LETTERS ON LOGIC 
 
 TO A YOUNG MAN 
 WITHOUT A MASTER 
 
 BY 
 
 HENRY BRADFORD SMITH 
 
 THE COLLEGE BOOK STORE 
 
 3425 Woodland Avenue 
 
 PHILADELPHIA, PA. 
 
 1920 
 
LETTERS ON LOGIC 
 
 TO A YOUNG MAN 
 WITHOUT A MASTER 
 
 BY 
 
 HENRY BRADFORD SMITH 
 
 THE COLLEGE BOOK STORE 
 
 3425 Woodland Avenue 
 
 PHILADELPHIA, PA. 
 
 1920 
 
 
■A* 
 
 PRESS OF 
 
 THE NEW ERA PRINTING COMPANY 
 
 LANCASTER, PA. 
 
 ^/V«2<? 
 
PREFATORY NOTE 
 
 Somewhat more than a year ago Professor Edgar A. Singer, Jr., 
 asked me to undertake the writing of a text-book for use in the 
 introductory course in logic at the University of Pennsylvania, 
 placing at my disposal the syllabus of a course of lectures, which 
 he had ceased to give some years before, and intimating that I 
 might use his materials in any way that I saw fit. The present 
 work is an effort, all too imperfect as I am aware, to meet his 
 request. The syllabus itself in its original form is placed at the 
 end of the book. 
 
 I shall make no apology for the changes, which have been 
 introduced into the text. They invariably involve the sacrifice 
 of a proper or methodical arrangement in order to gain a peda- 
 gogical advantage. And, finally, in fairness to Professor Singer 
 it must be understood that the chapters of the book bear no direct 
 relation to his own formal lectures. They have been suggested 
 by the outline alone. 
 
 The comparative brevity of the work is explained by the nature 
 of my task. It would have been risking too much to introduce 
 historical comments of my own or to venture upon questions of 
 application, when my purpose is only to furnish an approach in 
 popular form to a very original contribution to logical theory. 
 Those who use the book in the class room will be able to remedy 
 this defect in their own way. The syllabus itself will be found 
 indispensable for those who intend to pursue the theory still 
 further; in particular, for those who desire to express the whole 
 in the more elegant form of a symbolic technique. 
 
 H. B. S. 
 
LETTERS ON LOGIC 
 
 I 
 
 My dear Sir: 
 
 Some years have now passed since you first wrote to me, 
 modestly describing the desires of an intellect deprived from the 
 first of any adequate training and inquiring how such defects 
 as your mind possessed might best be remedied. I recall that 
 you had begun your effort for self-improvement in the field of 
 literature but that you had found such studies to yield a pleasure, 
 which charmed rather than satisfied a spirit set upon overcoming 
 obstacles. It was natural, then, that you should have turned 
 next to philosophy and certainly fortunate that Plato should 
 have been the first writer to altogether engage your attention. 
 I recollect now, however, that the event in your life, to which 
 you attached a prime importance, was the chance discovery of 
 a copy of Euclid's Elements, and it was at the period, when you 
 were absorbed in reflections of this genre, when you had become 
 aware that mathematics is the key to so many shrines, that you 
 had written to know if I could instruct you in the infinitesimal 
 calculus. It was here that our correspondence had begun. 
 
 I have often turned over in retrospect these and other details 
 of your career in the time that has elapsed, since the serious turn 
 in your financial affairs has compelled you for the moment to 
 more practical preoccupations. You had been good enough to 
 intimate that you had profited by that course of instruction 
 conveyed to you by letter and that you had profited was clearly 
 shown by the questions, which you raised. But, then, courtesy 
 would have required you to say as much. The proof was forth- 
 coming only yesterday and I confess that I feel flattered that 
 you should now ask me to introduce you to another science, no 
 less engrossing in itself and whose tradition is still more venerable. 
 That I should agree to undertake what you so delicately urge, 
 you yourself of all men must have been the last to doubt. 
 Surely none is unaware that those, whose profession it is to 
 impart knowledge have rarely to be urged to exercise what 
 
2 Letters on Logic to a 
 
 talents they possess; that they often act upon the slenderest of 
 pretexts, so that wise men will not provoke their comment unless 
 the occasion be grave and the need of their advice be urgent. 
 Need I add, then, that I accede to your request, nay, that I 
 have already in mind a plan of presentation? 
 
 It will not be necessary to remind a reader of the Dialogues, 
 that the order of ideas, which one follows, when he begins to 
 apprehend a science, is in no wise the order, in which those 
 ideas find themselves arranged, when that study has made a 
 certain progress; that, consequently, the pedagogical order, in 
 which the matter of science ought to be arranged, is rather the 
 reverse of the methodical. The latter in its perfection pertains 
 only to God, who geometrizes eternally; the former is a necessity 
 of those imperfections to which the flesh is heir. In the logical 
 ordering of the materials of thought those accidents, which had 
 come into being with the materials themselves and which had 
 become attached like barnacles to the proper subject-matter, 
 have been removed; conditions, which at first appeared as 
 necessities but had proved redundant, have been erased. It will 
 furnish you no surprise, therefore, if our plan should be to intro- 
 duce such and such a condition only to remove it later on, or to 
 postulate the truth of such and such a proposition only to discover 
 that its truth may be made to depend on others which enter at 
 a future date. I shall try with what art I possess to lay before 
 you the drama of a mind at work, rather than reveal at the 
 outset the finished system, which emerges for the first time, 
 when that work is done. 
 
 It only remains to state that we shall not be able to concern 
 ourselves with the applications of the science. Such matters are 
 of the most delicate sort and presuppose always the highest 
 degree of skill. You will find them treated in the current texts. 
 In any case you will wish to bring to their consideration an 
 adequate theory and it is this theory alone, which we propose 
 to supply. We shall not pretend that our doctrine is complete 
 but we shall develop it so far as is usual in the university in a 
 first semester course. 
 
Young Man without a Master 
 
 II 
 
 It will hardly be necessary to inform a student of mathematics 
 of the difference between defining an object or relationship by 
 the aid of synonyms, after the fashion of the dictionary, and 
 defining it by means of its formal properties, which is the pro- 
 cedure of science. The relations, which are the proper subject- 
 matter of a given compartment of thought, are commonly taken 
 to be defined by the set of axioms, necessary and sufficient to 
 yield all the theorems characteristic of the domain of application, 
 which is in question. One or more of the axioms, which taken 
 together determine unambiguously a given field of interpretation, 
 may hold true separately of other relationships than the ones, 
 which we are seeking to define. 
 
 Suppose, by way of illustration, that we wished to differentiate 
 three relations, which we will select as parallelism, perpendicu- 
 larity and implication. Two axioms, known as the axiom of 
 transitivity and the axiom of reciprocity, will then be enough to 
 effect our purpose. If all three of our relations be expressed as 
 transitive we should have: 
 
 (i) If x is parallel to y and y is parallel to z, then x is parallel 
 to z; 
 
 (2) If x is perpendicular to y and y is perpendicular to z, 
 
 then x is perpendicular to z; 
 
 (3) If x implies y and y implies z, then x implies z. 
 
 Of these three propositions the first and the last are true and the 
 second is false. If our three relations be represented as recip- 
 rocal, we should write: 
 
 (1) If x is parallel to y, then y is parallel to x\ 
 
 (2) If x is perpendicular to y, then y is perpendicular to x; 
 
 (3) If x implies y, then y implies x. 
 
 Here the first and the second propositions are true and the 
 third is false. Consequently, parallelism is both transitive and 
 reciprocal, perpendicularity is reciprocal but not transitive and 
 implication is transitive but not reciprocal. If it were only 
 desired to distinguish between these three, enough of their 
 properties would now have been enumerated and the formal 
 problem of constructing the definition of any one would have 
 
4 Letters on Logic to a 
 
 been solved. But, if a fourth relation, say that of being "sub- 
 sequent to" (in point of time), were among those of the set to be 
 distinguished inter se, our two principles would not suffice, for 
 "subsequent to" is transitive of any three events whatever, but 
 is not reciprocal of any two, and hence "subsequent to" and 
 "implication" would be identical in the sense, that what is 
 different in them is not revealed by the axioms of transitivity 
 and reciprocity alone. To discover a character of difference 
 between them it would be necessary to examine their behavior 
 in the context of some third principle. 
 
 In the light of this illustration our meaning will be clear, 
 when we say that a relationship is defined when enough of its 
 properties have been enumerated to completely distinguish it 
 from whatever other relationships are in question and that the 
 formal problem of logic, — i.e., its problem divorced from ques- 
 tions of application — is to define the relationships, of which it 
 treats, by constructing all the true and all the false propositions 
 into which these relationships enter exclusively. 
 
 The forms of relationship recognized in logic — we shall enumer- 
 ate them at the outset; their full meaning will appear in the 
 sequel — are as follows: 
 
 (i) the categorical, {adjective of quantity and copula), 
 
 A(ab) = All a is b (is true), 
 
 E(ab) = No a is & (is true), 
 
 I (ab) = Some a is b (is true) , 
 
 0(ab) = Not all a is b (is true), 
 
 = Some a is not b (is true), 
 A'(ab) = All a is b (is false), 
 E'(ab) = No a is b (is false), 
 
 V(ab) =, Some a is b (is false), 
 0'(ab) = Not all a is b (is false), 
 
 = Some a is not b (is false) , 
 
 (2) the hypothetical, (if, then), 
 
 x(ab) (is true) implies y(ab) (is true) is true, 
 = If x(ab) (is true) then y(ab) (is true) is true, 
 = x(ab) z y(ab), 
 x(ab) (is true) implies y(ab) (is true) is false, 
 = {x{ab)y(db)Y, 
 
Young Man without a Master 5 
 
 (3) the conjunctive, {and), 
 
 x(ab) (is true) and y(ab) (is true), 
 = x(ab)-y(ab), 
 
 (4) the disjunctive, (either, or), 
 
 Either x(ab) (is true) or y(ab) (is true), 
 = x(ab) + y(ab). 
 
 The notation x(ab), y(ab), etc., is used to denote indiffer- 
 ently any one of the propositions, A(ab), E(ab), I(ab), O(ab), 
 i.e., we say, x, y, etc., may take on any one of the four values 
 A, E,I,0. 
 
 In the categorical proposition x(ab) the terms are the subject a, 
 which is written first in the bracket, and the predicate b, which is 
 written second, i.e., the term-order in x(ab) is the order subject- 
 predicate. When we wish to indicate that the term-order is 
 not settled, we shall place a comma in the bracket between the 
 terms, x(a, b) standing either for x(ab) or x(ba). The terms a 
 and b, stand for classes, i.e., each stands for a group of ob- 
 jects, which are conceived by the aid of a common property, 
 every substantive in the language being the symbol for such a 
 group. 
 
 A(ab) asserts that all of the members of the a-class are con- 
 tained among the members of the &-class, leaving it undeter- 
 mined, whether the members of the subject-class are related to 
 the members of the predicate-class through identity or exhaust 
 only a part of the members of that class. The meaning of the 
 assertion, A(ab) may be illustrated by the following diagram 
 (Fig. 1), i.e., if All a is b is a true proposition, then the class a 
 is related to the class b in one of these two ways. 
 
 All a is b. 
 
 Either [a b 1 or 
 
 Fig. 1. 
 
 The diagrammatic representation of the other categorical proposi- 
 tions is given (in Figs. 2, 3, 4) below. 
 
Letters on Logic to a 
 
 Some a is b. 
 
 No a is b. 
 
 Some a is not b. 
 
 Either a 
 
 Fig. 2. 
 
 Fig. 3. 
 
 or 
 
 Fig. 4. 
 
 or 
 
 It will be intuitively clear, that any two classes whatsoever, 
 a and b, must be related in one, and cannot be related in more 
 than one, of the following five ways (see Fig. 5) : 
 
 QD 
 
 Fig. 5. 
 
