjnove the occa- 
 sion" from time to time, and lead a boy to think more accuratelv, when he 
 catches them out in such tricks. A few laughing words will often be of 
 much greater service on such an occasion than an hour's serious lecture. 
 There is a chance of the joke being remembered through life. The hour's 
 lecture, like other bores, will be certainly forgotten. See also the quota- 
 tion from Prof. Clifford, art. 74, last note, p. 66. 
 
 ARTS. 80, 8].] LOGIC FOE CHILDREN. 
 
 85 
 
 \i 
 
 acquainted with a number of animals, vegetables, and inorganic 
 substances, to render the classifications of animate and inanimate 
 quite distinct, and to make the separation of animal and plant 
 life easy. They will learn to appreciate the great laws relating to 
 all the three sets, and how these are aflfected by plant life, and still 
 more by animal life. But these special considerations belong 
 rather to science teaching, than to the few general remarks to 
 which I am forced to limit myself. 
 
 81. The so-called canons of induction need not occupy us much, 
 for school life does not afford many instances by which they can 
 be exemplified. They are (as already stated) merely elaborations 
 of the first principle of forming the simplest possible supposition 
 (art. 76), and serve to show how that supposition may be cor- 
 rected, and how it must almost always be considered as pro- 
 visional. They are, as you know, the methods of agreement, dif- 
 ference, concomitant variations and residues, which may be used sepa- 
 rately or jointly, and are chiefly applicable for experimental 
 research. The object is in every case to determine between which 
 two sets of events there exists an invariable unconditional rela- 
 tion. One of the first discoveries made is, that one set of events 
 may be preceded, and apparently invariably and unconditionally 
 preceded, by totally different sets of events. Thus, a man's death 
 is the consequence of certain diseases, certain poisons, certain 
 wounds, certain constrictions, &c., so that, given a dead man, to 
 find how he came by his death, is often a difficult, sometimes an 
 impossible problem to resolve, as shown by coroner's inquests. 
 In making deductions from inductions, we are obliged, from our 
 present ignorance of the logic of sequence, to reduce the problem 
 to one of consistency of assertions, thus : — Let the assertion, 
 
 " the events A come first," be X!', 
 
 " the events B come first," be Y", 
 
 "the events M come second" be Z." 
 Then fX.z, \Y.z represent the assertions, "if the events A (or B) 
 come first, the events M come second," (art. 59, b.) On develop- 
 ing the series of complexes (art. 54) we obtain 
 
 X.Y.Z X.7/.Z aj.Y.Z x.y.Z 
 
 fX.Y.z fX.i/.z faj.Y.z x.y.Z. 
 Consequently the occurrence of the events M as second are con- 
 sistent with the occurrence of either or both or neither of the 
 events A and B as first, and the only conclusion which can be 
 drawn is, that if the events M do not occur, neither of the sets of 
 events A or B can have preceded. This is the ambiguity which 
 arises from what is called the plurality of causes, and which besets 
 us in endeavouring to find the cause from the effect.* 
 
 * Dr. Carpenter [ibid.) gives as an example of this Mr. Lockyer's speak- 
 ing " as confidently of this sun's chromosphere of incandescent hydrogen 
 and of the local outbursts which cause it to send forth projections tens of 
 thousands of miles high as if he had been able to capture a flask of this 
 gas, and had generated water by causing it to unite with oxygen" — hydro- 
 gen being the only know n gas which will do so, whereas hydrogen has not 
 been proved to be the only extra-mundane gas which when incandescent 
 
86 
 
 LOGIC FOR CHILDREN. 
 
 [art. 81. 
 
 But here a new point has come to light. Both sets of e\'eiits 
 A and B might come first ; that is, they might occur simultane- 
 ously before the events M. Do sets of events which concur 
 modify each other or not ? Here we come to three laws, which 
 were first only known as laws of motion, but which science has 
 come to extend over the whole of its region. They are extremely 
 extensive inductions, and can be only partially illustrated. ITie 
 real proof is in the verification of the deductions made from them. 
 Succinctly stated, they are, — 
 
 i. The Law of Persistence. Every condition of things tends to 
 remain imaltered, and resists external attempts at alteration. 
 
 ii. The Law of Compomtlon. Every action upon a system or 
 condition of things produces its whole eflect, without regard to any 
 simultaneous action. 
 
 iii. The Law of Reaction. Every action upon a system of things 
 calls up a contrary, and, when properly measured, a precisely 
 equivalent action. 
 
