BC 
 
 UC-NRLF. 
 
 PRIMER o/ LOGIC 
 
 SMITH 
 
GIFT OF 
 
Primer of Logic 
 
 BY 
 
 HENRY BRADFORD SMITH 
 
 Assistant Professor of Philosophy in the 
 University of Pennsylvania 
 
 B. D. SMITH A BROS. 
 
 PULASKI, VA. 
 
 1917 
 
PREFACE 
 
 In the pages which follow will be found the outlines 
 of the logic of a set of categorical forms which are not 
 Aristotle's own. It is only the fragment of a general 
 theory, but the content of all the chapters of the old logic, 
 which are commonly regarded as essential, except that 
 one which deals with the calculus of classes, will be found 
 included. In the first appendix the relation of these new 
 forms to the traditional ones has been pointed out in detail. 
 
 I have at least one debt, which calls for a definite 
 acknowledgement. It is to my friend and teacher, Mr. 
 Edgar A. Singer, Jr., that I owe what training I have had 
 in the science. He has never failed, through hints thrown 
 out in conversation, to correct my misapprehensions. 
 But my indebtedness is more specific than this. I have so 
 far employed his own method, a method developed in his 
 academic lectures, that I could scarcely have ventured 
 upon the publication of these outlines of a theory without 
 his express permission. 
 
 H. B. S. 
 
TABLE OF CONTENTS 
 
 CHAPTER I PAGES 
 
 1-3. The fundamental properties of the categorical 
 
 forms. Exercises 1-9 
 
 CHAPTER II 
 
 4-8. The relationships of "better" and "worse." 
 Deduction of the moods of immediate infer- 
 ence and of the syllogism by rule. Exercises. . . 10-17 
 
 CHAPTER III 
 
 9-14. The relationships of "better" and "worse" 
 defined. Symbolic deduction of the moods 
 of immediate inference and of the syllogism. 
 Exercises 18-27 
 
 CHAPTER IV 
 15. The general solution of the sorites. Exercises.. .28-34 
 
 APPENDIX I 
 
 On the simplification of categorical expression 
 
 and the reduction of the syllogistic figures 35-44 
 
 APPENDIX II 
 
 Historical note on De Morgan's new preposi- 
 tional forms . . . . 45-48 
 
CHAPTER I 
 
 1. In 1846 Sir William Hamilton published the 
 prospectus of an essay on a "New Analytic of Logical 
 Forms,"* which revived the question as to whether or 
 not the quantity of the predicate of the categorical forms 
 shquld be stated explicitly. The chief difficulties of his 
 system result from the ambiguity of the meaning of some, 
 from the impossibility of making every form of categorical 
 expression simply convertible, and from the seemingly 
 curious effort to establish an order of better and worse 
 between the relations connecting subject and predicate. 
 Four of Hamilton's eight forms are redundant. The four 
 that remain will be represented here by the letters, a, 
 
 ft 7, e- 
 
 Accordingly let 
 
 a a b = all a is all b, 
 /3 ab = some a is some b, 
 7ab = all a is some b, 
 e a b=no a is b. 
 
 Here some, the some expressed explicitly in ft and 7, 
 means some at least, not all. This meaning of the word 
 is established unambiguously by the properties of the 
 forms. In addition to these abbreviations we will employ 
 the notation: 
 
 Xab =x ab (is true), 
 x' ab =x ab (is false), 
 
 x ab ' y & b =Xab (is true) and y ab (is true), 
 Xab + y & b =x ab (is true) or y ab (is true), 
 x ab Z y ab =x ab (is true) implies y ab (is true), 
 (x ab Z. y ab )' =x ab (is true) does not imply y ab (is true). 
 
 *Lectures on Logic, ed. by Mansel and Veitch, Boston, 
 Gould and Lincoln, 1863. 
 
2 A PRIMER OF LOGIC 
 
 2. The implications, which are given below, express 
 the chief characteristics of the forms. The theorems 
 follow by the principle of the denial of the consequent, 
 which may be written in the two forms: 
 
 (x Z y') Z (y Z x') and (x' Z y)' Z (y' Z x)'. 
 
 Postulates:* 
 
 tt ab Z 0'ab ab Z 7'ab (o'ab ^ ab)' (0'ab / Tab)' 
 
 dab Z T 'ab /?ab Z e' ab (d' ab Z Tab)' (jS'ab ^ Cab)' 
 
 ab Z e' ab 7 a b Z e' ab (tt' ab Z ab )' (7'ab ^ C a b)' 
 
 7ab Z 7'ba (7'ab Z 7 ba )' 
 
 Theorems: 
 
 ab Za' ab 7a b Zd' ab (c'abZttab)' (7'ab Z a ab )' 
 
 a b Z jS'ab 7a b Z jS'ab ( 'ab / jSabX (7'ab Z jSab)' 
 
 Ca b Z 7'a b ab Z a' ab (e'ab ^ Tab)' (0'ab Z tt a b )' 
 
 Let us postulate in addition that: 
 
 a ab Z (a'ab)' (o'ab)' ^ a ab 
 
 /3 ab Z (/3' a b)' O'l*)' ^ a b 
 
 7a b Z ( T ' ab )' (7'.b)' ^ Tab 
 
 6 ab Z (e' ab )' (c' ab )'Z e ab 
 
 Then, if k^rjiWafo and if k ab and w ab represent only 
 the unprimed letters, a ab , /3 ab , 7 ab , e ab , a complete induction 
 of the propositions given above yields the general result: 
 
 k ab Z (k' ab )', (k' ab )'Zk ab , I 
 
 k ab Z W'ab, (W' ab Z kab)', II 
 
 *These assumptions are in accord with those of the common 
 logic, but no longer hold when the terms are allowed to take on 
 the limiting values and 1 ; for 7 OI and e i are both true prop- 
 ositions. The assumptions (7 ab Z e' ab )' and (e ab Z 7' ab ) will 
 be characteristic of a more general logic, which will include 
 the classical logic as a special case. (See the concluding remarks 
 of chapter III.) 
 