 If we assert A'iab) to be a true proposition, {All a is b is false), 
 then a and b cannot be related in the first or second fashion and 
 must, consequently, be related in one of the other three ways 
 
Young Man without a Master 7 
 
 (Fig. 5). Glancing back at Fig. 4 you will see at once that the 
 denial of A(ab), i.e., the assertion of the falsity of A(ab), is 
 equivalent to asserting the truth of 0(ab). You should have 
 no difficulty in grasping the truth of the following equations, 
 each one of which should be intuitively verified (Figs. 1-5) : 
 
 A(ab) = 0'(ab), 0(ab) = A'(ab), 
 E(ab) = I'(ab), l(ab) = E f (ab). 
 
 If it be laid down as axiomatic, that A proposition must be 
 either true or false and cannot be both true and false, then a reference 
 to the equalities, that have just been written down, will make it 
 clear that A(ab) and 0(ab) cannot both be true and cannot both 
 be false and that the same applies to E(ab) and l(ab). Such 
 propositions as satisfy this condition are said to be contradictory. 
 Consequently A(ab) is the contradictory of 0(ab) and conversely, 
 while K(ab) is the contradictory of l(ab) and conversely. In 
 order to frame an image of this truth let me ask you to glance 
 again at Fig. 5. You will observe, since A(ab) and 0(ab) have 
 no diagrammatic representation in common, that they cannot be 
 true together. At the same time it will be apparent, that, since 
 taken together they exhaust all the modes of representation that 
 there are, they cannot both be false. You will have no difficulty 
 (referring to Fig. 5 again) in becoming aware that exactly the 
 same statements hold of K(ab) and l(ab). 
 
 These results are so important for our subsequent theory, 
 that it will be convenient to summarize them. In verifying the 
 forms, into which they may be cast, you cannot do better than 
 refer continually to the diagrams of Fig. 5. 
 
 A(ab) (is true) and 0(ab) (is true) is a false proposition, 
 
 T A(ab) (is false) and 0(ab) (is false) is a false proposition, 
 
 E(ab) (is true) and l(ab) (is true) is a false proposition, 
 
 E(ab) (is false) and l(ab) (is false) is a false proposition. 
 
 A(ab) (is true) or 0(ab) (is true) is a true proposition, 
 A(ab) (is false) or 0(ab) (is false) is a true proposition, 
 E(ab) (is true) or \(ab) (is true) is a true proposition, 
 E(ab) (is false) or \(ab) (is false) is a true proposition. 
 
 If we invent, as is customary, a symbol, (<?), to stand for a 
 proposition, that is false for all meanings of the terms, and 
 
 II 
 
8 Letters on Logic to a 
 
 another symbol, (i) to stand for a proposition, that is true for 
 all meanings of the terms, and if we recall the symbols, that have 
 already been introduced to stand for the conjunctive and the 
 disjunctive relationships, then the propositions, that have just 
 been enumerated, may be expressed very concisely thus: 
 
 I 
 
 II 
 
 A (ab) • O (ab) = o, E (ab) ■ I (ab) = o, 
 
 A f (ab) • 0'(ab) = o, E'(ab) ■ V(ab) = o, 
 
 A (ab) + O (ab) = i, E (ab) + I (ab) = i, 
 
 A f (ab) + 0'(ab) = i, E'(ab) + V(ab) = i. 
 
 The propositions, which have just been written down, are 
 fundamental in the classical logic, the science, which has de- 
 scended to us from Aristotle. The peculiar simplicity of the 
 common logic depends upon the fact, that, corresponding to 
 any member of the set of categorical forms there exists a single 
 other member of the set, which stands for its contradictory. Or 
 (what you will later come to recognize as the same thing) : 
 any categorical form, x(ab) (is false), may always in this system 
 of inference be replaced by another categorical form, y(ab) (is 
 true) (see the first set of identities, p. 7). There is small doubt 
 that this was the motive, which guided Aristotle to select his 
 four forms in this particular way. It was not because each one 
 happened to have a convenient verbal expression, as some not 
 too discerning depreciators of the ancient scheme of inference 
 have said. 
 
 About the middle of the last century a famous logician, Sir 
 William Hamilton, proposed to replace the four forms of Aristotle 
 by a new set of eight. All but four of these eight are unnecessary, 
 but we can easily show that the four that are essential to the 
 Hamiltonian system are logically equivalent to the four tradi- 
 tional ones, — i.e., each member of the new set can be represented 
 in the members of the old set and conversely. In the proposi- 
 tions, A, E, I, O, the word some, which is explicitly stated before 
 the subject of I and O and which is understood but not expressed 
 before the predicate of A and I, means some at least, possibly all. 
 Should we understand the word some to mean some at least, 
 not all, the manner in which the subject a is related to the predi- 
 cate b in each one of tihe diagrams of Fig. 5, can be expressed in a 
 simple verbal phrase. Thus, 
 
Young Man without a Master 
 
 All a is all b, 
 = a{ab) 
 
 Fig. 6. 
 
 Some a is some b, 
 
 Fig. 7. 
 
 All a is some b, 
 = y(flb) 
 
 Fig. 8. 
 
 No a is b, 
 = e(ab) 
 
 Fig. 9. 
 
 All b is some a, 
 = y(ba) 
 
 Fig. 10. 
 
 These four forms, which we have represented by a, #, 7 and e, 
 are the ones, which Hamilton insisted should be substituted for 
 the traditional ones, A, E, I, O. You should have little difficulty, 
 in the light of what has gone before, in understanding the follow- 
 ing equalities, which represent each member of the new set in 
 the members of the Aristotelian set and conversely : 
 
io Letters on Logic to a 
 
 A(ab) = a(ab) + y(ab) (see Figs, i, 6, 8), 
 
 E(ab) = e(ab) (see Figs. 3, 9), 
 
 l(ab) = a(ab) + fi(ab) + y(ab) + 7(60) (see Figs. 2, 6, 7, 8, 10), 
 
 0(ab) = e(ab) + /3(a&) + 7(60) (see Figs. 4, 7, 9, 10), 
 
 a(ab) = A(ab)-A(ba), 
 p(ab) = l(a&)-0(a&)-0(&a), 
 7 (a&) = A(a&).0(6a), 
 e(a&) = E(ab). 
 
 While these two sets of equalities express a certain equivalence 
 between the new and the old ways of formulating the same 
 system of inference, each set would be found to possess certain 
 advantages peculiar to itself. 
 
 You have now seen that the categorical forms, A, E, I, O, 
 are composed of the terms, (a and b), an adjective of quantity 
 {all, some, not all), and the copula (is), and that the word some 
 is always to be interpreted to mean some at least, possibly all 
 (this meaning of the word being unambiguously forced on us by 
 the propositions, which we say shall be true or untrue in our 
 science). In our next letter we shall begin to acquaint you 
 with some of the simpler types of inference^ — i.e., with those 
 types of proposition, into which the hypothetical relationship, 
 (if, then) implies, enters. 
 
Young Man without a Master ii 
 
 III 
 
 In the hypothetical proposition, x implies y, (If x is true then 
 y is true), or, in our abbreviated notation, x z y, the part, (x), 
 to the left of the implication sign, (/), is called the antecedent 
 and the part, (y), to the right of the implication sign, (/), is 
 called the consequent. 
 
 Here x or y may represent any sort of proposition, but, if each 
 one happens to stand for a single categorical form, then we should 
 replace x z y by the more definite notation, x(a, b) z y(a, b). 
 Any implication of this specific type is known as immediate 
 inference. 
 
 You will recall that the comma in the bracket between the 
 terms is used in order to indicate that the term-order is not settled. 
 The proposition, x(a, b) z y(a, b), may have either one of two 
 forms. If the term-order in the antecedent is the same as the 
 term-order in the consequent, i.e., if x{a, b) z y(a, b) be written 
 
 either (a) x(ab) Z y(ab), 
 or (b) x(ba) z y(ba), 
 
 then x(a, b) Z y(a, b) is said to be expressed in the first figure of 
 immediate inference. If the term-order in the antecedent is the 
 reverse of the term-order in the consequent, i.e., if x(a, b) Z y(a, b) 
 be written 
 
 either (c) x(ab) Z y(ba) f 
 
 or (d) x(ba) Z y(ab), 
 
 then x(a, b) z y(a, b) is said to be expressed in the second figure 
 of immediate inference. 
 
 Just as the comma between the terms means that the term- 
 order is not settled, so the x in x(ab) and the y in y{ab) is used to 
 indicate that the single categorical form, for which x(ab) or y(ab) 
 stands, is not specified. By giving specific values to x and y 
 we shall obtain sixteen distinct propositions and these sixteen 
 will be all that exist of the form, x(a, b) z y(a, b), i.e., 
 
 AO, b) z A(a, b) E(a, b) z A(a, b) 
 
 A(a, b) z E(a, b) E(a, b) z E(a, b) 
 
 A(a,b) Z l(a,b) E(a,b) z I (a, b) 
 
 A(a, b) z 0(a, b) E(a, b) z 0(a, b) 
 
12 Letters on Logic to a 
 
 l(a, ft) / A(a, ft) 0(a, b) z A(a, b) 
 
 I (a, 6) Z E(a, 6) 0(a, ft) z E(a, b) 
 
 I (a, 6) z I (a, b) 0(a, ft) Z I (a, ft) 
 
 I (a, ft) Z 0(a, ft) 0(a, ft) Z 0(a, ft) 
 
 These are obtained by taking all the permutations of the four 
 letters, A, E, I, O, two at a time and by taking each letter once 
 with itself. It will be convenient from time to time to leave 
 unexpressed the implication sign, (z), and the part, (a, ft), 
 and to write down the same set of sixteen implications in the 
 following more abbreviated fashion. 
 
 AA EA IA OA 
 
 AE EE IE OE 
 
 AI EI II 01 
 
 AO EO 10 00 
 
 Each proposition of the set may be expressed in either the 
 first or in the second figure and there are, consequently, thirty- 
 two possible propositions, x(a, ft) Z y(a, b). The entire set of 
 thirty-two is said to constitute the array of immediate inference. 
 Each member of the array is called a mood of the array. The 
 true propositions of the array are called valid moods of the array. 
 The remaining moods are called invalid moods of the array. 
 
 I shall now ask you to construct the array for yourself and to 
 pick out by inspection the valid moods in both the first and 
 second figures. For convenience of reference Fig. 5 is again 
 
 Fig. 5. 
 
 placed on the page before you and it is to this that you must 
 continually direct your attention. You will discover that the 
 array contains ten valid and twenty-two invalid moods. Three 
 illustrations of how to apply the method of inspection will furnish 
 you with a clue to the whole exercise, which I propose. 
 
 (1) Consider the mood, E(aft) z O(aft), or, in our abbreviated 
 notation, EO in the first figure. This asserts that if a is related 
 to ft as in the fifth diagram (Fig. 5), then a is related to ft either 
 
Young Man without a Master 
 
 13 
 
 as in the third diagram (Fig. 5) or as in the fourth diagram 
 (Fig- 5) or as in the fifth diagram (Fig. 5). It is intuitively 
 evident, then, that EO in the first figure is a valid mood. 
 
 (2) Consider the mood, I(ab) Z I(ba), i.e., II in the second 
 figure. 
 
 \(ab) is represented by 
 
 Either 
 
 or 
 
 or 
 
 and I (ba) is represented by 
 
 or 
 
 These two modes of representation are identical, except that 
 the diagrams do not appear in the same order. But, since the 
 order, in which the diagrams appear, is irrevelant, it is intuitively 
 clear that the mood is valid. If we had chosen to consider the 
 mood, AA in the second figure, it would have appeared at once 
 that the diagrammatic representation of the antecedent would 
 not have been the same as that of the consequent and that 
 the mood is invalid. 
 
 In general, if the mood x(ab) / x(ba) is valid, then x(ab) is 
 said to be a convertible form. The operation of simple conversion 
 
14 Letters on Logic to a 
 
 consists in the interchange of subject and predicate. You will 
 discover that this operation is permissible in the case of E(ab) 
 and \{ab) but not in the case of A(ab) or 0(ab). Employing 
 this language, it is customary to say that E{ab) and \{ab) are 
 convertible forms or that E(ab) and l(ab) are simply convertible. 
 