 The law of persistence is however partly due to an hypothesis 
 by which internal are replaced by external actions ; and that of c^m- 
 positi^m, to the consideration of so-called generated forces as new 
 forces, bearing in mind the great law, quite recently evolved by 
 men still living, the conservation of energy, showing that no really 
 new force is ever generated, but that such apparent new forces 
 are but transformations of others which have superficially disap- 
 peared.* These matters can of course only be touched on in 
 passing. It will simply be the teacher's business, in any investi- 
 gation, to bring back the pupils' thoughts when they wander from 
 these secure paths. 
 
 Analogy is not so much an inductive method, as a means of sug- 
 gesting the direction for inductive investigations. Two sets of 
 events agree in a*nuraber of particulars. The first has certain 
 consequences. We conceive that the second will have conse- 
 quences also agreeing in many particulars with the first. This is 
 merely a simple supposition, founded on induction based on such 
 observations. But observation also shows that the analogy fre- 
 quently breaks down, and hence we now use it sim])ly as a sugges- 
 tion. The analogies of sound and light suggested the undulatory 
 
 ■will produce a certain coloured line on being viewed through the spectro- 
 scope, on which Mr. Lockyer's conclusions were based. For the same 
 reason he shows that a new yellow line which has been ohserve<^l in the 
 Hpectroscope need not necessarily be duo to a new metal helium. In fact, 
 the state of temperature at the sun's surface so far exceeds any on which 
 "we can experiment on earth, that the incandescent substances in the sun 
 may affect our spectroscope differently from those on earth. 
 
 • Perhaps this conservation of energ\' may be considered as merely a 
 phase of the law of persistence, when stated as by Dr. Caq:)enter {ibid.), 
 " Energy of any kind, whether manifested in the * molar' motion of masses, 
 or consisting in the 'molecular' motion of atoms, must continue under 
 some form or other without abatement or decay." The forms, however, aro 
 •o different, that it is only by the precisely equivalent mechanical effects 
 that we infer the identity or conservation of the energy. The law of 
 persistence, however, extends to social relations. 
 
 ARTS. 81^ 82j i. ii.] logic for children. 
 
 87 
 
 fA 
 
 n( 
 
 p^( 
 
 theory for the latter ; but this has to repose upon its own merits, 
 and the whole nature of wave motion in the latter had to be con- 
 ceived so difierently from that in the former, that the analogy 
 rapidly ceased to be useful. Ordinary language is full of analogies 
 arising . from vague metaphors or partial resemblances hazily 
 seized, and the results of" pushing these home" are often sufficient 
 to discredit them. But, in all our formations and applications of 
 inductive laws, we assume that there are events and objects so 
 similar, that if one be substituted for the other, the results could 
 be substituted one for the other. This inter changeahility is dis- 
 tinct from analogy, although it probably suggested it, because the 
 interchangeability is only partial, not total. Thus, in respect of 
 being an object of thought separately conceivable, a mouse is in- 
 terchangeable with an elephant, and on such grounds reac all the 
 inductions respecting number. In respect of being fluid at ordinary 
 temperatures, water is interchangeable with mercury. By inves- 
 tigations of waves formed in mercury, important conclusions were 
 c()nse((ueritly obtained for waves of water. But here the resem- 
 blance was so great, we should hardly call the effects analogous. 
 The resemblance between the government of a family by the 
 father, and of a state by the sovereign, is however merely an ana- 
 logy, suggesting much, but not giving a ground for transference 
 of conclusions. 
 