A PRIMER OF LOGIC 
 
 and by denial of the consequent, 
 
 (xZ y) Z(y'Z x'), 
 (xZy)'Z(y'Zx')', 
 
 it follows that: 
 
 (k'a b )'Z W'ab, (w' ab Z (k'ab)')'. HI 
 
 If now we postulate: 
 
 (d ab Z tt'ab)' (a'ab / a ab )' 
 
 (/3ab Z /3'ab)' (/3'ab^0ab)' 
 
 (7ab^7'ab)' (7'ab/7ab)' 
 
 (e ab ^ ' ab )' ('ab/ e ab )' 
 
 and, consequently, 
 
 (kab / k'ab)', (k'ab/kab)', 
 
 it will follow by* 
 
 (xZ y) (xZ z)'Z(yZ z)', 
 
 (xZz) / (yZz)Z(xZy) / , 
 that (k ab Z (w'ab)')', ((w'ab)' Z k ab y. IV 
 
 Finally if we assume 
 
 d a b dba> ^ab -^ ^ba Tab Z 7 & b, Cab 
 
 it will follow, by 
 
 (x Z y) (y Z z) Z (x Z z) and (x Z y) Z (y' Z x'), 
 that 
 
 kab / kab, k'ab Z k' ab . 
 
 Definitions. 
 
 If x ab Z y'a b and y' ab Z x ab , x ab is said to be contra- 
 dictory to y a b- By I, k ab is contradictory to k' ab and, by 
 I', k'ab is contradictory to k a b. 
 
 "These are obtained from (x Z y) (y Z z) Z (x Z z) by 
 (xy Z z) Z (xz ; Z yO and xy Z yx. 
 
4 A PRIMER OF LOGIC 
 
 If x ab Z y'ab and (y' ab Z x ab )', x a b is said to be contrary 
 to y ab . By II, k ab is contrary to w ab , and conversely, 
 since k a b and w a b are interchangeable. 
 
 If (x ab Z y' ab )' and y' ab Z x ab , x ab is said to be subcon- 
 trary to y ab . By III, k' ab and w' ab are subcontrary pairs. 
 
 If (x ab Z y'ab)' and (y' ab Z x ab )', x ab is said to be sub- 
 alternate to y ab . By IV, k' ab and w ab are subalternate 
 pairs. 
 
 3. Having classified the categorical forms under 
 these heads, it remains to differentiate them by means of 
 their formal properties. If we assume as valid, 
 
 a aa Z a aa , e aa Z e afi , (a aa Z a aa ) , ( aa Z e aa ) , 
 
 where a represents the class contradictory to a, (non-a), 
 the other propositions given below may be derived.* 
 
 a aa Z a aa (ct aa Z 0, aa ) tt aa Z tt aa (tt aa Z O, aa ) 
 
 0aa Z /3' aa (/3' aa Z )' ft afi Z iS'a, (^ afi Z /3 aa )' V 
 
 7aa Z T 'aa (Vaa Z Taa)' 7 aa Z ^aa (Vafi Z 7 aa)' 
 
 aa Z e' aa (e' aa Z 6 aa )' e' aa Z aa ( aa Z e'aa)' 
 
 The results of V, together with the non-convertible 
 character of 7**, are enough to establish the definitions 
 of the four forms. 
 
 *Under the conditions mentioned above, in note, p. 2, we 
 shall have to write (7 aa Z 7' a a)'. Implications V are an extension 
 of the meaning of implication, made necessary by our having 
 to call a aa and e a a true propositions. (See Boole, Investigation 
 of the Laws of Thought, chap. XI, p. 169.) 
 
 **The operation of simple conversion consists in interchang- 
 ing subject and predicate. By the principle, (xZ z/(yZ z) Z 
 (xZ y)', and what has gone before, we have: 
 
 (TabZ Vab/CTbaZ T ' ab ) Z ( T ab Z T ba)'. 
 
A PRIMER OF LOGIC 5 
 
 Definitions. 
 
 A form which is the contrary of itself is called a null- 
 form. By V, aa , Taa, c aa , a a& , /3 aa , 7 &a , are null-forms. 
 
 A form which is the subcontrary of itself is called a 
 one-form. By V, a aa , e aa , are one-forms. 
 
 If x ab is unprimed and x aa a one-form then x ab is 
 called an a-form. 
 
 If x ab is unprimed and simply convertible and if 
 x aa and x aa are null-forms, then x ab is called a /3-form. 
 
 If x ab is unprimed and not simply convertible, then 
 x a b is called a 7-form. 
 
 If x ab is unprimed and x aa a one-form, then x ab is 
 called an -form. 
 
A PRIMER OF LOGIC 
 
 EXERCISES 
 
 (1) Assuming k ab = k ab * k ab , k ab Z w' ab , show that, 
 
 Of ab Z j'ab 7'ab ab T'ba 
 
 a ab 7 ab e ab 7 ba 
 7 ab Z a'a b /3' ab e' ab 7' ba 
 e ab Z a' a b |8 ab 7 ab 7 ba 
 by the aid of, 
 
 (xZy) (yZz)Z(xZz), 
 (xZy)Z(zxZzy). 
 
 Equality is defined by 
 
 (xZy)(vZx)Z(x = y), 
 (x = y)Z(xZ y) (yZ x). 
 
 If now 
 
 0'ab 7ab c'ab T'ba Z ab 
 
 ^ 
 
 a^ab ^'ab C^ab Vba Z 7 ab 
 ab jS'ab 7 ab 7'ba Z ab 
 
 it will follow that 
 
 a a b = /^ab T^b ^ ab 7^ ba ^ab=/3 a b + 7ab + e ab + 7 ba 
 
 j8 a b ='ab T'ab 'ab 7'ba /3' ab = ab + 7 ab + ab + 7 ba 
 
 7ab='ab ^ ab ^ ab 7^ ba 7 / 'ab=ab + /3 ab + ab + 7 ba 
 
 6 ab = 'ab j8 ab 7 ab T ba e' ab =QJ ab + /3 ab + 7 ab + 7 ba 
 
 The second set of equations follows from the first by the 
 principle, that the contradictory of a product is the sum of the 
 contradictories of the separate factors, and by substituting k ab 
 directly for (k' ab )'. 
 
 (2) Show by the method of the last example that 
 
 aab=ab /3' ab =a ab 7'ab=ab c' ab 
 
 ab = /3 ab a' ab = /3 ab 7' ab = |8 ab e' a b 
 
 7ab=7ab ^ab=7ab ^ab = 7ab e' ab 
 Cab = Cab 'ab = ab ^'ab = ab 7' ab 
 
 Show too that 
 
 ab=ab 0'ab 7'ab=ab j^'ab t / ab=ab ?'ab c' ab , etc., etc. 
 
 and that a ab =a ab /3' ab 7' ab c' &b , etc., etc. 
 Derive the analogues of the first set: 
 
 a'ab = C/ab + /3 tt b = 'ab + Tab ='ab + C a bi etc., etc, 
 
A PRIMER OF LOGIC 7 
 
 Accordingly, since k a b = k a b ' w' a b and k / a b = k / a b + w a b, any 
 primed letter is a modulus of multiplication with respect to 
 any unprimed letter not itself and any unprimed letter is a 
 modulus of addition with respect to any primed letter not itself. 
 