 (3) Consider the mood 0(ab) z E(ba), or OE in the second 
 figure. If a and b are related as in the third diagram (Fig. 5) 
 or as in the fourth diagram (Fig. 5), then 0(ab) is true and E(ba) 
 is false. Consequently the mood is invalid. 
 
 This case leads us to make an important observation. Since 
 true means necessarily true and untrue means not necessarily true, 
 it is enough to point out one diagrammatic representation of the 
 antecedent, which at the same time is not a diagrammatic repre- 
 sentation of the consequent, in order to become aware that the 
 mood is invalid. 
 
 It is said of certain treatises of the Hindoos on geometry, 
 that the master, instead of offering a proof of the separate 
 theorems, was content, after stating the proposition, to draw the 
 figure and write under it the word, "Ecce." The pupil was thus 
 expected to gather intuitively the abstract or general truth from 
 the observation of a single illustration. You are well aware that 
 the ideal of the Greek geometers was to deduce the theorems of 
 the science from the fewest possible number of initial assumptions. 
 Whether this ideal be a mistaken one or not, it has at least 
 inspired the procedure of all science to the present day. For two 
 thousand years the mathematical genius of the race was spent 
 in the effort to show that the truth of the fifth postulate of 
 Euclid could be made to depend on that of the other four, and 
 the proof of the independence of this fifth postulate emancipated 
 mathematical speculation from many of the misapprehensions, 
 which had previously stood in the path of its progress. 
 
 If we were to apply this historical contrast of the Greek and 
 the Hindoo geometers to ourselves, we might say that up to now 
 our study of logic has been carried out on the Indian plan. Up 
 to now we have been Hindoo logicians, for we have been content 
 merely to write "Ecce" beneath the diagrams of Fig. 5 — a sort 
 of Cartesian test, an application of the dare et distincte percipio. 
 But from this moment forth we shall fashion our doctrine after 
 the Helladian model. We shall deduce all the true and all the 
 untrue variants of immediate inference by the aid of certain 
 principles from the fewest possible number of initial postulates. 
 
Young Man without a Master 15 
 
 The meaning of the symbol (0) has been explained already. 
 You will do well, however, in interpreting the postulates, which 
 follow, to translate it by the words, an impossibility is true. 
 The first implication below will read, accordingly, if A{ab) is 
 true and 0(ab) is true, then an impossibility is true. 
 
 Postulate 1. — A(ab)-0(ab) z 0, 
 
 Postulate 2. .—A'(ab)-0'(ab) Z 0, 
 
 Postulates- — E(ab)-l(ab) z 0, 
 
 Postulate 4. — E'(ab)'I'(ab) Z 0. 
 
 Definition. — Two propositions, which cannot both be true and 
 cannot both be false, are said to be contradictory. By postulates 
 1-4 it follows that A(ab) and O(ob) and that E(ab) and \{ab) 
 are contradictory pairs. 
 
 Principle i. — If in any valid mood the antecedent and the 
 consequent be interchanged and each be replaced by its contra- 
 dictory, a valid mood will result. 
 
 Postulate 5. — A(ab) Z A{ab) is a valid mood, 
 Postulate 6. — A(ab) Z l(ab) is a valid mood, 
 Postulate 7. — \{ab) z l(ba) is a valid mood. 
 
 Theorem 1. — 0(ab) z 0(ab) is a valid mood (from postulate 5 
 upon application of principle i), 
 
 Theorem 2. — E(ab) z 0(ab) is a valid mood (from postulate 6 
 and principle i), 
 
 Theorem 3. — E(ba) Z E(ab) is a valid mood (from postulate 7 
 and principle i) . 
 
 Definition. — If x Z y is a valid implication, then x is said to 
 be a strengthened form of y and y is said to be a weakened form 
 of x. By postulate 6, A(ab) is. a strengthened form of \{ab) and 
 l(ab) is a weakened form of A(ab); by postulate 7, I(a6) is a 
 strengthened form of I (ba) and I iba) is a weakened form of \{ab) ; 
 by theorem 2, E(ab) is a strengthened form of 0(ab) and O(afr) 
 is a weakened form of E(a6); and by theorem 3, E(ba) is a 
 strengthened form of E(ab) and E(a6) is a weakened form of 
 E(ba). 
 
 Principle ii. — If in any valid mood an antecedent be 
 strengthened or a consequent be weakened, a valid mood will 
 result. 
 
 Theorem 4. — A(ab) z I (ba) is a valid mood (by weakening 
 the consequent of postultae 6). 
 
1 6 Letters on Logic to a 
 
 Theorem 5. — E(ba) Z 0(ab) is a valid mood (by strengthening 
 the antecedent of theorem 2 or by contradicting and inter- 
 changing antecedent and consequent in theorem 4) . 
 
 Theorem 6. — l(ab) z l(ab) is a valid mood (by strengthening 
 the antecedent or by weakening the consequent in postulate 7). 
 
 Theorem 7. — E(ab) Z E(ab) is a valid mood (by strengthening 
 the antecedent or by weakening the consequent in theorem 3 or 
 by contradicting and interchanging antecedent and consequent 
 in theorem 6). 
 
 We have, accordingly, by postulating the validity of three of 
 the moods of immediate inference, deduced the remaining seven 
 by the aid of two principles. The deduction of the invalid 
 moods I shall leave to you as an exercise. Since it will be neces- 
 sary to postulate four of these moods as invalid, you will have 
 eighteen theorems to deduce. The postulates and the principles 
 of deduction are given below. It is only necessary to add that 
 the additional results of theorems 4-7 (above) must be kept in 
 mind, when you come to apply principle iv (below). 
 
 Postulate 8. — A(ab) Z A(ba) is an invalid mood, 
 Postulate 9. — A(ab) Z 0(ba) is an invalid mood, 
 Postulate 10. — A(ab) Z 0(ab) is an invalid mood, 
 Postulate 11. — E(ab) Z l(ab) is an invalid mood. 
 
 Principle iii. — If in any invalid mood the antecedent and the 
 consequent be interchanged and each be replaced by its contra- 
 dictory, an invalid mood will result. 
 
 Principle iv. — If in any invalid mood an antecedent be 
 weakened or a consequent be strengthened, an invalid mood will 
 result. 
 
 Theorems. — The other (18) invalid moods. 
 
 I shall conclude this letter by introducing you to a form of 
 implication, which is closely allied to immediate inference, and 
 I shall ask you to pick out by inspection (by a reference again to 
 the diagrams of Fig. 5) the valid and the invalid moods. 
 
 Already (postulates 1-4) we have had to interpret specific 
 instances of the hypothetical proposition, x(a, b)y(a, b) Z 0. In 
 constructing its array you have only to notice that the order, in 
 which the two categorical forms conjoined in the antecedent 
 occur, is indifferent, which was not true in the case of 
 x{a, b) Z y(a, b) ; i.e., if x (is true) and y (is true), then y (is true) 
 
Young Man without a Master 17 
 
 and x (is true) — if two propositions are represented as true to- 
 gether, then this representation may be expressed with the 
 propositions in either order. Employing a more technical lan- 
 guage, we should say that logical multiplication is commutative. 
 There will, accordingly, be fewer distinct moods in the array, 
 x(a, b)y(a, b) /_ 0, than in the array of immediate inference. 
 
 In order to establish the intuitive validity of any mood it will 
 be enough to understand that x(a, b) and y(a, b) cannot be repre- 
 sented in any diagram as true together. The intuitive invalidity 
 of any mood will appear, when at least one diagram represents 
 x(a, b) and y(a, b) as true together, i.e., if x (is true) and y 
 (is true) does not imply an impossibility, then the mood is invalid. 
 Clearly a valid mood will result, whenever the representations 
 of each one of the two forms conjoined in the antecedent do not 
 overlap in Fig. 5, as in the case of A(ab)K(ab) /_ 0. Otherwise 
 an invalid mood will result, as in the case of l(ab)0(ab) Z 0. 
 When you have finished the exercise, which I have proposed to 
 you, you will have found that the array contains twenty distinct 
 moods, of which five are valid and fifteen invalid. At another 
 time we shall show how the moods of this array may be deduced 
 by the aid of a principle, which will be introduced later on. 
 
1 8 Letters on Logic to a 
 
 IV 
 
 We have now to study an array of a more general character 
 than that of immediate inference and I shall begin, not by de- 
 scribing it in abstract terms but by directing your attention to a 
 few specific instances. 
 
 Consider the proposition, A(ba)A(cb) / A(ca). Suppose that 
 we desire to represent the antecedent as a whole. The diagrams 
 below will evidently exhaust all the modes of expression that are 
 possible. 
 
 Either 
 
 or 
 
 or 
 
 You will observe that each one of the four ways of representing 
 the antecedent is at the same time a way of representing the conse- 
 quent. Accordingly if a, b and c are related as in the antecedent, 
 then it follows that a and c are related as in the consequent, 
 so that the proposition, A(ba)A(cb) /_ A(ca), is a valid implica- 
 tion. 
 
 The rule for constructing the diagrams, which represent the 
 antecedent as a whole, is this: If the second form in the ante- 
 cedent has (say) three modes of representation, then represent 
 the first form completely three times (on three separate lines) and 
 add to the first line the first way of representing the second form 
 in the antecedent, to the second line the second way, to the third 
 line the third way. The antecedent will then be completely 
 represented as a whole. 
 
 For example, consider A(ab)0(cb) /. 0(ca). Since 0(cb) is 
 represented in three ways, we represent A(ab) three times, thus: 
 
Young Man without a Master 
 
 19 
 
 Now supply to the first line the first way of representing 0(cb), i.e., 
 either 1 I or 
 
 wj 
 
 and to the second line the second way of representing 0(cb), i.e., 
 
 either 
 
 or 
 
 and, finally, to the third line the third way of representing O(cb), 
 i.e., 
 
 either 
 
 c a 
 
 b or 
 
20 
 
 Letters on Logic to a 
 
 You will perceive at once that each separate manner of denoting 
 a, b and c as related in the antecedent, is also a manner of denoting 
 a and c as related in the consequent so that it is intuitively 
 evident as in the last illustration that the implication is valid. 
 It will be necessary, perhaps, for you to study the more com- 
 plicated diagram given above for some time, in order to satisfy 
 yourself that, together with the others, it exhausts all of the 
 possibilities that there are. 
 
 Since we have understood true to mean necessarily true and so un- 
 true to mean not necessarily true, in order to perceive the invalidity 
 of any proposition of the form under consideration, it will be 
 enough to point to a single representation of the antecedent, 
 which at the same time is not a representation of the consequent. 
 
 Consider the proposition, E(ba)A(bc) z E(ca). The complete 
 representation of the antecedent is: 
 
 either & 
 
 or 
 
 But in the last diagram we have two separate instances of the 
 untruth of E(ca). Consequently, the implication is invalid. 
 