 82. i. The method of agreement is simply this. A great number 
 of diiferent sets of autucedcnts (or events which come first) having 
 been naturally or experimentally observed, in all of which the 
 same events A occur, and it being found, on noting all the corre- 
 sponding different sets of consetpients (or events which come 
 second,) that the same set of events M occur in all of them, it is 
 assumed as the simplest (of course provisional) supposition that 
 there is an invariable, unconditional relation between the two sets 
 of events A and M, so that wUenevev A occurs (and not merely in 
 the cases hitherto observed) we shall look to see M, and whenever 
 M occurs, we may possibly learn that A preceded. Having made 
 this assumption, we reduce it to a formal assertion, and use it as 
 in our previous deductive logic, in connection with other asser- 
 tions, and then compare, if possible all, but at any rate some of 
 the conclusions with accurate observation. If the two again 
 agree, we feel more confident. But a great degree of hesitation 
 will always attach to this method owing to tlie plurality of causes 
 (art. 81, p. 85). 
 
 ii. The method of difference is much more satisfactory. A set 
 of antecedents and consequents being known and observed, sup- 
 pose that when one of the antecedents is omitted, the consequents 
 also are observed to become different by the omission of one ; or 
 when an additional antecedent is introduced, the consequents are 
 likewise changed by the addition of one. In these cases the sim- 
 plest supposition is that there was an invariable unconditional re- 
 lation between the omitted antecedent and consequent, and also 
 between the additional antecedent and consequent. But numerous 
 experiments aro necessary to force this scientifically home to the 
 mind, as it is evident some intermediate links may have been 
 
^^ I-OGIC FOR CHILDREN. [aRT. 82, iii. iv. 
 
 overlooked, or some other additions or omissions unobserved It 
 has however given rise to the popular use of the word 'cause' 
 (art. 79, iv.)* 
 
 iii. The method of concomitant variations is a common variety 
 of the method of difference. Here no antecedent is entirely re- 
 moved or added, but one antecedent is varied in intensity, and a 
 corresponding variation of intensity in one consequent is ob- 
 served. We must guard against expecting these variations to be 
 ot the same kind, or that if one takes pla^e in one direction, the 
 other will also constantly take place in one direction ; or that 
 the antecedents will alter in the same ratio as the conse- 
 quents. It IS very commonly said that the effect varies as the 
 cause. I his is an attempt to give mathematical accuracy of 
 expression to what has been only vaguely observed. As thus 
 expressed, the proposition is far from being generally true. If you 
 spill a glass of water over a person's hand, the effect is very different 
 ti-om letting a small quantity drop upon the hand gradually. This ii 
 an excellent experiment for boys. If you diminish the heat, 
 alcohol continually diminishes in volume ; water diminishes to a 
 certam extent, and then increases. This should be shown, for the 
 lact is valuable. 
 
 iv. The method of residues consists simply in deducting from a 
 set of consequents those known, from previous complete inductions, 
 to be the result of a portion of the antecedents, and then making 
 the simplest supposition that the remaining consequents are due 
 to tlie remaining antecedents. 
 
 We might go on to the combination of the different methods, 
 but the only real point of importance in all these methods for our 
 present purpose, is to understand that they are all referable to the 
 one law of forming the simplest supposition, and that the conclu- 
 sions are therefore all provisional, requiring constant verification. 
 ±rom this we see the necessity of registering all such conclusions 
 
 ARTS. 82, 83.J LOGIC FOR CHILDREN. 
 
 89 
 
 * Dr- Carpenter says (ibid.): "While fully accepting the logical defi- 
 f^^pV .'•'''" "' ^^I'^^^-r"^^^^ ^'^ <^«°^'^rence of antecedents'on which 
 l^e onf nnlT"^^^^ unconditionally consequent,' we can always 
 
 angle out one dynamical antecedent-the power that does the work-from 
 the aggregate ot material conditions under which that power may be dis- 
 tributed and apphed." This distinction of the ^ot^v./Jand th^J^^.I . 
 " No donh?''!'>,^^ ^'"r ^^''^'^T^ ^°d theoretical. It is never obiTgatory. 
 No doubt, he continues, " the temi cause is ver>' loosely employed hi 
 
 which^^l^f, '' power which acts m each case, from the conditions under 
 from Vi«h1 a *^' ."^'"^l^^ ^^^^'^^ "P ^y gunpowder, a man falling 
 ^uTes" " f h^' r ^i^^'v T ^^ ?*"^' ^'' ^^^n>enter make^ the « efficient 
 causes "the force locked up in the gunpowder, ... the gravity which 
 was equ^y pulling him down while ho rested upon it [the ladder 1 
 