 The prepositional zero is defined by 
 
 (xyZ (xy)')Z(xyZ 0), 
 (xyZ 0)Z(xyZ (xy)'), 
 
 and the prepositional one by 
 
 ((xy/Zxy^lZxy), 
 (1Z xy)Z((xy)'Z xy). 
 
 From the principles, 
 
 (xZ y')Z(xyZ (xy)'), 
 (xyZ(xy)')Z(xZy'), 
 it follows that 
 
 (xZyOZ(xyZO) 
 (xjZO)Z(xZ y') 
 
 (3) Derive: 
 
 k ab ' k' ab / (k ab ' k'ab)'; (k ab + k' ab )' Z k ab + k' ab ; 
 k ab ' w ab Z (k ab ' w ab )'; ((k ab + w ab )' Z k ab + w ab )'; 
 (k'ab ' w' ab Z (k^ ' w'ab)')'; (k 7 ab + w' ab )' Z. k' ab + w' ab ; 
 (k ab * w'ab ^ (kab ' w'ab)') '; ((k ftb + w f &b YZ. k ab + w'.b)'. 
 
 (4) Show that 
 
 ab |8ab ^ 0, ab 7 ab Z 0, etc. 
 
 1 Z a' ab + 0'ab, 1 ^ a'ab + 7'ab, etc. 
 and that 
 
 'ab #'ab T'ab 'ab T'ba -^ 
 
 1 Z a:' ab + j3' ab + 7'ab + e' ab + T'ba 
 
 (5) Assuming x ab = x ab ' x ab , x ab = x ab + x abf 
 show that 
 
 a'ab ^'ab=7ab + ab +^7ba, 'ab 7'ab = /3 a b + ab + 7ba, CtC. 
 ab + /3ab=7'ab C a b 7'ba, ab + 7ab = /3'ab c' a b 7'ba etc. 
 
 (6) Derive the general result: 
 
 (ka.b/ W a , b )'. 
 
 The comma between the terms indicates that the term order is 
 not fixed. Thus k a ,b stands for either k ab or k ba . 
 
8 A PRIMER OF LOGIC 
 
 (7) From the principle, 
 
 (xZz)'(yZz)Z(xZy)', 
 and the postulate (a aa Z a' aa )', 
 derive (a aa Z /3 aa )', ( aa Z Taa)', ( aa Z e aa )'. 
 
 (8) From the principles, 
 
 (xZy)(yZz)Z(xZz), 
 (x'Z x)Z(yZ x), 
 (xZxOZ(xZ y), 
 
 and the postulate, c/ aa Z a aa , show that all propositions of the 
 form x aa Z y aa , except the three cases in the last example, are 
 valid implications, x aa and y aa representing only the unprimed 
 letters. 
 
 (9) By the method of the last example, prove that 
 (a aa Z a' aa y is the only invalid implication of the form x aa Z y' aa . 
 
 (10) Derive seven valid implications in each one of the 
 forms x' aa Z y aa and x' aa Z y aa and nine invalid implications of 
 each one of the same forms. 
 
 (11) From ( aa Z e' aa )' by the method of example 7 derive 
 (e afl Z a aa )', ( aa Z aa )', (c aa Z 
 
 (12) From e' aa Z a5 by the method of example 8 derive 
 thirteen valid implications. 
 
 (13) Show that (6 aa Z e' aa )' is the only invalid implication 
 of the form x aa Z y'aa. 
 
 (14) Derive the following implications: 
 
 izaa 
 
 0/aaZ 
 
 1 Z a' aa 
 
 aa Z 
 
 1 Z 0'aa 
 
 j8 aa Z 
 
 1 Z j8' aa 
 
 /3 aa Z 
 
 izy aa 
 
 Taa Z 
 
 1 Z T' a a 
 
 Taa Z 
 
 1Z e' aa 
 
 6aa Z 
 
 1Z e aa 
 
 ' afi Z 
 
 (lZo/ aa y 
 
 (aa / 0)' 
 
 (iza aa y 
 
 ('aaZO)' 
 
 
 03'aazoy 
 
 az/3 aa y 
 
 tf'aaZ 0)' 
 
 (IZT.a)' 
 
 (VaaZO)' 
 
 (14 7s.) 
 
 (T'aaZO)' 
 
 (1Z 6aa)' 
 
 (e'aaZO)' 
 
 (1Z 6^)' 
 
 (e aa Z 0)' 
 
 Some of the postulates in the text (p. 2) of the form, 
 (k' ab Z w ab )', (k ab Z k'ab)', (k' a b Z k ab )', may be established by 
 
A PRIMER OF LOGIC 9 
 
 reducing them to one of the forms, (k' aa Z w aa )', (k' aa Z w aa )' 
 (k aa Z k' aa )', etc., cases already considered in preceding exercises 
 
 (15) Establish the invalidity of 
 
 'ab Z ab 'ab Z Tab Vab ^ Tba 
 
 /3'ab^ Tab /3'ab^ a b Vab ^ ab 
 
 'ab ^ ab jS'ab ^ l^ab V ab ^ Tab 
 
 'ab a b Z c'ab C a b Z a b 
 
CHAPTER II 
 
 4. At this point in our theory it will be necessary 
 to introduce certain indefinables, which we shall call the 
 distinctions of better and worse, following a suggestion of 
 Sir William Hamilton's.* For our immediate purpose 
 it will be enough to define better than and worse than de- 
 notatively, establishing an order among the four forms by 
 a simple enumeration. Better than and worse than are not 
 transitive relations. When we wish to express the rules 
 for the deduction of the moods symbolically, we shall 
 have to invent symbols to represent worse than (/), doubly 
 worse than (//), and trebly worse than (///). This necessity 
 is avoided in the verbal statement of the principles of de- 
 duction by the words "in the same degree" (see rule 1 
 below). 
 
 Definitions. An 6-form is worse than an a-, a 0-, or 
 a 7-form. 
 
 A 7-form is worse than an a- or a /3-form. 
 A /3-form is worse than an a-form. 
 