 The first example, which we examined above, was A(ba)A(cb) 
 Z A(ca). Consider now A(ab)A(bc) Z A(ca) and be careful to 
 notice that the term-order, which this form presents, is not the 
 same as that of the one first mentioned. The complete repre- 
 sentation of the antecedent is 
 
 or 
 
 or 
 
Young Man without a Master 21 
 
 and you will observe three distinct instances among these dia- 
 grams of the untruth of the consequent, so that the implication 
 is invalid, i.e., the validity of an implication of the type under 
 consideration depends not only upon the particular categorical 
 forms which enter into it, but also upon the particular manner, in 
 which the terms are arranged. We shall now determine all the 
 possible ways of arranging the terms. These will evidently be 
 not more than eight in number, viz., 
 
 ba 
 
 ab 
 
 ba 
 
 ab 
 
 cb 
 
 cb 
 
 be 
 
 be 
 
 ca 
 
 ca 
 
 ca 
 
 ca 
 
 ba 
 
 ab 
 
 ba 
 
 ab 
 
 cb 
 
 cb 
 
 be 
 
 be 
 
 ac ac ac ac 
 
 In our last letter we spoke of the conjunctive relation of logic 
 as being commutative. The two categorical forms conjoined 
 in the antecedent may therefore be written in either order and 
 we may, if we wish, always write a specific one of the two first. 
 We agree, as a matter of convention, always to write first the form, 
 which contains the predicate of the consequent, thus, A(ba)A(cb) 
 Z. A(ca), not A(cb)A{ba) £ A{ca). To accord with this con- 
 vention, the second line above will have to be rearranged thus, 
 
 cb 
 
 cb 
 
 be 
 
 be 
 
 ba 
 ac 
 
 ab 
 ac 
 
 ba 
 ac 
 
 ab 
 ac 
 
 We shall now show that the arrangements of this set are only a 
 repetition of those in the first line above, but in a different order, 
 so that there will turn out to be only four distinct ways of arrang- 
 ing the terms. 
 
 Suppose that we were to draw two lines, one connecting the 
 terms in the categorical form written first in the antecedent and 
 another connecting the term, which does not appear in the 
 consequent. Then the eight varieties of term-order will appear 
 thus: 
 
22 
 
 Letters on Logic to a 
 
 It becomes apparent that the arrangements in the second re 
 are only a restatement of those in the first, for the figures 
 
 give a very clear geometrical image of the number of possible 
 term-orders. You will do well to commit to memory at once 
 the four variations in the first line, which we shall constantly 
 refer to as figures I, 2, 3 and 4, respectively. The four figures 
 are easily remembered as combined in an isosceles triangle 
 standing on its vertex (see below). 
 
 We proceed now to summarize these results and to define a 
 certain number of technical terms. 
 
 The syllogism is a form of implication belonging to one of the 
 types, 
 
 1. x(ba)y(cb) Z z(ca), 
 
 2. x(ab)y(cb) z z(ca), 
 
 3. x(ba)y(bc) Z z(ca), 
 
 4. x(ab)y(bc) z z{ca). 
 
 These differences are known as the first, second, third, and 
 fourth figures of the syllogism respectively. The two forms 
 conjoined in the antecedent are called the premises and the 
 consequent is called the conclusion. The predicate of the con- 
 clusion is called the major term and points out the major premise, 
 which by convention is written first in the antecedent. The 
 
Young Man without a Master 23 
 
 subject of the conclusion is called the minor term and points out 
 the minor premise. The term, which is common to the premises 
 and which does not appear in the conclusion, is called the middle 
 term. 
 
 Since x, y and z may take on any one of the four forms, A, E, 
 I, O, there will be sixty-four syllogistic variations of the form, 
 x(a, b)y{b, c) Z z(ca), obtained from the permutations of the 
 four letters taken three at a time. Each one of these sixty-four 
 variations may be expressed in each one of the four figures, so 
 that we shall have two hundred and fifty-six cases to consider. 
 Each one of these two hundred and fifty-six cases are known as 
 moods of the array x(a f b)y(h, c) / z(ca). True propositions of 
 the array are known as valid moods of the array. Those remain- 
 ing are known as invalid moods of the array. 
 
 In representing the array of the syllogism, it will prove con- 
 venient, as in the case of immediate inference, to omit the 
 symbol, (z)» and the parts, (b, a), (c, b), (ca), and to exhibit 
 each mood as a simple combination of the three letters. The best 
 method for you to employ will be to add to each one of the sixteen 
 permutations of the four letters, A, E, I, O, taken two at a time, 
 each one of the four letters in succession. The array under each 
 figure will then appear thus: 
 
 AAA 
 
 EAA 
 
 IAA 
 
 OAA 
 
 E 
 
 E 
 
 E 
 
 E 
 
 I 
 
 I 
 
 I 
 
 I 
 
 
 
 
 
 
 
 
 
 AEA 
 
 EEA 
 
 IEA 
 
 OEA 
 
 E 
 
 E 
 
 E 
 
 E 
 
 I 
 
 I 
 
 I 
 
 I 
 
 
 
 
 
 
 
 
 
 AIA 
 
 EIA 
 
 IIA 
 
 OIA 
 
 E 
 
 E 
 
 E 
 
 E 
 
 I 
 
 I 
 
 I 
 
 I 
 
 
 
 
 
 
 
 
 
 AOA 
 
 EOA 
 
 IOA 
 
 OOA 
 
 E 
 
 E 
 
 E 
 
 E 
 
 I 
 
 I 
 
 I 
 
 I 
 
 
 
 
 
 
 
 
 
24 Letters on Logic to a 
 
 I shall now ask you to construct the array and to examine each 
 member of it in each one of the four figures, in order to determine 
 the validity or invalidity of the mood in question, by the method 
 of inspection. I shall furnish you no clue as to which moods 
 are valid, except to remark that six true propositions will be 
 found under each one of the four figures. 
 
 That A(ba)A(cb)0(ca) Z o is a true implication will be intui- 
 tively clear to you, as soon as you have ascertained by trial that 
 the three forms conjoined in the antecedent cannot be represented 
 in any diagram as true altogether, i.e., the product of AAO in the 
 first figure does imply an impossibility. 
 
 If you were to construct the array corresponding to 
 x(a, b)y(b, c)z(c, a) Z o by taking the permutations of the four 
 letters, A, E, I, O, three at a time, you would discover that not 
 all of the two hundred and fifty-six moods so obtained are distinct. 
 By way of illustration consider the three valid moods, 
 
 A(ba)A(cb)0(ca) Z o, 
 A(ab)0(cb)A(ca) z o, 
 0(ba)A(bc)A(ca) Z o, 
 
 Construct a triangle, with the term a at the end of the base to 
 the right, the term c at the left and the term b at the vertex 
 above. Let the arrow indicate the direction of "flow" from 
 subject to predicate, or the order subject-predicate. Then the 
 three moods will be represented as in the figures below. 
 
 b b 
 
 A 
 
 a 
 
 Now slide the second and the third figures around so that O 
 will appear on the base, thus : 
 
 b 
 A 
 
 a 
 
 You will notice that in each instance the direction of "flow," 
 as indicated by the arrows, is continuous and in one direction 
 
Young Man without a Master 25 
 
 from the subject of O to the predicate of O. The formal identity 
 of the three cases will appear more clearly, if the second and 
 third figures be taken out of the plane of the paper and turned 
 over, thus: 
 
 In this example we have considered the case of three valid 
 moods, which at first blush appeared to be distinct but which 
 turned out to be identical. Suppose, by way of further illustra- 
 tion of this geometric method of ascertaining sameness and differ- 
 ence, we compare four invalid moods, viz., A(b, a)A(c, b)A(c, a) 
 Z 0, in each one of the four figures. 
 
 In order to show that each one of these implications is invalid 
 it is enough to point out that the three forms conjoined in 
 antecedent can be represented as true together in each figure, 
 when the circles that stand for a, b and c are the same. Em- 
 ploying again the image of the triangles, we should have: 
 
 b 
 
 b y i 
 
 A a x \A x u 
 
 A 
 
 c J7 a c a c A* s a c a a 
 
 Here the second and the third figures may be moved around as 
 before so as to show an identity with the first, i.e., a one-direc- 
 tional "flow" from the subject of one of the A-s to its predicate, 
 but the fourth cannot by any moving about be made to appear 
 as other than a clockwise or a counter-clockwise "flow." Ac- 
 cordingly, of the four apparent differences, with which we began, 
 only two remain. 
 
 The exercise, which I now propose to you, is to construct the 
 array, x(a, b)y(b, c)z(c, a) /_ 0, and to pick out the valid moods 
 by the method of inspection. You will recall, that in order to 
 establish the invalidity of any mood, it will be enough to point 
 to a single diagram, which represents the three forms conjoined 
 in the antecedent as true together. Afterward you should reduce 
 
26 Letters on Logic to a 
 
 the apparent to the essential differences by the aid of the tri- 
 angles, as described above. Later on we shall show how all the 
 valid and invalid moods of this array, as well as those of the 
 syllogism, may be deduced, as in the case of immediate inference, 
 by postulate and principle. 
 
Young Man without a Master 27 
 
 V. 
 
 I take it for granted that you have made a list of the valid 
 moods of the syllogism, having applied the method of inspection 
 to the two hundred and fifty-six possible cases. In order that 
 you may verify your results, the six that are valid under each 
 figure are placed on the page before you. They are : 
 
 I 
 
 II 
 
 III 
 
 IV 
 
 AAA 
 
 AEE 
 
 AAI 
 
 AAI 
 
 AAI 
 
 AEO 
 
 All 
 
 AEE 
 
 All 
 
 AOO 
 
 EAO 
 
 AEO 
 
 EAE 
 
 EAE 
 
 EIO 
 
 EAO 
 
 EAO 
 
 EAO 
 
 IAI 
 
 EIO 
 
 EIO 
 
 EIO 
 
 OAO 
 
 IAI 
 
 We shall now, as in the case of immediate inference, by postu- 
 lating the truth of the smallest possible number of these moods, 
 deduce the remainder by the aid of two principles. The assump- 
 tions, which we shall have to make, are as follows : 
 
 Postulate 1. A(ba)A(cb) z A(ca) is a valid mood, 
 Postulate 2. E(ab)A(cb) Z E(ca) is a valid mood, 
 
 Principle i. If in any valid mood either premise and the con- 
 clusion be interchanged and each be replaced by its contradic- 
 tory, a valid mood will result, 
 
 Principle ii. If in any valid mood a premise be strengthened 
 or the conclusion be weakened, a valid mood will result, 
 
 Theorems. — The remaining (22) valid moods. 
 
 When you have carefully studied the examples, which I shall 
 set down below, you should be able to carry out the entire deduc- 
 tion without further aid, and the work of doing this should have 
 been completed before reading the remainder of the letter. 
 
 (1) Suppose that we were to combine the first postulate and 
 the first principle. Interchanging the minor premise and the 
 conclusion of A(ba)A(cb) Z A{ca) and replacing each form by its 
 contradictory, we obtain A{ba)0{ca) Z 0(cb). You will notice 
 that the major term is b, the minor term is c, and that the middle 
 
28 Letters on Logic to a 
 
 term has become a. The figure is now determined in the way 
 already described, viz. 
 
 b 
 
 i.e., AOO in the second figure is a valid mood. 
 
 Similarly, by contradicting and interchanging the major prem- 
 ise and the conclusion and replacing each by its contradictory, 
 we should have obtained the theorem : 
 
 OAO in the third figure is a valid mood. 
 You must in every case, before examining the figure, be sure that 
 the major premise has been written first. 
 
 (2) AOO in the second figure being now established as a valid 
 mood, we may apply to it either one of the principles in the same 
 sense as to the postulates. Let us begin by writing the mood 
 with the terms ordered as in the original convention, i.e., 
 A(ab)0(cb) z 0(ca), and, applying principle ii, let us strengthen 
 the minor premise O(cb) to E(bc). This will be possible by 
 applying a result of immediate inference, which has already been 
 established, viz., E(bc) z 0(cb). Accordingly our third theorem 
 becomes : 
 
 AEO in the fourth figure is a valid mood. 
 
 (3) Suppose that we were to turn back now to the first prin- 
 ciple and apply it to the result, which has just been obtained. 
 Contradicting major and conclusion and interchanging in 
 A(ab)E(bc) z 0(ca) we obtain immediately A (ca)E (be) z 0(ab). 
 
 It is important you should not fail to observe that the premises 
 are no longer in the normal order and that the normal order 
 must be restored, before the figure can be ascertained. Failure 
 to make this change might result, as you will readily see, not 
 only in a mistake in the mood but also in the figure. Our theorem 
 is, accordingly: 
 
 EAO in the fourth figure is a valid mood. 
 