 ^ndiffnn^T ""fft^ °''/f ^^'" ^l«P^«vely» and all thereat, " the material 
 ^^^ ' '' ^^«*^\"^}°§. (*!> "se the old scholastic term) only ' their formal 
 ^^?H ,.^%^J«1« distinction scientifically appreciable, excluding the 
 o?aA 7^^ Ti'^^S" ^f^^^' metaphysical te^rJns, is sh;>wnat the^close 
 vLnVJnl!: 5> \l .^^ ^^^ *^® quotation from Prof. Chflford respecting the 
 vagubnees of the term cause m the note f to art. 79, iv., p. 76. 
 
 
 
 as have been satisfactorily verified, and this registration takes the 
 form of a -treatise upon a particular branch of science, as distinct 
 from the general investigation of the principles which lead to the 
 necessity of such investigation. Induction is the father of science, 
 not science itself. But science cannot be properly appreciated till 
 its parentage is understood.* 
 
 83. In the preceding remarks I have endeavoured to indicate 
 the principal points which" a teacher must attend to in teaching 
 induction. On account of the great difficulties which attend the 
 preparation for verification in all great inductions, it is not pos- 
 sible to obtain a complete view of the subject without long and 
 arduous study of the principal inductive sciences. Hence treatises 
 upon Inductive Logic drift more or less into a philosophy of the 
 elementary principles of the special sciences, which cannot be 
 even partially understood without something more than an ordi- 
 nary household acquaintance with them. Thus almost half of 
 Prof. Bain's work on Induction is devoted to the " Logic of the 
 Sciences," and yet each science is treated with almost bald suc- 
 chictness. Now it is evident that all this is out of the question in 
 schools, and that it is only in advanced schools with very capable 
 masters, that it can be even partially taught, in respect, for ex- 
 ample, to geometry, mechanics, chemistry, sociability. But a 
 few great principles can be always inculcated, on innumerable 
 occasions, so that when the child grows up he may find the forest 
 cleared for the subsequent cultivation of science. These are the 
 principles which I have been trying to present to your notice. If 
 a child leaves school with a firm conviction of the uniformitij of 
 nature, that is, the invariahility and unconditumality of relationSy 
 and a constant habit of making the simplest suppositions consistent 
 with a representation of the whole of the facts hnown, he will be 
 a totally different being, intellectually, from the child who has 
 been taught the usual routine, in which no notice of these great 
 
 * Prof. Cliflford concluded his admirable lecture as follows (p. 511, c. 2) : 
 " By scientific thought we mean the application of past experience to new 
 circumstances, by means of an observed order of events. By saying that 
 this order of events is exact, we mean that it is exact enough to correct 
 experiments by, but we do not mean that it is theoretically or absolutely 
 exact, because we do not know. The process of inference we found to be 
 in itself an assumption of uniformity, and that, as the known exactness of 
 the uniformity became greater, the stringency of the inference increased. 
 By saying that the order of events is reasonable, we do not mean that 
 everything has a purpose, or that everything can be explained, or that 
 everything has a cause ; for neither of these is true. But we mean that 
 to every reasonable question," [previously defined (p. 611, c. 2) as one 
 which is asked in terms ot ideas justified by previous experience, without 
 itself contradicting that experience,] " there is an intelligible answer, 
 which either we or posterity may know by the exercise of scientific thought. 
 . . . . Scientific thought is not an accompaniment or condition of human 
 progress, but human progress itself. And for this reason the question 
 what its characters are, ... is the question of all questions for the 
 human race." 
 
90 
 
 LOGIC FOR CHILDREN. 
 
 [art. 83. 
 
 ART. 84.] 
 
 LOGIC FOR CHILDREN. 
 
 91 
 
 princi])Ies occurs * And there seems to be no good reason why 
 a child should not attain to such habits and convictions. The 
 material is all there, and it remains with the teadier to employ it. 
 A few occasions have been pointed out as they arise, from which 
 the method I would adopt generally may be inferred. But some 
 additional special observations seem desirable. 
 