 Best. a~ /3-- 7 e Worst. 
 
 5. Immediate inference is a form of implication 
 belonging to one of the types: 
 
 1. x ab / y a b- 2. x ab Z y ba . 
 
 These differences are known as the first and the sec- 
 ond figures of immediate inference respectively. 
 
 The part to the left of the implication sign is called 
 the antecedent; the part to the right is called the consequent. 
 
 *Lectures on Logic, Appendix, p. 536. 
 
A PRIMER OF LOGIC 11 
 
 Since x and y may take on any of the forms, a, /3, 
 7, c, there will be sixteen propositions of each type, ob- 
 tained from the permutations of the letters two at a time 
 and by taking each letter once with itself. Each one of 
 the sixteen distinct propositions in each one of the two 
 figures is called a mood of immediate inference. The 
 rules which follow below, applied to the postulates, will 
 yield all the true and all the false propositions of each type. 
 
 Valid Moods. 
 
 1. In any valid mood of the first figure make ante- 
 cedent and consequent worse in the same degree. 
 
 2. In any valid mood convert simply in any form 
 but 7. 
 
 Postulate: a ab Z. a ab . Theorems: The other (6) valid 
 
 moods. 
 Invalid Moods. 
 
 1. In any invalid mood of the first figure make 
 antecedent and consequent worse in the same degree. 
 
 2. In any invalid mood of the first figure interchange 
 antecedent and consequent. 
 
 3. In any invalid mood convert simply in any form 
 but 7. 
 
 Postulates:* 
 
 {a ab /0ab}'; {dab/Tab}'; (a a b/ Cab}'; {7ab/7ba}'. 
 
 Theorems: The other (21) invalid moods. 
 
 6. We may also formulate rules for the detection 
 of the invalid moods. These are: 
 
 *The mark over the bracket is interred to indicate that 
 the mood is invalid, 
 
12 A PRIMER OF LOGIC 
 
 1. If the antecedent be worse than the consequent, 
 the mood is invalid. 
 
 2. If the antecedent be better than the consequent, 
 the mood is invalid. 
 
 Definition. Distributed terms are those modified, 
 either implicitly or explicitly, by the adjective all, i. e. 
 the subject of the a-, 7- and e-form, and the predicate of 
 the a- and e-form. 
 
 3. it a term be distributed in the consequent but 
 undistributed in the antecedent, the mood is invalid. 
 
 7. A syllogism is a form of implication belonging 
 to one of the types: 
 
 1. x ba y cb Z z ca =(xy z) x 
 
 2. x ab y cb Z z ca =(xy z) 2 
 
 3. x ba y bc Z z ca =(x y z) 3 
 
 4. x ab y bc Z z ca =(xyz) 4 
 
 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 predi- 
 cate of the conclusion is called the major term and points 
 out the major premise, which by convention is written 
 first, and the subject of the conclusion is called the minor 
 term and points out the minor premise. The term common 
 to both premises and which does not appear in the con- 
 clusion is called the middle term. 
 
 Since x, y and z may have any one of the values, 
 a, ft 7, e, there will be sixty-four ways in each one of the 
 four figures, called the moods of the syllogism, in which 
 x y Z z can be expressed. There will be consequently 
 
A PRIMER OF LOGIC 13 
 
 two hundred and fifty-six cases to consider. Twenty- 
 nine of these are valid implications; the remaining two 
 hundred and twenty-seven are invalid. From the rules 
 and postulates below, all the moods, valid and invalid, 
 may be deduced. 
 
 Valid Moods. 
 
 1. In any valid mood of the third figure make a like 
 major premise and conclusion worse in the same degree. 
 
 2. In any valid mood of the second figure make a 
 like minor premise and conclusion worse in the same degree. 
 
 3. In any valid mood convert simply in any form 
 but 7. 
 
 Postulates : Theorems : 
 
 (aaa) x (aaa) 2 , 3(4 (/3aj8) If2f3i4 
 
 (7^)2,4 
 
 Invalid Moods. 
 
 1. In any invalid mood of the third figure make a 
 like major premise and conclusion better in the same degree. 
 
 2. In any invalid mood of the second figure make 
 a like minor premise and conclusion better in the same 
 degree. 
 
 3. If the premises and conclusion of an invalid mood 
 in the fourth figure are all alike, make them all worse in 
 the same degree. 
 
 4. If the premises and conclusion of an invalid mood 
 are all alike make the conclusion any degree better or any 
 degree worse. 
 
14 A PRIMER OF LOGIC 
 
 5. If the premises and conclusion of an invalid mood 
 are all unlike, interchange them in any order. 
 
 6. In any invalid mood convert simply in any form 
 but 7. 
 
 Postulates:* 
 
 (aa/3)', (tfc)', tfry)', (777)'. W, 
 
 (aa)', 
 (aPy)', 
 
 Theorems: 
 The other (208) invalid moods. 
 
 8. As in the case of immediate inference we may 
 formulate rules for the detection of the invalid moods of 
 the syllogism. These are five in number. 
 
 1. A mood is invalid if the conclusion differ from the 
 worse premise. 
 
 2. A mood is invalid if an a- and a 7- premise be 
 conjoined in the antecedent and the middle term be un- 
 distributed in the major premise. 
 
 3. A mood is invalid if the middle term be undis- 
 tributed in both premises. 
 
 4. A mood is invalid if a term which is distributed 
 in the conclusion be undistributed in the premise. 
 
 5. A mood is invalid if each premise be in the 
 e-form.** 
 
 *The mark (') over the bracket is intended to indicate that 
 the mood is invalid. 
 
 **These rules are, of course, not sufficient to declare (766)2,4 
 and (7c) I)2 invalid, in case we decide to so regard them. See 
 the concluding remarks of chap. III. 
 
A PRIMER OF LOGIC 15 
 
 EXERCISES 
 
 If x a ,b^ y a ,b be denoted simply by (xy), (the comma be- 
 tween the terms indicating that the term order and so the figure, 
 is not determined), the array of sixteen propositions may be 
 constructed thus: 
 
 aa fia yy. ea 
 
 "^ 00 70 $ 
 
 or? 07 77 7 
 
 ae j8c 76 ee 
 
 the moods valid in both figures being underlined twice, the one 
 valid only in the first figure being underlined once. Applying 
 the first rule to the postulate, we obtain in succession, j3/3, 77, 
 ee, in the first figure; and converting simply in the consequent 
 of those valid in the first figure, except 77, we obtain aa, /3/3, ce 
 in the second figure. 
 