 Had we chosen to contradict and interchange the minor and 
 conclusion of AEO in the fourth figure, we should have obtained, 
 in the same way, the theorem : 
 
 AAI in the fourth figure is a valid mood. 
 
Young Man without a Master 29 
 
 We observe in this connection a general rule to this effect: 
 the application of principle i to any mood in the fourth figure 
 places the premises out of the normal order but leaves the figure 
 unchanged. Employing a more technical language we should 
 say, that the fourth figure is invariant under principle i. 
 
 Having deduced the twenty-two theorems, you should now 
 set yourself the exercise of deriving the valid moods under each 
 figure separately and you should strive to arrive at each result 
 by the fewest possible number of steps. In deducing those 
 under the fourth figure, it will economize steps and so add to the 
 elegance of your demonstration, if you keep in mind the rule, 
 which has been stated in the last paragraph. The following 
 rules, whose correctness you will do well to verify for yourself, 
 show the effect on mood and figure of contradicting and inter- 
 changing either premise and the conclusion. 
 
 Major Premise and Conclusion 
 
 (1) The first figure changes to the third and conversely and 
 the premises remain in normal order. 
 
 (2) The second figure changes to the third with the normal 
 order of the premises reversed. 
 
 (3) The fourth figure remains invariant with the normal order 
 of the premises reversed. 
 
 Minor Premise and Conclusion 
 
 (1) The first figure changes to the second and conversely and 
 the premises remain in normal order. 
 
 (2) The third figure changes to the second with the normal 
 order of the premises reversed. 
 
 (3) The fourth figure remains invariant with the normal order 
 of the premises reversed. 
 
 It will also be found advantageous to state in the form of rules 
 the effect of simple conversion in either premise or in the con- 
 clusion. These rules are: 
 
 (1) Simple conversion in the major premise changes the first 
 figure to the second and conversely, the third figure to the fourth 
 and conversely, 
 
 (2) Simple conversion in the minor premise changes the first 
 figure to the third and conversely, the second figure to the fourth 
 and conversely, 
 
30 Letters on Logic to a 
 
 (3) Simple conversion in the conclusion changes the first 
 figure to the fourth and conversely and leaves the second and 
 third figures unchanged. 
 
 The symbol, (0), we have employed to denote a proposition 
 that is false for all meanings of the terms. The definition of 
 zero, (0), is given by the following implications: 
 
 Z*', (o'zo)'. 
 
 It is usual to employ the symbol, (i), instead of 0', which accord- 
 ingly, stands for a proposition that is true for all meanings of 
 the terms. With this substitution the definition of zero, or, as 
 it is sometimes called, the null-proposition, becomes: 
 
 Z i, (i Z 0)'. 
 
 The one-proposition (i.e., i) is so called, because it acts like a 
 unit multiplier in ordinary algebra. When it appears as & factor, 
 that is, when it appears conjoined with one or more propositions, 
 it is usually not expressed, since it neither adds to nor subtracts 
 from the information contained in the product. Its full meaning 
 will appear, when the use to which it may be put is once realized. 
 In order to illustrate this use as well as to obtain certain con- 
 structive results, we shall assume the following postulates, which 
 are to play a very important r61e in our subsequent theory. 
 
 Postulates: i Z A(aa), i z I(aa). 
 
 These postulates mean nothing more than that A{aa) and \{aa) 
 are to be regarded as true propositions for all meanings of a. 
 They may be interpreted to read : it is necessarily true that all a 
 is a, it is necessarily true that some a is a. 
 
 We shall now introduce a principle, which has been tacitly 
 assumed up to now and which we shall have to make use of 
 subsequently. 
 
 Principle. — A valid implication will remain valid, when as 
 many terms have been identified as we desire. 
 
 Two illustrations of the use, to which this new concept of the 
 one-proposition may be put, are set down below. 
 
 (1) By the principle above the valid syllogism, A(ba)A(cb) 
 Z A(ca), will remain valid, when the terms in the major premise 
 have been identified. Accordingly, A(aa)A(ca) Z A{ca) is a 
 valid implication. Now strengthen the part, A(aa), to i by the 
 
Young Man without a Master 31 
 
 first postulate above, and we have i-A(ca) Z A(ca). But the i, 
 as already explained, may be omitted. The part that remains, 
 A(ca) / A(ca), is one of the postulates under the valid moods 
 of immediate inference. This postulate has, consequently, been 
 saved, since its validity has been made to depend upon that of 
 another, which was later introduced. 
 
 (2) By the same principle above the valid syllogism, E(ab)A(cb) 
 Z E(ca), will remain valid, when the terms in the minor premise 
 
 have been identified. Accordingly, E(ab)A(bb) Z E(ba) is valid 
 and this becomes E(ab) z E(ba), when we suppress the part, 
 A(bb) f as before. E(ab) z E(ba) yields I (ba) z I(ab), when 
 antecedent and consequent are contradicted and interchanged, 
 and the truth of this last result, one of the moods postulated 
 under immediate inference, has, accordingly, been made to de- 
 pend upon that of one of the moods postulated later on. 
 
 The only assumption under the valid moods of immediate 
 inference that remains is, consequently, A{ab) Z l(ab). It 
 should be observed in this connection that one of the postulates 
 just enunciated can be made to depend upon the other. For 
 A(ab) Z I (#6) yields A(aa) Z l(aa), for a = b. Now weaken 
 the consequent of i Z A{aa) to \{aa) and we have the theorem 
 i z l(aa). 
 
 Similarly, if we assume 
 
 Postulate: 0(aa) z 0, 
 then there follows immediately the 
 
 Theorem: E{aa) z 0, 
 which is obtained by strengthening the antecedent in the postu- 
 late — E(aa) Z 0(aa) being obtained from E(ab) Z 0(ab) through 
 the identification of a and b. 
 
 The use, to which E{aa) z and 0(aa) z may be put, will 
 be illustrated by a reduction of the number of postulates for the 
 deduction of the invalid moods of immediate inference. 
 
 (3) Suppose that A{ab) Z 0(ab) and A(ab) Z 0(ba) were 
 valid implications; then A(aa) Z 0(aa) would be a valid implica- 
 tion by the principle, which permits of the identification of the 
 terms. If the antecedent of A(aa) z 0(aa) be strengthened to i t 
 then i z 0{aa) results, and if the consequent of i Z 0(aa) be 
 weakened to 0, then i /. results. Consequently, if A{ab) 
 
 Z 0{ab) and A(ab) Z 0(ba) were valid, then i /_ would be 
 
32 Letters on Logic to a 
 
 valid. But (i Z o)' is part of the definition of o. Accordingly, 
 A(ab) Z O(ab) and A(ab) Z 0(ba) are invalid and two of the 
 postulates for the derivation of the invalid moods of immediate 
 inference have been saved. 
 
 It is important, in this connection, that you should become 
 aware of the fact, that, while a valid implication must remain 
 true for all special values of the terms (for example, when any 
 number of the terms have been identified), it is not true that an 
 invalid implication will remain invalid for all special values of 
 the terms. Thus, l(ab) Z A(ab) is invalid, but it becomes valid 
 for a = b. This fact underlies all of our theory. You will 
 easily grasp its significance, when I remind you that true (valid) 
 means necessarily true, true in general, or true for all special 
 cases; whereas, untrue (invalid) means not necessarily true, not 
 true in general, i.e., there exist at least some instances of the 
 untruth of the implication in question, although not all special 
 cases are necessarily such instances. 
 
 I propose now, as an exercise, that you should examine again 
 the invalid moods of immediate inference in order to determine 
 in how many cases their invalidity may be established by reduc- 
 tion to the form i z o, as explained above. 
 
 The introduction of the null-proposition and its contradictory 
 the one-proposition has now made it possible for us to deduce 
 the valid moods of the two arrays, x(a, b)y(a, b) Z o and 
 x(a, b)y(b, c)z(c, a) z o. The examples, which are added below, 
 will furnish you the clue to the whole derivation. It will only 
 be necessary to point out the reason for identifying i with the 
 contradictory of o and conversely o with the contradictory of i. 
 This will appear evident, if we assume (o')' = o, which is a 
 specific case of (#')' ~' x > an equality, which is universally true 
 in logic. Thus, since i = o' by convention, i' = {o')' '— o. 
 
 (i) It has already been pointed out that the one-proposition 
 may be conjoined to any other proposition; or suppressed, when 
 it appears as a factor in any logical product. Accordingly, the 
 valid mood, A{ab) z I(a#), of immediate inference may be 
 written in the form, A(ab)-i z l(ab). Now, conceiving the 
 part, (i), of the antecedent, as if it were a minor premise, apply 
 the rule for contradicting and interchanging. We obtain at once 
 A(ab)V(ab) z i', or, what is the same thing, A(ab)E(ab) Z o, 
 a valid mood of the array, x(a, b)y(a, b) Z o. It is obvious that 
 
Young Man without a Master 33 
 
 this result might have been obtained by the same process from 
 E(ab) / 0(ab). 
 
 The principle of contradiction and interchange, which was 
 employed in the deduction of the valid moods of immediate 
 inference and of syllogism, may be expressed in a more general 
 form, viz. : 
 
 If in any valid implication the consequent and any factor in the 
 antecedent be contradicted and interchanged, a valid implication 
 will result. 
 
 This principle has been tacitly assumed in the preceding ex- 
 ample. The statement of it above, its most general expression, 
 is the one we shall have to employ later, when we approach the 
 solution of the sorites, a form of implication, whose antecedent 
 contains, not two, but any number of premises. 
 
 (2) In analogy with the method of the last example, the valid 
 syllogism, A(ba)A(cb) /_ A(ca), may be written A(ba)A(cb)i 
 Z A(ca). Contradicting and interchanging the ^-factor and the 
 conclusion by the principle just enunciated, we obtain imme- 
 diately, A(ba)A(cb)Q(ca) z. 0. 
 
 Simple as the process is, you should now set yourself the task 
 of deducing all of the valid moods of the two arrays, which we 
 have just been considering, assuming the valid moods of imme- 
 diate inference and of the syllogism as a point of departure, and 
 you should further strike out the repetitions by the diagrammatic 
 method explained in the fourth letter. 
 
34 Letters on Logic to a 
 
 VI 
 
 It remains, in order to complete the solution of the syllogism, 
 to deduce all of the two hundred and thirty-two invalid variants 
 from the fewest possible number of initial assumptions. The 
 present letter will be given over to the consideration of this 
 problem. We shall find that two moods will have to be postu- 
 lated as invalid and that all of the others may be derived from 
 these or else reduced to invalid moods of immediate inference. 
 The most elegant way to proceed will be to begin with a single 
 postulate and a single principle and to introduce further assump- 
 tions only when we are compelled to do so. We introduce, 
 accordingly, 
 
 Postulate i. — E(ba)E(cb) z l(ca) is an invalid mood. 
 
 Principle i. — If in any invalid mood a premise be weakened or 
 the conclusion be strengthened, an invalid mood will result. 
 
 Let us begin by weakening, in succession, the major premise to 
 E(ab), the minor premise to E(bc) f and finally each premise to 
 E(ab) and E(bc) respectively. We shall then have established 
 by postulate and theorem the invalidity of EEI in all four figures. 
 
 If, now, the premises be weakened and the conclusion be 
 strengthened in every possible way, the untru ( th of 
 
 EEI, EOI, OEI, OOI, 
 EEA, EOA, OEA, OOA, 
 
 will have been established in each one of the four figures. The 
 invalidity of thirty-one moods has, accordingly, been made to 
 depend on that of EEI (in the first figure) alone. It should be 
 noted in this connection that the application of principle ii 
 (below) to any mood in this set of thirty- two will yield no mood 
 that is not already contained in the set; that postulate 2 (below) 
 will yield no mood of the set by either principle; and that no 
 mood of the set can be established as invalid by any of the 
 methods that are given later on. We now introduce the second 
 postulate and the second principle. 
 
 Postulate 2. — A(ab)A(cb) Z l{ca) is an invalid mood. 
 