 ♦ In the preceding articles I have made much use of the philosoj)hic 
 laws as stated by Augnste Comte. As these laws are not easily accessible 
 in the form he has assigned, it seems expedient to annex them, as far as 
 possible, in the words of his Politique Positive, vol. 4, pp. 173 — 180, pub- 
 lished in 1854. As they are there always stated in an oblique form, I 
 have thought it best to alter the construction, and also to disinter them 
 from the body of the paragraphs in wliich they occur, and arrange them 
 in the order which Comte has i-ather indicated than actually carried out. 
 The greater part of these laws are generally recognized, and it is only the 
 form which has been given to them that is peculiar to Cemte ; this form, 
 however, is of consider.ible importance. Laws 7, 8, 9 are peculiar to 
 Comte's philosophy, and have been vehemently disputed. Laws 10, 11, 
 12, 13, in their original form, are Kepler's, Galileo's, and Newton's laws of 
 motion, and D' Alembert's principle, respectively. The law of the Conserva- 
 tion of Energy, due to Mayer, Helmlioltz and Joule (art. 81), was unknown 
 to Comte, but he would probably have included it under law 10. At the end 
 of each law here referred to, I have added the number of the article in 
 which it is used and explained, and this will dispense with the necessity 
 of any translation. The law is generally printed in italics in the articles 
 cited. ^Ticre no article is referred to, the law has not been used. 
 
 " Principes TJniversels snr le concours desquels doit reposer la systema- 
 tisiition finale du dogme positif. Les quinze lois universelles do la 
 ]>hilosophie prijmiere, reparties entre trois groupes, dont les deux demiers, 
 doubles chacun du j)remier, se decomposent respectivement en deux 
 series cgales. 
 
 I. Le premier groupe, non moins relatif a la constitution interieui-e 
 de nos speculations qu'ii lour destination exterieure. 
 
 1. Principe fundamental. Fomiez I'hypothi se la plus simple que com - 
 porte I'ensemble des documents a representor. (Art. 76, p. fiS.) 
 
 2. Reconnaissez rimmuabilite des lois (juelconques, qui reginsent les etres 
 d'apres les evenements, quoique I'ordre abstrait permette seul de les ap- 
 precier. (Art. 78, p. 70.) 
 
 3. Les modifications quelconques de I'ordre imiversel se trouvent bor- 
 nees a I'intensito des pheuomenes, dont rarrangement demeure inalt*^- 
 rable. (Art. 78, p. 71.) 
 
 IL Le second groupe, directement relatif a I'entendement, se decompose 
 
 en deux, statique et dynamique. 
 a. Statique. L'ordro consiste dans I'etablissemont de I'unite. 
 
 4. Subordonnez les constructions subjectives aux materiaux objectifs. 
 (Art. 80, p. S3.) 
 
 5. Les images inttrieures sont moins vivcs et moins nettes que les im- 
 pressions exterieiu*e8. (Art. 80, p. 82.) 
 
 6. Que I'image normale aio la preponderance sur celles que I'agitation 
 cerebrale fait simultanement surgir. 
 
 b. Dynamique. Le progres consiste dans le devcloppement de I'ordre. 
 
 7. Chaque entendement pr^scnte une succession de trois etats, fictif, 
 abstrait et positif, envers des conceptions quelconques, mais avec ime vi- 
 teasc proportionnee a la genenilite des pheuomenes conespondauts. 
 