 (1) From the rules and postulates for the derivation of the 
 invalid moods deduce the remaining twenty-one invalid moods. 
 
 The rules for the immediate detection of the invalid moods 
 are all necessary if we can point to at least one example which 
 falls uniquely under each rule. They are sufficient if they de- 
 clare all the invalid moods to be invalid. 
 
 (2) Construct the set of propositions of immediate inference 
 and place after each invalid mood the number of a rule which 
 declares it to be invalid. 
 
 (3) Make a list of moods which are declared invalid by the 
 first rule and by no other rule, and a list of moods which are 
 declared invalid by the second rule and by no other rule. 
 
 |) 
 
 (4) Find an invalid mood which is declared invalid by the 
 third rule and by no other rule and prove that it is the only 
 unique illustration of this rule. 
 
 For those who approach the study of the syllogism for the 
 first time, it may facilitate manipulation to point out the general 
 effect of conversion in the form of certain rules. 
 
 1, Simple conversion in the major premise changes the 
 first figure to the second and conversely, the third figure to the 
 fourth and conversely. 
 
16 A PRIMER OF LOGIC 
 
 2. Simple conversion in the minor premies, changes the 
 first figure to the third and conversely, the second figure to the 
 fourth and conversely. 
 
 3. Simple conversion in the conclusion changes the first 
 figure to the fourth and conversely and leaves the second and 
 third figure unchanged. 
 
 It must of course not escape the beginner's notice that the 
 effect of converting simply in the conclusion is to interchange the 
 premises, since the major term then becomes the minor term and 
 the minor term becomes the major term. The conjunctive 
 relation of logic being commutative, the order of the premises 
 is indifferent, but we agree, as a matter of convention, always 
 to write the major premise first. 
 
 (5) From (eye)i, and the third rule under the valid moods 
 alone, deduce (eye) 2 , (7ee) 2 and (yee) 4 . 
 
 (6) From the rules and postulates deduce the remaining 
 valid moods. 
 
 (7) Assuming only the third rule under the valid moods 
 and the rule : in any valid mood of the first figure make a like 
 major premise and conclusion worse in the same degree, deduce 
 all the remaining valid moods from (aaa) z , (777)1 and (077)1. 
 
 (8) Assuming only the second and third rules under the 
 valid moods deduce the remaining valid moods from (aaa) lt 
 (777)i, (7<*7)i and 
 
 (9) From (efieYt alone deduce seventy-eight other invalid 
 moods. 
 
 (10) From (afieYt alone deduce twenty-three other invalid 
 moods. 
 
 (11) Deduce the invalid moods in the first figure which have 
 a 7-minor premise. 
 
 The rules for the immediate detection of the invalid moods 
 are sufficient, if they declare all the moods not already found 
 to be valid to be invalid. They are all necessary if we can point 
 to at least one example which falls uniquely under each rule. 
 
 (12) Construct the array of the syllogism and place after 
 each invalid mood the number of a rule that declares it to be 
 invalid, 
 
A PRIMER OF LOGIC 17 
 
 (13) Show that it follows from one of the rules alone that 
 two jS-premises do not imply a conclusion. 
 
 (14) Prove that there are only two moods which illustrate 
 the second rule uniquely. 
 
 (15) Make a list of examples which fall uniquely under 
 each one of the rules. 
 
18 A PRIMER OF LOGIC 
 
 CHAPTER III 
 
 9. In this third chapter it is proposed to completely 
 define the relationships of better and worse by deducing all 
 the true and all the false propositions into which these 
 relationships may enter and then to give a complete ex- 
 pression in the language of symbols of the rules for the 
 deduction of the moods of immediate inference and the 
 syllogism. 
 
 10. Let us, first of all, invent symbols to denote 
 worse than, doubly worse than, and trebly worse than, i. e. 
 
 x / y =x is worse than y, 
 
 x // y =x is doubly worse than y, 
 
 X ///Y =x is trebly worse than y, 
 
 and let us add the following: 
 
 Definition. In the propositions, x / y, x//y, and 
 x///y, x is called the inferior, y the superior form. 
 
 Since x and y may take on any of the four forms 
 a, /3, 7, e, there will be sixteen possible propositions of each 
 type, x / y, x //y and x///y, obtained by permuting the 
 letters two at a time and by taking each letter once with 
 itself. The following postulates and principles will yield 
 all the valid moods of each type. We have assumed four 
 principles here because the principles for the deduction of 
 the invalid moods may be derived from these four as 
 theorems. 
 
 Principles: 
 
 i. (x / y) (y //z) Z (x///z) iii. (x / z) (y///z) Z (y //x) 
 ii. (x / y) (z //x) Z (z///y) iv. (x / z) (y // z) Z (y / x) 
 
 Postulates: /3/a; 6/7; 7//a. 
 
 Theorems: 7/0; e//0; e///a. 
 
APRIMER OF LOGIC w 
 
 We may also formulate rules for the derivation of 
 the moods. It will then be necessary to assume one pos- 
 tulate only. 
 
 Definition: In the propositions, x / y, x //y and 
 x///y, the relation connecting x and y is known as the 
 worse-relation. 
 
 Definition: Trebly worse (///) is worse than doubly 
 worse (//) and doubly worse than worse (/). Doubly 
 worse (//) is worse than worse (/). 
 
 The rules are: 
 
 1. In any valid mood make superior and inferior 
 form one degree better or one degree w r orse. 
 
 2. In any valid mood make inferior form and worse- 
 relation one degree better or one degree worse. 
 
 Postulate : Theorems : 
 
 /3 / a. The other (5) valid moods. 
 
 The invalid moods of each type may be derived from 
 the following postulates and principles: 
 
 Principles:* 
 
 i. (x / y) (x///z)'Z(y//z)' 
 
 (x///z)'(y//z) Z(x / y)' 
 ii. (x / y) (z///y)'Z(z//x)' 
 
 (z///y)' (z//x) Z(x / y)' 
 iii. (x / z) (y//x)'Z(y///z)' 
 
 (y//x)'(y///z) Z(x / z)' 
 
 iv. (x / z) (y / x)'Z(y//z)' 
 (y / x)'(y//z) Z(x / z)' 
 
 These principles follow from those used for the deduction 
 of the valid moods by (xyZ z) Z (xz'Z y'). 
 