Young Man without a Master 35 
 
 Principle ii. — If in any invalid mood either premise and the 
 conclusion be interchanged and each be replaced by its contra- 
 dictory, an invalid mood will result. 
 
 The application of this principle will offer no difficulty that 
 has not been already overcome and I have no doubt that your 
 practice in the derivation of the valid moods has been enough to 
 enable you to dispense with further illustrations here. Thus 
 we should obtain at once the theorems : 
 
 (a) A(ba)E(c, b) Z 0(ca) by 2, ii, 
 
 (b) A(fo)E(c, b) Z E(ca) by a, i, 
 
 (c) A(ab)I(c,b) Z l(ca) byb, ii, 
 
 (d) 1(6, a)A(cb) Z l(ca) by c, i, 
 
 (e) E(6, a)A(bc) Z E(ca) by d, ii. 
 
 Other moods, which follow from the second postulate and 
 whose invalidity you will easily establish in all four figures, are 
 
 EIE, IEE, IEO, III. 
 
 Of this set of twenty-six theorems, whose invalidity depends 
 on that of A(ab)A(cb) z I(ca), it can be said, that each one is 
 independent of our original set of thirty-two and that none can 
 be reduced by the methods that we are about to introduce. 
 
 You will recall that since a valid implication must remain true, 
 when as many terms have been identified as we desire, it follows 
 that the invalidity of any implication is established, whenever 
 we can point to a special instance of its being untrue. The 
 invalidity of a mood of the syllogism would be proven, accord- 
 ingly, if we could reduce it to the particular case of an invalid 
 mood of immediate inference. The examples, which I have set 
 down below, will be enough to suggest to you a general method 
 of reduction, that will yield the moods not yet resolved. 
 
 (1) Suppose that I (a, b)0(bc) Z 0{ca) were valid implications. 
 Identifying terms in the major premise and suppressing the part, 
 l(aa), (i.e., strengthening l(aa) to i), it would follow that 
 O(oc) z 0(ca) is a valid implication. But 0(ac) Z 0(ca) is 
 an invalid mood of immediate inference and, consequently, 
 I (a, b)0{bc) Z 0(ca) are invalid moods of the syllogism. 
 
 (2) By the method of the last example, A(ab)A(cb) z A(ca) 
 will reduce to A(ac) z A(ca), for b = c, and A{ba)A(bc) Z A{ca) 
 for b = a. The moods, A{ba)0{cb) z 0{ca) and 0(ba)A(cb) 
 
36 Letters on Logic to a 
 
 Z 0(ca), cannot be reduced by the method in question, but they 
 may be derived from the two moods just established by the aid 
 of principle ii. Thus, 
 
 A(ab)A(cb) Z Ate) yields A(ab)0(ca) Z 0(cb), on inter- 
 changing contradictories of minor and conclusion, and 
 
 A(ba)A(bc) Z A(ca) yields 0(ca)A(bc) z 0(ba) t on inter- 
 changing contradictories of major and conclusion. 
 
 (3) Suppose E(ba)A(cb) z Ite) were a valid mood and iden- 
 tify terms in the minor premise. The result is an invalid mood of 
 immediate inference. Accordingly, E(ba)A(cb) z Ite) is an 
 invalid mood of the syllogism. Now 
 
 E(ba)A(cb) Z Ite) yields E(ba)E(ca) Z 0(cb), on interchang- 
 ing contradictories of minor and conclusion. 
 
 This last result, whose invalidity in the other figures follows 
 at once by principle i, will yield invalid moods of the syllogism 
 that remain to be established. We obtain immediately from 
 EEO, by principle i, each one of the following moods in each 
 one of the four figures, viz. 
 
 EEE, EOE, OEE, OOE, 
 EEO, EOO, OEO, OOO. 
 
 The invalid moods of the arrays, x(a, b)y(b, c) £ and 
 x(a, b)y{b, c)z(c, a) z 0, axe gotten at once from results already 
 obtained by the principle of interchanging contradictories, as 
 illustrated in the examples below. 
 
 (1) A(ab) Z A(&a) may be written A(ab)-i Z Ate). 
 Contradicting and interchanging, there results at once 
 
 A{ab)A'(ba) z i\ or A(ab)0(ba) Z 0. 
 
 (2) A(ab)A(bc) Z Ate) may be written A(ab)A(bc)-i Z Ate). 
 Consequently, as before, A(a^)Ate)Ote) z 0. 
 
Young Man without a Master 37 
 
 VII 
 
 The type of implication, which we are now to consider, is one, 
 in which the number of terms is greater than three and, as in 
 immediate inference and syllogism, the number of premises one 
 less than the number of terms. Accordingly, it will be more 
 convenient to employ in place of the term-symbols, a, b, c, etc., 
 the ordinal numbers, 1, 2, j, etc. 
 
 The sorites is an implication of the general form : 
 
 x(i, 2)y(2, 3)z(3, 4) • • • u(n - 1, n) / w(ni), 
 
 following the convention of writing the major premise first, so 
 that the term-order in the conclusion is fixed as (ni). 
 
 We shall begin by illustrating the manner of constructing a 
 valid sorites from a chain of valid syllogisms. 
 
 (1) Suppose that we were to be given the chain of valid 
 syllogisms, 
 
 A(2/)A(32) Z A(jj), 
 A( 3 i)A(43) Z A( 4 i), 
 A( 4 I)A(S4) Z Afcz), 
 
 and were asked what valid mood of the sorites is thereby implied. 
 It is clear that the major premise of the last syllogism, being the 
 same as the conclusion of the second, may be strengthened to 
 A(ji)A(4j). The immediate result of this strengthening is a 
 valid mood of the sorites, viz., 
 
 h(3i)A{43)A{54) Z A{ 5 i). 
 
 The major premise of this last implication may in turn be 
 strengthened to A(2i)A(j2), by the first syllogism, and we have 
 
 A{2i)h{ 3 2)A{43)A{54) Z A(yj). 
 
 (2) The valid mood of the sorites, which has just been built up, 
 may in turn be reduced successively to each member of the chain, 
 upon which it depends. If the terms in the fourth premise be 
 identified, the sorites becomes 
 
 K{2i)h{ 3 2)A(43)A{44) Z A{ 4 i), 
 or, when we suppress in the usual way the part A (44), 
 A(2i)A(32)A( 43 ) z A( 4 i). 
 
38 Letters on Logic to a 
 
 Similarly, identifying terms in the last premise of the last mood, 
 we obtain 
 
 A(«)A(ja) Z A(3i), 
 
 which is the first syllogism of the chain. 
 
 The second syllogism will evidently be gotten by identifying 
 terms in the first and last premises and the third syllogism by 
 identifying terms in the first and second premises. 
 
 (3) Another method of constructing a valid mood of the sorites 
 from a chain of valid syllogisms depends upon an application of 
 the following : 
 
 Principle. — If in any valid implication the same factor be 
 conjoined to both antecedent and consequent, a valid implication 
 will result. 
 
 Let our chain of syllogisms be 
 
 E(2i)A( 3 2) z ECjj), 
 E(jz) 1(34) Z 0(4i), 
 0( 4 i)A(45) Z 0(51), 
 
 and suppose that we conjoin to antecedent and consequent of the 
 first member, E(2i)A(32) Z E(jj), the minor premise of the 
 second member, 1(34), and so obtain 
 
 E(2i)A(32)l(34) Z E( 3 i)I(34)- 
 
 The second syllogism allows us to weaken the consequent of this 
 result to 0(41). Accordingly, we obtain 
 
 E(3i)A( 3 2)I(34) Z 0(41). 
 
 Now conjoin to antecedent and consequent of this sorites the 
 minor premise of the third syllogism, A (45), i.e., 
 
 E(2i)A( 3 2)l(34)A(4S) Z 0( 4 i)A( 4 5), 
 
 and weaken the consequent of this implication to 0(51) by the 
 last member of the chain. Consequently, 
 
 E(2i)A( 3 2)\(34)A(45) Z 0( 5 i) 
 
 is the valid mood of the sorites, which was to be built up. 
 
 (4) Suppose, being given a valid mood of the sorites, we should 
 be asked to find the chain of syllogisms, upon which it depends. 
 Let the mood be 
 
 A(i2)A(2 3 )0(43)A(45)A( 5 6) z 0(6/). 
 
Young Man without a Master 39 
 
 The premises of the first syllogism of the chain will be the same 
 as the first two premises of the sorites and the minor of the 
 second syllogism will be the same as the third premise of the 
 sorites, and so on. The fragment of the chain so far ascertained 
 will be 
 
 A(i2)A(23) / — 
 
 - 0(43) Z - 
 
 - A(45) Z - 
 
 - A( S 6) z - 
 
 Now the conclusion of the first syllogism — whose premises appear 
 out of the normal order — which is evidently A (13), must be the 
 major of the following syllogism, whose conclusion in turn is 
 determined as 0(41). Following out this same process, each 
 member of the chain will be unambiguously determined as, 
 
 A(i2)A(2 3 ) z A(jj), 
 A(i 3 )0(43) Z 0(4/), 
 0( 4 i) A(4S) Z 0(51), 
 0(5i)A(s6) z 0(6i). 
 
 The invalidity of any mood of the sorites will be established 
 at once, if it can, through the identification of terms, be reduced 
 to an invalid syllogism. The examples, which are given below, 
 will illustrate all of the methods, which it will be necessary to 
 employ later on. 
 
 (1) To establish the invalidity of the sorites, 
 
 A(2i)A(2 3 )A(43)A(34) Z Afci). 
 
 If the terms in the last premise and in the next to the last premise 
 be identified and the parts, A(44) and A(33), be suppressed, we 
 should obtain in succession, 
 
 A(2i)A(2 3 )A(4 3 ) z A( 4 i), 
 A(2i)A(2 3 ) Z A(ji). 
 
 Now the last result, AAA in the third figure, is an invalid syllo- 
 gism and, consequently, the mood of the sorites is invalid. 
 
 Had we identified terms in the first and last premise we should 
 have obtained, in the same way, 
 
 A(23)A(43) Z A(42), 
 an invalid mood in the second figure. You will remember that 
 
40 Letters on Logic to a 
 
 the reduction of a mood of the syllogism to a valid mood of 
 immediate inference proves nothing as regards either its validity 
 or its invalidity. The same observation of course holds true 
 of a mood of the sorites. If in the case in question we had 
 identified terms in the second and third premises, there would 
 have resulted 
 
 A(2i)A(S2) Z A(5i), 
 
 a valid mood of the first figure, and the invalidity of the original 
 sorites would not have been established by this process. 
 (2) To establish the invalidity of the sorites, 
 
 0(i2)0(2 3 )A(34)0(4S)0(56) Z A(6i). 
 
 Let us begin by identifying terms in the A-premise and sup- 
 pressing the part, A (33), and we have 
 
 0(K)0(aj)0(j5)0($<S) Z A(6i). 
 
 Now strengthen each one of the O-premises to E-premises and 
 our result will be 
 
 E(i2)E(2 3 )E(35)E(s6) z A(tfz). 
 
 Finally, identifying 6 and 3 and 5 and 2, and converting simply 
 in the third premise, the mood becomes 
 
 E(i2)E(2 3 )E(2 3 )E(2 3 ) z A( 3 i). 
 
 It will be necessary now to postulate the implication, 
 
 E{ab) Z E(ab)E(ab), 
 
 which we shall have occasion to employ continually later on. 
 By its aid we may strengthen E(2j)E(2j) to E(2j), so that, by 
 two steps, we arrive at 
 
 E(I2)E(2 3 ) z A(jz), 
 
 an invalid mood of the syllogism. The mood of the sorites, with 
 which we began, is, accordingly, invalid as well. 
 
 In the light of these illustrations it will be clear that: 
 
 (a) The validity of a mood of the sorites containing A- or 
 I-premises may be made to depend on the validity of a mood, in 
 which these premises are absent. 
 