 ( 
 
 84. Deductive logic can be taught by a continual recurrence 
 to one single class of instances, the occurrence of letters in words. 
 Inductive logic cannot be taught without recurrence to a great 
 variety of classes of instances, as the great variety alone can give 
 suflRcient sanction to its conclusions. Deductive logic requires a 
 very close attention to a large number of particulars, which in- 
 volves a great strain on the mind unless assisted by something 
 approaching to mathematical severity of notation, while the very 
 abstractness of that notation makes it formidable to many 
 minds. Inductive logic, in its widest and simplest principles, can 
 be studied at first without the same attention to minute details, 
 without any approach to a mathematical notation, and by a con- 
 tinual recurrence to facts actually present to the eye or mhid, 
 which arouse and fix attention by their concrete character. It is 
 quite true that we cannot proceed even a moderate distance in 
 strict induction, without close deduction (art. 72); but in the 
 familiar instances of daily life, the deductions are usually so 
 simple that they may be assumed as self-evident. The habit of 
 making them by aid of the simplest supposition principle (which 
 the teacher will take care is properly applied) will be an excellent 
 introduction to that stricter form already explained, which is, 
 after all, merely an application of the same principle. Hence we 
 must not wait for instruction in deductive logic before we proceed 
 to induction, and we must therefore not begin by teaching induc- 
 tion systematically. The object is to lead the young mind by 
 mere work to appreciate some of the great principles upon which 
 
 8. On reconnait une progression analogue pour I'activite, d'abord con- 
 queranto, puis defensive, enfin industrielle. 
 
 9. On eteud aussi la meme marchc a la sociabihte, d'abord domestique 
 puis civique, enfin universelle. * 
 
 III. Le troisieme groupe, ou domine robjcctivite, se divise en deux series, 
 
 celle des plus objectives, et celle des plus subjectives. 
 a. Les plus objectives ne furent d'abord appreciees qu' envers les pheno- 
 
 menes mathematiques. 
 
 10. Tout etat, statique ou dynamique, tend a persister spontan^^ment, 
 sans aucune alteration, en resistant aux perturbations exterieures. 
 (Art. 81, p. 26.) 
 
 11. Un systeme quelconque a I'aptitude a maintenir sa constitution, 
 active ou passive, quand ses elements eprouvent des mutations siniul- 
 tan^'es, pourvu qu'elles soient exactement communes. (Art. 81, p. 26.) 
 
 12. II existe partout une equivalence necessaire entre la reaction et Tac- 
 tion, si leur intensite se trouve mesuree conformement a la nature de 
 chaque conflit. (Art. 81, p. 26.) 
 
 b. Les plus subjectives, oti I'origine mathematique devient moins appre- 
 ciable. 
 
 13. Subordonnez partout la thoorie du mouvement a celle de 1' exis- 
 tence, en concevant tout progres comme le devcloppement de I'ordre corre- 
 spondant, dont les conditions quelconques regissent les mutations qui con- 
 stituent revolution. 
 
 14. Le classement positif s'effectue toujours d'apres la generaHte crois- 
 sante ou decroissante, taut subjective qu'objective. (Art. 80, p. 24.) 
 
 15. Subordonnez tout intern ediare aux deux cxtiOmes dont il opere la 
 Uaisou." 
 
92 
 
 LOGIC FOR CHILDREN. [arts. 84, 85, i. ii. 
 
 indnctions are made, but not itself to make larj^e or accurate in- 
 ductions. The following are merely suggestions, which an intel- 
 ligent teacher will readily seize, and which indeed do not in them- 
 selves present any novelty. 
 
 85. i. The first care must be to put a child into the proper con- 
 dition of mind to make or appreciate inductions. Hence accurate 
 ohsei-vation must be cultivated. The children must see things and 
 describe them. Enter a room, remain a short time, retire and tell, 
 or better still write down, the names of all the objects you can re- 
 collect, (this was one of the exercises of the celebrated French con- 
 jurer, Robert Houdin). The room may be exchanged for a shop, and 
 entering for passing. Take a walk, name all the roads turning 
 to the right or left, the bridges, rivers, streets, isolated houses. 
 Pass a house, describe the number of storeys, the nature of the 
 roof, the number and situation, or any peculiarity in the shape of 
 the windows. Describe a person seen, height, sex, age (child, 
 adult, old), colour of hair and eyes, complexion, hands, boots or 
 shoes, dress (especially the colours of its parts, or any peculiarity 
 in its shape). In a walk mention the number of persons seen, 
 and divide them into male and female, and subdivide according to 
 age; also divide first by age, and then subdivide by sex; also 
 mention number of quadrupeds, and number of each kind of 
 quadruped observed. Describe each child in the school (if not 
 large) or in the class, or class-room. Such descriptions and sta- 
 tistics should be reduced to writing, and stated in a tabular form. 
 The above arc merely specimens to guide the teacher. Occasions 
 for such observations arise constantly in the course of instruction, 
 in languages, in arithmetic, in geometry, in lessons on objects, 
 especially when concerning minerals, crystals, shells, flowers, 
 animals. Extreme accuracy should be always appreciated. The 
 natural powers of observation, however, differ materially. Beware 
 of thinking a quick observer clever, or a slow observer dull. A 
 teacher can hardly make a greater mistake. 
 