20 A PRIMER OF LOGIC 
 
 Postulates: Theorems: 
 
 (a/VAX; (0//A)'; (7///e)'; The other (3 6) invalid moods. 
 
 (///)'; (e///7)'; 
 
 As in the case of the valid moods, rules may be form- 
 ulated for the derivation of the invalid moods. Here it 
 will be necessary to assume only three postulates. The 
 rules are: 
 
 1. In any invalid mood make superior and inferior 
 form one degree better or one degree worse. 
 
 2. In any invalid mood make inferior form and worse- 
 relation one degree better or one degree worse. 
 
 3. In any invalid mood make superior form three 
 degrees worse. 
 
 Postulates: Theorems: 
 
 / 7)'; (7 / a)'; (e / a)'. The other (39) invalid moods. 
 
 11. Having now completely defined the relationships 
 of better and worse by deducing all the prepositional forms 
 into which these relationships may enter, there remains 
 for this chapter only one other task, which is to deduce 
 symbolically the moods of immediate inference and the 
 syllogism. 
 
 12. From the postulates and principles, which are 
 given below, all the moods, valid and invalid, of immediate 
 inference may be deduced. 
 
 Principles: 
 
 i. (y / x) (x b Z x a b ) Z (y a b ^ y a b). 
 iv. (xZ y) (yZ z)Z (xZ z). 
 
 Postulates : Theorems : 
 
 aab^ a ba*, jSabZ /3 ba ; &b Z. e ba . The other valid moods. 
 
A PRIMER OF LOGIC 21 
 
 Principles:* 
 
 ii. (x / y) (x a b / x a b ) Z (x a b Z y a b)' 
 (x//y) (x a b ^ x a b ) Z (x a b Z y a b )' 
 
 iii. (y / x) (x a b / x a b ) Z (x a b Z y a b )' 
 (y//x) (x a b / x a b ) Z (x a b Z y a b )' 
 
 v. (xZ y) (xZ z)'Z (yZ z)' 
 (xZ z)' (yZ z) Z (xZ y)' 
 
 Postulates: Theorems: 
 
 (dab^ c ab )'; (e ab Z a ab )'; (Tab^ 7ba)'. The other invalid 
 
 moods. 
 
 13. All the valid and invalid moods of the syllogism 
 may be deduced from the assumptions which follow. 
 The right to convert simply in any form but 7 is ex- 
 pressed under v and vi. It will be evident that some of 
 the postulates might have been saved at the expense of 
 introducing new principles, and conversely. The first 
 two principles for the deduction of the invalid moods 
 under iv are theorems from the ones that have gone be- 
 fore under i, by (xy Z z) Z (xz' Z y'). 
 
 Principles: 
 
 i. (y / x) (x ba z bc / x ca ) Z (y ba z bc Z y ca ) 
 (y / x) (z ab x cb Z x ca ) Z (z ab y c b ^ y ca ) 
 
 v. (xy Z z) (z Z w) Z (xy Z w) 
 (xy Z z) (w Z x) Z (wy Z z) 
 (xy Z z) (w Z y) Z (xw Z z) 
 
 *These principles, except the second under iii are really 
 special cases of principles i and iii under the syllogism, obtained 
 from the latter by making b = c, the primed part of the ante- 
 cedent in iii becoming unprimed in the special case. 
 
22 A PRIMER OF LOGIC 
 
 Postulates: (aaa) z ; (777) r 
 
 Theorems: The other (26) valid moods. 
 
 Principles: 
 
 ii. (y / x) (z I y) (xyZ z)' Z (xz Z y)' 
 (y / x) (z//y) (xyZz)'Z(xzZ y)' 
 (y//x) (y / z) (xyZz)'Z(xzZy)' 
 (y///x) (y / z) (xyZ z)'Z (xzZ y)' 
 (y /x) (z / y) (xyZz)'Z(zyZx)' 
 (y / x) (z//y) (xyZz)'Z(zyZx)' 
 (y//x) (y / z) (xyZz) / Z(zyZx) / 
 (y///x) (y / z) (xyZ z)'Z (zyZ x)'. 
 
 iii. (x / y) (x a , b x b , c Z x ca )' Z (x a , b x b , c Z yea)' 
 (x // y) (x a , b x btC Z Xca)' Z (x a , b x b , c Z y ca )' 
 (y / x) (x a , b x b ,o Z x ca y Z (x a , b x b ,c Z y ca )' 
 
 iv. (x / z) (y ab x cb Z Xca)' Z (y ab z cb Z z ca )' 
 (x / z) (x ba y bc Z x ca )' Z (z ba y bc Z z ca )' 
 (y / x) (x ab x bc Z x ca )' Z (y ab y bc Z y ca )' 
 
 vi.* (xy Z z)' (w Z z) Z (xy Z w)' 
 (xy Z z)' (x Z w) Z (wy Z z)' 
 (xy Z z)' (y Z w) Z (xw Z z)' 
 
 (aetf)', (a/3 7 )' t (ae 7 )'x 
 
 (aa 7 )'x (ajSe)', (^a 7 )' x (jSeyX, ( 777 )' 2 (e^ 7 )' x (e 7 e)' 3 
 
 (aac)' z (a 77 )' 3 (jSjSe)', ( 7 a 7 )' 2 ( 7 c 7 )' 
 
 The other (206) invalid moods. 
 
 *Principles v and vi are of course not independent. The 
 first under v is a variation of transitivity, the third a variation 
 of the second by xyZ yx. Those under vi follow from those 
 under v by (xy Z z) Z (xz' Z y') . Principles v under immediate 
 inference follow from transitivity by the same principle. 
 
A PRIMER OF LOGIC 23 
 
 14. We have already pointed out, (note p. 2), that 
 the product 7 a ,b e a , b does not vanish in general if we allow 
 the possibility of the limiting values and 1 for the terms. 
 Under these conditions, (7ee) 2 , 4 and (ej) I>2 are not valid 
 moods of the syllogism, for they become 7<>i e 0tl /. 0, for 
 a = c =0 and b = 1. A logic, which recognizes these limiting 
 values of the terms, will have to postulate (eye)'^ say, 
 which yields, (eye)' 2 and (yee) ' 2 , 4 . 
 
 The only change, which we should then have to make 
 in chapters II and III, would be to replace (777) ' 2 among 
 the postulates by (eye)*!, from which (777) ' 2 follows as a 
 theorem, and to subtract (eye) It2 and (yee) 2 , 4 from the list 
 of valid moods. 
 