 (b) Since an O-premise may always be strengthened to an 
 E-premise, the validity of a mood of the sorites containing E- 
 
Young Man without a Master 41 
 
 or O-premises may be made to depend on the validity of a mood, 
 in which only one E-premise occurs. 
 In general, if the chain of syllogisms, 
 
 XiO, i)x 2 (3, 2) / x s (3i), 
 x z (3i)x A (4,3) Z x b (4i), 
 x 5 (4i)x 6 (5, 4) Z xi(ji), 
 
 X2n-s(n — 1 i)x 2n -i(n, n — 1) Z x 2n -z(ni) T 
 
 be valid throughout, then 
 
 #i0, i)x 2 (3, 2)xi(4, 3) •-• x 2n -±(n, n — 1) Z x 2v -z{ni) 
 is a valid mood of the sorites. Hence a certain number of valid 
 sorites may be constructed from chains of valid syllogisms. 
 // remains to be proven that the only valid moods of the sorites that 
 exist may be built up from chains of valid syllogisms in the manner 
 described. 
 
 We proceed to enunciate certain theorems, which will abbre- 
 viate the work of establishing the general solution that follows. 
 
 Definition. — A form, x(ab), which becomes unity, when its 
 terms are identified, is said to be affirmative. By results already 
 obtained, A(ab) and l(ab) are affirmative while K(ab) and 0{ab) 
 are not affirmative forms. 
 
 Definition. — A form, x(ab), which becomes null, when its terms 
 are identified, is said to be negative. By results already obtained, 
 K(ab) and 0(ab) are negative, while A(ab) and l(ab) are not 
 negative forms. 
 
 Definition. — If x(ab) Z y(ab) and t^fa^) Z x(ab)) f then x{ab) 
 is said to be universal and y(ab) is said to be particular. Accord- 
 ingly, A(ab) and E(ab) are universal, \{ab) and 0(ab) are par- 
 ticular forms. 
 
 Theorem 1. — If the conclusion be affirmative, then all of the 
 premises are affirmative. For, if there were present one or more 
 negative premises, the mood of the sorites would be reducible to 
 an invalid syllogism of one of the forms, 
 
 AEA, EAA, AEI, 
 EAI, EEA, EEL 
 
 Theorem 2. — If the conclusion be negative, then one and only one 
 premise is negative. For, if all of the premises were affirmative, 
 the mood of the sorites would be reducible to an invalid syllogism 
 of one of the forms, 
 
42 Letters on Logic to a 
 
 AAE, AIE, IAE, HE, 
 AAO, AIO, IAO, IIO, 
 
 and, if more than one premise were negative, to one of the forms, 
 
 EEE, EOE, OEE, OOE, 
 EEO, EOO, OEO, 000. 
 
 Theorem 3. — // the conclusion be universal, then all of the 
 premises are universal. 
 
 Theorem 4. — If the conclusion be particular, not more than one 
 premise is particular. 
 
 It will be convenient to take the conclusion successively in 
 each one of the four forms. 
 
 Conclusion in the A-form 
 
 The conclusion is universal and affirmative. Accordingly, the 
 premises are all universal {theorem 3) and all affirmative {theorem 
 1). The sorites, if it be valid, must, consequently, be of the form, 
 
 A(j, 2)A{2, 3) • • ■ A{n - i,n) z A{m). 
 
 Now the term-order in each premise is established as (s s — 1), 
 i.e., with the larger ordinal number coming first, for, suppose 
 that the term-order {s — 1 s) should occur in any premise. 
 Then the mood of the sorites would be reducible to an invalid 
 syllogism, 
 
 Either A{ s — 1 , 5 — 2)A{s — is) z AQ s — 2 ), 
 or A{s - 1 s)A{s + j, s) Z A{s -fj s - 1). 
 
 The term-order is, accordingly, no longer ambiguous and the 
 sorites is of the form, 
 
 A{2i)A{32)A{43) • • • A(n n - 1) Z A(m), 
 
 which can be generated from the chain of valid syllogisms, 
 
 A{2i)A{ 3 2) z A(jj), 
 A{ 3 i)A{ 43 ) Z A( 4 i), 
 A{ 4 i)A{ 5 4) Z A( 5 i), 
 
 A{n - 1 i)A{n n - 1) Z A{ni). 
 
Young Man without a Master 43 
 
 Conclusion in the I-form 
 
 The conclusion is affirmative, so that all of the premises are 
 affirmative {theorem I), and particular, so that not more than 
 one premise is particular {theorem 4). 
 
 Case I. — A(i, 2)A{2, 3) • • • A{n — i,n) Z I(ni), a form which 
 contains no I-premise. 
 
 The first premise, which presents its terms in the order 
 {s — 1 s), establishes that order in each premise, which follows. 
 For, suppose a premise coming after the premise in question 
 would present the term-order {s s — 1). Then the sorites would 
 be reducible to an invalid syllogism of the form, 
 
 A(s - 2 s - i)A{s s - 1) Z I{s s - 2] 
 
 Let us suppose that the term-order {s s — 1) is preserved as far 
 as the rth premise and is then reversed. Then the order 
 {s — 1 s) is established from the rth premise to the end and the 
 sorites becomes 
 
 A{2i)A{32) • V- A(r r - i)A(r r + 1) ■ ■ ■ A(» - 1 n) z K>i), 
 which is constructible from the chain, 
 
 A{2i)A{ 3 2) z A(jj), 
 
 A(r - 1 i )A{r r - 1) Z A{n), 
 
 A {n)A {r r + 1) Z I(r + 1 1), 
 
 10 + 1 i)A{r + 1 r + 2) Z I(r + 2 1), 
 
 I(» - 1 z)A(w - in) Z l(ni). 
 
 If the term order (55 — 1) is preserved from the first premise to 
 the last, so that the form of the sorites is 
 
 A{2i)A{32) • • • A(» n - 1) Z l(ni), 
 then the generating chain of syllogisms will be 
 
 A{2i)A{32) z A(ji), 
 A{ 3 i)A{ 4 3) Z A( 4 i), 
 
 A{ n -2 i)A{n -in - 2) Z A(» - j /), 
 A(» - 1 i)A{n n- 1) Z I(»/). 
 
 Case II. — Suppose that the /th premise is in the I-form and 
 that the sorites is 
 
 A(j, 2) • • • A(/ - 1, t)l{t, t + i) • • • A(w - j, w) Z I(»/). 
 
44 Letters on Logic to a 
 
 The term-order in the first t — I premises is established as 
 (s s — i). Otherwise, by the identification of terms, we should 
 come upon an invalid syllogism of the form, 
 
 A(s - i s)l(s, s + i) Z 10 + i s - j). 
 
 Similarly, the order of terms in the last n — t — I premises is 
 established as (s — I s). For, should any premise following the 
 I-premise present the term-order (s s — i), the mood of the 
 sorites would reduce to an invalid syllogism, viz., 
 
 1(5 — 2, s — i)A(s s — 1) Z l(s s — 2), 
 Consequently, the sorites takes the form, 
 
 A(2i) • • • A(tt - i)l(t, t + j)A(* + 1 t + 2) • • • 
 
 A(n - 1 n) Z 1(^7), 
 which may be built up from the chain of syllogisms, 
 
 A(2i)A( 3 2) z A(ji), 
 
 A(t - 1 i)A(tt - 1) Z A(/j), 
 A(ti)l{t, Hh7) Z !(/ + 1 1), 
 
 l(t + i i)A(7+~i T+~2) z l(T+2 1), 
 
 I(« - 1 i)A{n - in) Z l{m). 
 
 Conclusion in the E-form 
 
 Here there must occur a single E-premise {theorems 2, 3) and 
 all of the other premises are in the A-form (theorem 3). All valid 
 moods of this form are to be obtained by contradicting and inter- 
 changing the I-premise and the I-conclusion in the type of valid 
 sorites, which has just been established (case II, above). For, 
 every mood of the sorites with an E-conclusion not so obtained 
 would be reducible to an invalid mood already established by 
 contradicting and interchanging the E-premise and the con- 
 clusion. 
 
 Conclusion in the O-form 
 
 One and only one premise is a negative form. All valid moods 
 of this type are obtained by contradicting and interchanging a 
 premise and conclusion in one or other of the valid moods already 
 
Young Man without a Master 45 
 
 established. For, otherwise, as in the last case, we could reduce 
 any mood not so derived to one of the invalid moods already 
 established. 
 
 Accordingly, all of the valid moods of the sorites are deter- 
 mined as of certain specific types and each one of these moods 
 may be constructed from a chain of valid syllogisms. 
 
46 Letters on Logic to a 
 
 PROFESSOR SINGER'S SYLLABUS 
 
 (I) 
 Classification of Sciences into 
 
 Empirical: Physics, Chemistry, Biology, Sociology, etc. 
 Non-Empirical: 
 
 (a) In which judgment of truth of propositions involves knowl- 
 edge of the meaning of terms: (Mathematics). 
 
 (b) In which judgment of the truth of propositions does not 
 involve knowledge of the meaning of terms : (Logic) . 
 
 Definition of Logic: Logic is the science whose problem it is to 
 construct all propositions whose truth is independent of the 
 meaning of terms. 
 
 (id 
 
 Grammarians recognize six kinds of sentence: Declarative. 
 Optative, Exclamatory, Interrogative, Hortatory, Imperative, 
 These fall into two classes : 
 
 (a) Sentences that are either true or false. (First three.) 
 
 (b) Sentences that are neither true nor false. (Last three.) 
 Definition of Proposition: A proposition is a sentence that is 
 
 either true or false. The logician recognizes the following forms 
 
 of propositions as necessary and sufficient for the expression of 
 
 any truth. 
 
 Categorical All a is b = A (a b) 
 
 No a is b = E (a b) 
 
 Some a is b = I (a b) 
 
 Some a is not b = O (a b) 
 
 Hypothetical X implies Y =X/Y 
 
 X does not imply Y = (X / Y)' 
 
 Conjunctive X (is true) and Y (is true) = X-Y 
 
 Disjunctive Either X (is true) or Y (is true) = X + Y. 
 
 an) 
 
 Categorical forms composed of terms and relation. 
 Terms are subject (a) and predicate (b). 
 Relations are composed of 
 
Young Man without a Master 47 
 
 A 
 
 Adjective of quantity 
 all 
 
 and 
 
 Copula 
 is 
 
 E 
 
 no 
 
 
 is 
 
 I 
 
 some 
 
 
 is 
 
 
 
 not all 
 
 
 is 
 
 The array of propositions of categorical forms in which terms 
 are identical : 
 
 X(aa) 
 A (a a) E (a a) 
 
 I (a a) O (a a) 
 
 Definition: True propositions of a given array are called valid 
 moods of that array. 
 
 False propositions of a given array are called invalid moods 
 of that array. 
 
 Post. 1. A(a a) is valid mood. 
 
 Post. 2. I (a a) " " 
 
 Post. 3. E(a a) " invalid mood. 
 
 Post. 4. 0(a a) " " 
 
 (IV) 
 
 Definition 1. — In the hypothetical forms X Z Y and (X z Y) r , 
 X is called the antecedent and Y is called the consequent. 
 
 Definition 2. — If X/Y' and Y' / X, X is called the contra- 
 dictory of Y. 
 
 Principle i, (X z Y') Z (Y Z X') 
 Principle ii, (Y' Z X) z (X' z Y) 
 
 Theorem: If X is the contradictory of Y, then Y is the contra- 
 dictory of X. 
 
 Post. 1. A(a b) Z O' (a b). Post. 2. 0'(a b) Z A (a b). 
 
 Post. 3. E(a b) Z I' (a b). Post. 4. I'(a b) Z E (a b). 
 
 Historical note: There are no other pairs of contradictories 
 among categorical forms. Contrary forms, sub-contrary forms, 
 subalternate forms. 
 
 (V) 
 The array X(a, b) Z Y (a, b) 
 Two figures 1. X(a b)/Y(a b) 
 2. X(a b) Z Y (b a). 
 