 ii. Next take invariable sequences of common occwTetice^ with a 
 view of making the /ac< of the existence of such sequences familiar. 
 The rising of the sun and all heavenly bodies in the east, their 
 passing to the south, and setting in the west. Light following 
 sunrise, and disappearing at sunset. Power of vision following 
 lighting a candle in the dark. Falling of a stone unsuppoi-ted ; 
 weariness resulting from supporting a stone in the hand ; weari- 
 ness from any continued exertion. Sleepiness following wakeful- 
 ness. Wakefulness following excitement. Pain following burns 
 or blows. Hunger and thirst following much exertion, or long 
 abstention from eating and drinking. Heat from approaching 
 fire, or from exercise. Solution of salt and sugar, and non-solu- 
 tion of stones in water. Sinking of stones and non-sinking of 
 cork placed on surface of water. Children growing taller as they 
 grow older ; adults not altering in height. Adults growing weak 
 as they grow old. Death following life with or without disease, 
 in vegetables, domestic animals, and man. Budding in spring, fall 
 of leaf in autumn. Meat rendered fit to eat by cooking. Improve- 
 ment in sports and work by practice. And so on, and so on, the 
 
 ART. 85, iii. — v.] LOGIC FOR CHILDREN. 
 
 93 
 
 great point being by extremely numerous examples, sure to be 
 familiar to children, to prepare them for the acceptance of the 
 general principle of invariability throughout nature, inanimate 
 and animate, involuntary and voluntary, individual and social. 
 
 iii. Accidents of common occurrence should also be noted, and it 
 should be shown that in several cases an invariability was unex- 
 pectedly brought into action. A boy runs, trips, falls, cuts his 
 hand ; here the accident was his tripping, that is, his having so 
 wrongly measured his distance as to strike the stone ; this was 
 " avoidable," the rest " unavoidable," an important distinction. 
 Two boys running round a corner in opposite directions collide 
 with pain; here there was ignorance; see whether it was alto- 
 gether unavoidable. A boy writing is startled by a sudden noise 
 and blots his book ; which is the accident — the noise, the start, or 
 the blot ? After many such notions of accident, proceed to others, 
 as the state of the weather on certain days of the month, at cer- 
 tain changes of the moon, &c. As these changes are badly re- 
 membered, make boys record them, as well as they can. It is 
 very important to upset the prophecies of weather almanacs, espe- 
 cially in agricultural districts. Show, however, that weather obeys 
 the law of averages, so that we expect a certain kind of weather in 
 certain months. See the former exercises (art. 79, ii. p. 73.) 
 
 iv. Slimple inductions may now be introduced. The fir st kind 
 which I would suggest aie not the most simple in themselves, 
 but are most simple to the children, because they have probably 
 been in the habit of making them. They consist of investigations 
 into what has happened, inquiries into circumstances of school 
 and social life. Thus, " Has John been here ? " Some children 
 may have seen something which John usually brings, a book, a 
 hat, a parcel, and hence conclude he has been here. This should 
 be followed up. " It is a very good simple supposition, but could 
 not the things have come otherwise? Are they always brought 
 by John, and no one else ? " We have here the plurality of causes 
 exemplified. Again, a dispute arises in a game; the evidence on 
 each side is taken, the importance of accurate observation appears 
 in contradictions where good faith is assumed; the necessity of 
 deliberation is shown ; inferences will have been probably substi- 
 tuted for the statement of direct observation ; the differences be- 
 tween eyesight and hearsay will be displayed; the impossibility 
 of decision, that is, of making a safe induction, may become appa- 
 rcTit, or hasty decisions may be shown to be false. An event hap- 
 pens in a town, there is a trial reported, a question of identity or 
 alibi arises ; the evidence must be scrutinized. And so on. 
 