 This logic, which might be called non-Aristotelian, or 
 semi-Aristotelian, or imaginary logic, is more general than 
 the ordinary or classical logic and includes the latter as 
 a special case, becoming, in fact, identical with it when the 
 field of its application is narrowed so as to exclude ''nothing" 
 and "universe" as limiting values of the terms. One 
 principle, which is true in the special, but not in the general 
 case, is: 
 
 (y / x) (x ba z cb / x ca ) Z (v ba z cb Z y ca ), 
 
 and this principle may be regarded as the differentiating 
 character of the two cases. If we had chosen to assume 
 it, instead of the first principle under i, we could have saved 
 the third postulate, but the second principle under iv 
 would not then have followed as a theorem. 
 
 The definitions of chapter I, 3, hold for both cases; 
 the only change to be made in implications V in order to 
 make them true in the more general logic, will be to replace 
 7aa =0 by 7aa4 : 0, and this property has not been made use 
 of in defining the 7-form, 
 
24 A PRIMER OF LOGIC 
 
 The Aristotelian forms, A, E, I, O, (see Appendix I), 
 will yield only eight valid moods of the syllogism, under the 
 new condition, instead of the twenty-four valid moods 
 commonly recognized. They satisfy all the conditions of 
 maximum simplicity in the special or classical instance 
 they are the best possible forms to choose for the con- 
 struction of an Aristotelian logic but they fail in the gen- 
 eral instance, for they then lose their peculiar advantage, 
 that, corresponding to any member of the set there should 
 be another member of the set which represents its contra- 
 dictory. 
 
A PRIMER OF LOGIC 25 
 
 EXERCISES 
 
 By the aid of the principles, 
 
 (xy Z z) (z Z w) Z (xy Z w) 
 (xy Z z) (w Z x) Z (wy Z z) 
 (xy Z z) (w Z y) Z (xw Z z) 
 
 we are enabled to convert in either premise or the conclusion. 
 The example which follows will illustrate the method. 
 
 (7ba <*cb Z Tea) (o%Z Otj\c) Z (7b a bc ^ Tea) 
 
 (1) From (eye) x derive (7e) 2 , 4 . 
 
 (2) From the principles and the postulates in the text de- 
 duce the remaining valid moods. 
 
 (3) From the postulates, (aaa) lt (777)1, (OTY)I and the prin- 
 ciple (y/x) (x ba z cb Z x ca ) Z (y ba z c b ^ y ca ) deduce the remaining 
 valid moods. 
 
 If we identify the subject and predicate of the conclusion 
 in the mood, (Pafi) 3 , we obtain |3babaZ 0, (chapter I, impli- 
 cations v). By the aid of (xyZ 0) Z (xZ y') it follows that 
 
 abZ /3'ab- 
 
 (4) Deduce as many as possible of the propositions of the 
 form, x ab Z y' a b, (chapter I, p. 2) by identifying subject and 
 predicate in the conclusion of the valid moods of the syllogism. 
 
 The principles, 
 
 (xyZ z)'(wZ z)Z (wyZ w)' 
 (xyZ z)'(xZ w)Z (wyZ z)' 
 (xyZ z)'(yZ w) Z (xwZ z)' 
 
 enable us to convert in either premise or the conclusion. 
 
 (5) From (fi8/3)'x, derive the invalidity of this mood in the 
 other figures. 
 
 (6) From (e/3e)'i alone and principles iii, iv and vi deduce 
 seventy other invalid moods. 
 
 (7) From (ajfryX, alone and principles ii deduce nineteen 
 other invalid moods, 
 
26 A PRIMER OF LOGIC 
 
 (8) Deduce the invalid moods in the third figure, whose 
 conclusion is in the 7-form. 
 
 (9) Deduce the invalid moods in the fourth figure, whose 
 major premise is in the 7-form. 
 
 (10) From 7771 alone, deduce forty-six valid implications 
 of the form L a ,b Mb.c N' ca , L, M and N representing only 
 the unprimed letters. 
 
 (11) Assuming eye lt2 and yee 2t4 to be invalid, show that 
 777! yields only thirteen valid implications of the form given 
 in the last exercise. 
 
 (12) Assuming (/3ftJ')'i (B^')\ (/S/Se 7 )'. (770')'= (rtfO'j (777')'. 
 
 (77*')'* (erf% (rO'i ("0')'. (O'. 
 
 deduce sixty-nine other non-implications of the same form. 
 
 Any non-implication of the form, Lb, a M c ,b ^ N' ca , which 
 contains an a-form may be proven invalid by identifying terms 
 in the a-form. Thus /3b a Pbc oi c& reduces to /3b a / /3'ba for 
 c = a; 7b a acb^ 7' ca reduces to 7 ba Z 7 r ba for c = b, etc. 
 
 (13) Establish the invalidity of the thirty-four non-impli- 
 cations of the form Lb, a M a ,b/ N' ca not accounted for in the 
 preceding exercise. 
 
 (14) Show that there are thirty-six, and only thirty-six, 
 distinct valid implications of the form L a>b Mb, c N c , a Z 0, 
 L, M and N representing only the unprimed letters, a, /3, 7, e. 
 
 (15) Derive indirectly the (777') 'i of exercise 12. 
 
 A certain number of the postulates for the derivation of the 
 invalid moods of the syllogism (p. 21) may be shown indirectly 
 to be invalid by reducing them to invalid moods of immediate 
 inference. Thus (aa/3)i reduces to a^Z. c/ a b when the terms 
 in the conclusion are identified, and (a&y) i reduces to /3 ca ^ Tea, 
 when we identify terms in the major premise and suppress the 
 part a && (see chap. IV). 
 
 (16) Establish the invalidity of 
 
 (#)! ( 7) i 
 
 (copy)! (0:77)3 (707) 
 
A PRIMER OF LOGIC 27 
 
 Most of the other postulates for the deduction of the invalid 
 moods of the syllogism may be reduced by the method of the 
 following example: 
 
 Suppose that (epy)i is valid. Now (e/3y')i is valid by a 
 preceding exercise. 
 
 * ( ba &b Z Tea) ( ba 0cb Z T'CB) Z (*ba fob ^ 0) 
 
 since (xZ y) (xZ y') Z (xZ 0). 
 
 Consequently ba 0cb ^ c ca . 
 
 If now we postulate (ej8e)' x , it follows that 
 
 (17) Establish the invalidity of 
 
 (77)3 (707) 2 (777)4 
 
 (7*7)2 (77)3 (cea)! 
 