48 Letters on Logic to a 
 
 Sixteen moods in each fig. 
 
 The valid moods. 
 
 Prin. i, (X / Y) Z (Y' Z X') Denial of consequent. 
 
 Prin. ii, (X Z Y)(Y Z Z) Z (X Z Z). Transitivity. 
 
 Post. I, A(a b) Z I (a b) (A I)i is valid mood. 
 
 Post. 2, I(ab) Z I (ba) (I I) 2 " " 
 
 Deduction from Post, i and 2 and from Posts, of (IV) of valid 
 moods by means of Prins. i and ii. 
 
 Valid moods: (A A)i (A fy (E E) x (EO)i (I I)i (O 0)i 
 
 (A I) 2 (E E) 2 (EO) 2 (I I) 2 
 The invalid moods: 
 
 Prin. iii. (X z Y)' Z (Y' Z X')'. 
 
 Prin. iv. (X z Z)'(Y Z Z) Z (X Z Y)'. 
 
 Prin. v. (X z Y)(X Z Z)' Z (Y Z Z)'. 
 
 Post. i. {A(a b) z 0(a b)}' (A 0)i is invalid. 
 
 Post. 2. (E(ab) Z I(ab)}' (E I) x " 
 
 Post. 3. {A(a b) z A(b a)}' (A A) 2 " " 
 
 Post. 4. {A(a b) z 0(b a)}' (AO) 2 " " 
 
 Rules for immediate detection of invalid moods : 
 
 Def. I. A form which yields a true proposition when the 
 terms are made identical, is called an affirmative form. 
 
 From (III), A (a b) and I (a b) are affirmative forms. 
 
 Def. 2. A form which yields a false proposition when terms 
 are made identical is called a negative form. From III, E(a b) 
 and 0(a b) are negative forms. 
 
 Rule I. An affirmative form does not imply a negative form, 
 ex. (A 0) 2 
 
 Rule II. A negative form does not imply an affirmative form, 
 ex. (E I)x 
 
 Def. 3. If X(a b) Z Y(a b) and {Y(a b) Z X(a b)}' X(a b) 
 is called a universal form and Y(a b) is called a particular form. 
 
 From results of present chapter A (a b) and E(a b) are universal 
 forms, I (a b) and 0(a b) are particular forms. 
 
 Def. 4. The subject of a universal and the predicate of a 
 negative form is called a distributed term, other terms undis- 
 tributed. 
 
 Aff. Neg. 
 
 Universal A (a b) E(a b) 
 
 Particular I (a b) 0(a b) 
 
Young Man without a Master 49 
 
 Rule III. A form in which a given term is undistributed does 
 not imply a form in which that same term is distributed. (A A) 2 
 Proof of the necessity and sufficiency of these rules. 
 
 (VI) 
 
 The array X(a, b) Y(b, c) z Z(c, a) 
 
 Def. 1. When forms are conjoined in antecedent, each is 
 called a premise and the consequent is then called the conclusion. 
 
 Def. 2. The order of terms in present array called cyclical 
 order. 
 
 Def. 3. A cyclical order of terms with one premise called 
 immediate inference. X(a, b) Z Y(a, b). With more than one 
 premise, mediate inference. Mediate inference with two premises 
 called syllogism ( Z syn and logos) ; more than two forms called 
 Sorites (z soros). 
 
 (1) In order of premises. 
 
 X(a, b) Y(b, c) Z Z(c, a) 
 Y(b, c) X(a, b) Z Z(c, a) 
 
 (2) In order of terms. 
 
 I. X(b a) Y(c b) Z Z(c a) 
 
 II. X(a b) Y(c b) z Z(c a) 
 
 III. X(b a) Y(b c) z Z(c a) 
 
 IV. X(a b) Y(b c) z Z(c a) 
 
 (3) In XY Z Z, X, Y and Z may each take on the four forms 
 A, E, I, O. 
 
 (1) Prin. i. XY z YX 
 
 Theorem (XY z Z) Z (YX z Z) i and ii, Ch. V. 
 
 We may consequently confine our attention to one order of 
 forms. 
 
 Def. 4. The term occurring in both prems. called middle term. 
 
 The predicate of conclusion called major term. 
 
 The subj. of conclusion called minor term. 
 
 The prem. containing major term called major premise. 
 
 The prem. containing minor term called minor premise. 
 By convention we study array in which major premise is written 
 first. 
 
 (2) The four ways in which the terms may be ordered yield 
 four figures of syllogism. 
 
50 
 
 Letters on Logic to a 
 
 By convention we number figures in the order given. 
 
 (3) In each figure there are as many moods as there are com- 
 binations of the four forms A, E, I, O taken three at a time 
 (two prems. and the conclusion) = 4 3 = 64 moods. 
 
 The array is then constructed under each fig. as follows : 
 AA(A,E,I,0) EA(A,E, 1,0) IA(A,E,I,0) OA(A,E,I,0) 
 
 AE " EE 
 
 n 
 
 IE 
 
 a 
 
 OE 
 
 AI " EI 
 
 ti 
 
 II 
 
 a 
 
 01 
 
 AO " EO 
 
 a 
 
 10 
 
 a 
 
 00 
 
 The valid moods. 
 
 
 
 
 
 Prin. ii. Prin. iii. 
 
 (XYzZ)z(XZ'z Y') (XYzZ)z(Z'Yz X') 
 
 Prin. iv. (WzX)(XYzZ)z (W Y Z Z) 
 Prin. v. (X Y / Z) (Z / W) / (X Y Z W) 
 
 Post. 1. A(b a)A(c b) Z A(c a) (A A A)i is valid mood. 
 
 Post. 2. E(b a)A(c b) Z E(c a) (E A E)i is valid mood. 
 
 Deduction of valid moods. 
 
 For convenience of application principles of deduction may be 
 stated in form of following rules : 
 
 Rule I. Interchange contradictories of either premise and of 
 the conclusion. 
 
 Rule II. Strengthen premise or weaken conclusion. 
 
 Rule III. Convert either prem. or conclusion. 
 
 Historical Note: The problem of reduction. 
 
 (VII) 
 The invalid Moods. 
 
 Prin. i. (X Y Z Z)' z(XZ'z Y')' 
 
 Prin. ii. (X Y Z Z)' Z(Z'YZ X')' 
 
 Theorem. (W Z X) (W Y Z Z)' Z (X Y z Z)' 
 
 Theorem. (X Y z W)' (Z z W) Z (X Y z Z)' 
 
 Post. 1. (A(ba)A(cb) zO(ca)}' (AAO)i is an invalid mood. 
 
 Post. 2. {A(ba)E(cb) Z I(ca)} 7 (AEI)i " 
 
 Post. 3. { A(ba) E(cb) Z 0(ca) } ' (A E 0)i " 
 
 Post. 4. (E(ba)E(cb) Z I(ca)}' (EEI)i " 
 
 Post. 5. {0(ba) A(cb) Z 0(ca) } ' (O A 0)i " 
 
 Post. 6. { A(ab) A(bc) Z A(ca) V (A A A) 4 " 
 
 Post. 7. { A(ab) A(bc) Z 0(ca) } ' (A A 0) 4 " 
 
Young Man without a Master 51 
 
 Deduction of invalid moods. For convenience of application 
 principles of deduction may be stated in form of following rules. 
 
 Rule I. Interchange contradictories of either premise and of 
 the conclusion of invalid mood. 
 
 Rule II. Weaken premise or strengthen conclusion of invalid 
 mood. 
 
 Rule III. Convert either premise or conclusion of invalid 
 mood. 
 
 Rules for immediate detection of invalid mood : 
 
 Rule I. Two negative premises do not imply a conclusion 
 (EEI)l 
 
 Rule II. Two affirmative forms do not imply a negative 
 conclusion (AAO) 4 . 
 
 Rule III. An affirmative and a negative premise do not 
 imply an affirmative conclusion (AEI)i. 
 
 Rule IV. Two prems. in neither of which middle term is 
 distributed do not imply a conclusion (OAO)i. 
 
 Rule V. Premises in which a given term is undistributed do 
 not imply a conclusion in which that term is distributed. (AEO)i 
 major term; (AAA) 4 minor term. 
 
 Proof of the necessity and sufficiency of these rules. 
 
 (VIII) 
 
 The Sorites (vid. Ch. VI, Def. 3.) 
 
 Xi (1, 2) X 2 (2, 3) • • • X„ (n, n + 1) Z X n+ i(n + 1, 1) 
 
 The valid moods : 
 
 Prin. i. (Y/Z)z (XY / XZ). 
 Theorem: If X^i, 2) X 2 ( 2 , 3) Z X 3 (3, 1) 
 X 3 ( 3 , 1) X 4 ( 3 , 4) Z X 5 ( 4 , 1) 
 
 X 2n _ 5 (n, i)X 2n _ 4 (n, n + 1) z X 2 „_ 3 (n + 1, 1) 
 
 Then X x (1, 2) X 2 (2, 3) • • • X 2n _ 4 (n, n + 1) Z X 2n _ 3 (n + 1, 1). 
 
 The invalid moods : 
 
 Prin. ii. A mood is invalid if the mood obtained by identifying 
 any two terms can be shown to be invalid. 
 
 Prin. iii. A mood containing the prem. A(aa) or I (aa) is 
 invalid if the mood from which this prem. is omitted can be 
 shown to be invalid. 
 
52 Letters on Logic to a 
 
 Prin. iv. A mood containing the prems. X(ab)X(ab) is invalid 
 if the mood containing but one of these prems. can be shown to 
 be invalid. 
 
 By means of these principles, the invalid moods of sorites 
 may be deduced from previous results without further postulates. 
 We find that the valid moods constructed under Prin. i are the 
 only valid moods. 
 
 (IX) 
 The Zero cycle. 
 
 Prin. i. (X / Y) z {XY' z (XY')'}. 
 Prin. ii. {(XY' Z (XY')'} Z (X Z Y). 
 Def. I. {(XY') Z (XY')'} Z (XY' z O). 
 (XY' ZO) z {(XY') Z (XY')'}. 
 
 Def. 2. When the antecedent is conjunctive of categorical 
 forms with terms in cyclical order the form called zero cycle. 
 Have solved zero cycle with exception of form 
 
 X(aa) Z O 
 Post. I. 0(aa) Z O 
 Theor. I. E(aa) Z O 
 Post. 2. {A(aa) Z O}' 
 Theor. 2. { I(aa) Z O}' 
 
Young Man without a Master 53 
 
 APPENDIX 
 
 Note on the Relation of Subalternation 
 
 The relation of subalternation being all but universally denied 
 in recent times, it will not be inappropriate to point out in what 
 sense this denial rests upon a misapprehension. The following 
 solution is due to Professor Singer. 
 
 If we employ the symbol, z , for inclusion, the four categorical 
 forms might supposedly be represented as follows : 
 
 (A) All a is b = {a z b) 
 
 (E) No a is b = {a z V) 
 
 ( I) Some a is b = {a z b')' 
 
 (O) Some a is not b = (a z b) r 
 
 A is now the contradictory of O and E is the contradictory of 
 I but A Z I and E/Ono longer hold true. 
 
 This interpretation of Aristotle's four forms, however, is in 
 no way forced upon us, for we may assume : 
 
 (A) All a is b = (a zb) 
 
 (E) No a is b = (a z b')(a Z a')'(b z b')' 
 
 ( I) Some a is b = {a z b')' + (a Z a') + (b Z V) 
 
 (O) Some a is not b = (a Z b)' 
 
 These equalities fulfill the essential conditions : 
 
 AE z 0, i Z A(aa), 
 
 the first of which contains 
 
 A z I and E z O, 
 
 since the members of the pairs, E, I and A, O are contradictory. 
 It will be observed, too, that E and I retain their characteristic 
 property of simply convertibility. 
 
NON-ARISTOTELIAN LOGIC 
 
 BY 
 
 HENRY BRADFORD SMITH 
 
 Assistant Professor of Philosophy in the University of Pennsylvania 
 
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