 V. The next class of inductions are really more simple, but re- 
 quire more careful and exact treatment, and hence have to be 
 taken last. Two stones are let fall from the same height, and 
 reach the ground at the same time. Does this depend on their 
 weights, actual or relative, on the particular height? Vary 
 both. Very light weights, pieces of paper, or feathers, take a 
 longer time in falling. Show that this time can be increased by 
 creating a wind : show that if, by attaching a little weight, the 
 paper be made to fall sideways in calm air, it more nearly falls in 
 the same time as a heavy stone. Show that if upon a flat weight 
 (as a penny) a smaller piece of paper be laid, both the paper and 
 
94 
 
 LOGIC FOR CHILDEEN, 
 
 [art. 85, V. 
 
 wciKbt will fall in the same time. Make simple suppositions with re- 
 Tpect to time of tall ing being independent of weight but dependent 
 on air Experiment on resistance of air generally, by movmg 
 hand or flat body, full face or sideways. Similarly m water Ex- 
 Sent with kites and oars. Experiment with throwing bodies 
 E" motions cf rotation. Experiment with throwing and 
 droppTng in water. Two weights slung over a pulley, give a 
 rou-h Atwood's machine, and lead to many valnable results, as to 
 rateof falling in equal times, as to time of falling depending on 
 I^ra and difference' of weights. Make pendnluni of thread and 
 buUet of stone, find law of variation of its length as compared 
 w"th its time of vibration. Find length of seconds pendulum, by 
 Counting oscUations-swings is the child's word-and difference 
 
 o?rates^of different pendulums by observing th^Sn'of '^v^s 
 Apply to clocks, beating time m music, &C; Reflection of faves 
 balls carefully observed and measured, giving average angle of 
 reflection equal to average angle of incidence, and leading to the 
 induction that it is only obstacles (such as rough vvall or ground 
 and imperfect elasticity) which prevent exact equality. ReA/f ''?" 
 of light" showing the law to be exact. Refraction of light Limits 
 to both by polarisation, as before explained (art. 79, v. p. 77) 
 Ma-Tetic attraction, formation of needles into magnets, polanty 
 of Seedles, the north of the magnet compared with position ol 
 polar star Determination of south from the position of sun 
 Ened by observing shadows of a pole in the P^^ygff^; 
 com^rison of this with magnetic south, and comparisons of time 
 of s^n's southing with the clock noon. Electric attraction of 
 rubbed sealing wax and rubbed glass for chips or pieces of cork, 
 e pectuy suspended by silk strings and the Bubsequent repulsion. 
 Experiments on sound, law of reflection of sound as in light, 
 echoes, whispering tubes, scratch at end of long pole heard at 
 ^therrates of light and sound. This last should be reauced to 
 measurement. Children at different distances clap hands when 
 Another ^-^ises his hand, the sounds are heard m succession. 
 Measure distance (about 1120 feet, at a te^P^f '"[k "^ ^, 
 Fahrenheit) at which sound is heard one second after the hands 
 were seen to be clapped. Apply to thunder and lightning, to 
 determine the distance of the storm. 
 
 It seems scarcely necessary to repeat that no one can have a 
 proper notion of the meaning and power of induction who has not 
 Sone through at .east some course of some particular science, 
 fr° ated in sSch a manner as to show.the elementary ob«.ervationB 
 the primary inductions, the experiments for gendering those 
 inductions more precise, their statement as definite assertions, 
 Ihe deductions dmwn from these assertions, the verifications of 
 the conclusions thereby obtained, and the F«'-*""°°^,.'f ^" 
 throughout to avoid error either from maccurate observation, or 
 imperfect ratiocination. Success in teaching depends on the 
 teacher's own knowledge and power °/ communication. Book 
 learning can at most suggest. He must have hand ed the tools 
 hTmself He must know what induction is, practically as well as 
 theoretically. A few such teachers are even now to be touna. 
 May they soon be rife. , 
 
 Printed by C. P. Hodgson k Son, Gough Square, Fleet Street, B.C. 
 
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