 The postulates (Pey)'i (ej8e)'x and (eye)' 3 that remain (p. 21) 
 may be reduced by the following method: 
 
 (jSba &b / e' ca )' (|8ob ^ Vcb) ^ (^ba 7'cb / e'ca)' 
 
 by a principle under vi and the postulate of a preceding exercise 
 
 GSba 7'cb/ OVtfba tcaZTcb)' 
 
 which yields (Pey)^ by simple conversion in the major premise. 
 (18) Establish the invalidity of ( c/3 e) z and ( ey e) 3 . 
 
 Any non-implication of the form L a , b M b , c ^ N ca , which 
 contains an a-premise, may be reduced to an invalid mood of 
 immediate inference, and so shown to be invalid, by identifying 
 terms in the a-premise. All of the other invalid moods may be 
 derived from the postulates of exercise (12), the forms of im- 
 mediate implication given in chapter I, the principles iv and vi 
 of chapter III, together with (xy Z z) Z (xz'Z y') and (xy Z z) Z 
 (z'y^ A 
 
 19. From the postulates of exercise (12) deduce all the 
 non-implications of the form L a ,b Mb, ^ N ca , without making 
 use of principles ii and iii of this chapter. 
 
 20. Show that there exist no valid implications of the 
 form L ; a ,b Mb, c Z N ca or L a ,b MV C ^ N ca and consequently 
 none of the form L ; a , b M'b.c Z N Qa or L' a ,b M^.o Z N' ca . 
 
CHAPTER IV 
 
 15. The sorites is a form of implication of the 
 general type:* 
 
 Xx(i |2 ) x 2 ( 2 ,3) x 3 ( 3f 4) x n -x(n7T, n) Z x n (n i), 
 
 in which the number of terms is greater than three. 
 
 Certain valid moods of the sorites can be constructed 
 from chains of valid syllogisms. Thus the chain of syl- 
 logisms: 
 
 a(i,2) 0(2,3) Z a(3i), 
 
 a(3i) 0(3,4) Z a(4i), 
 
 0(41) 0(4,5) Z a(si), 
 
 will yield a valid sorites, viz : 
 
 0(1,2) 0(2,3) 0(3,4) 0(4,5) Z 0(51), for 
 
 { 0(1,2) 0(2,3) Z 0(31) } Z { 0(1,2) 0(2,3) 0(3,4) Z 0(31) 0(3,4) } 
 
 .'. 0(1,2) 0(2,3) 0(3,4) Z 0(41), by the second syllogism and 
 
 the principle of transitivity. 
 
 { 0(1,2) 0(2,3) 0(3,4) Z 0(41) } Z 
 
 { 0(1,2) 0(2,3) 0(3,4) 0(4,5) Z 0(41) 0(4,5) } 
 
 . *. 0(1,2) 0(2,3) 0(3,4) 0(4,5) Z o(si), as before. 
 
 Consequently in general, if 
 
 X I (l,2) X a (2.3) Z X 3 (3l) 
 X 3 (3l) X 4 (3,4> Z X s () 
 X s (4l) X 6 (4,5) Z X 7 (si) 
 
 X 2n - s (n-i i) X 2n . 4 (n-i, n) Z X 2n - 3 (ni) 
 
 *In this chapter it will be more convenient to employ the 
 notation x(ab) for x a b or x(i,2) for x lf2 . The comma between 
 the terms means that the term order is not settled. 
 
A PRIMER OF LOGIC 29 
 
 be a chain of valid syllogisms, then 
 
 x z (i, 2 ) x 2 ( 2|3 ) x 4 (3, 4 ) x 2n _ 4 (iiT7, n) Z x 2n . 3 (ni) 
 
 is a valid mood of the sorites. It remains to be proven 
 that the only valid moods that exist can be constructed 
 from chains of valid syllogisms. The proof depends on 
 the following principles. 
 
 Principle i. A valid mood of the sorites, which has 
 one premise of the same form as the conclusion, will re- 
 main valid, when as many of the other premises as we 
 desire are put in the a-form. 
 
 Principle ii. A valid mood of the sorites will remain 
 valid, when as many terms have been identified as we desire. 
 
 Principle iii. An a-premise, whose subject and predi- 
 cate are identical, may be suppressed as a unit multiplier. 
 
 Principle iv. A valid mood of the sorites, none of 
 whose premises has the same form as the conclusion, will 
 remain valid, when as many premises as we desire are put 
 in the a-form. 
 
 Theorem i. There exists no valid mood of the sorites, 
 in which none of the premises has the same form as the 
 conclusion. 
 
 For (principle iv) put all the premises after the first 
 in the a-form. Then by identifying terms (principle ii) 
 the mood of the sorites can be reduced (principle iii) to an 
 invalid syllogism of the form: 
 
 X x (l,2) a(2,3) Z X n (3l). 
 
 Conclusion in the a-form. 
 
 At least one of the premises is in the a-form (theorem i). 
 If one of the remaining premises, x r (s - i, s), be not in the 
 a-form, put each one of the other premises in the a-form, 
 
30 A PRIMER OF LOGIC 
 
 if all but x r be not already in that form (principle i). 
 Then by identifying terms (principle ii) the mood of the 
 sorites will reduce (principle iii) to an invalid syllogism 
 of the form: 
 
 x r (s - i, s ) a(s, s + 1) Z a(s + 1 s - i), 
 or a(s - 2, s - i) x r (s - i, s) Z a(s s - 2). 
 
 Consequently all the premises are in the a-form if the 
 mood of the sorites is valid and the sorites is of the general 
 type: 
 
 0(1,2) 01(2,3) a(n- i, n) Z a(ni), 
 
 which can be constructed from the chain of valid syllo- 
 gisms : 
 
 a(2,i) a(3, 2 ) Z a(3i), 
 
 a(3i) 01(4,3) Z a(4i), 
 
 a(4i) 01(5,4) Z a(si), 
 
 a(n - 1 i) a(n, n - i) Z a(n i). 
 
 Conclusion in the /3-form. 
 
 At least one premise, x t , is in the /3-form (theorem i), 
 and all the other premises are in the a-form. For suppose 
 one of the other premises x r (s - 1, s) were not in the a-form. 
 Put all the premises (principle i) except x t and x r in the 
 a-form. Then by identifying terms (principle ii) the mood 
 of the sorites will be reducible to an invalid syllogism 
 (principle iii) of the form: 
 
 0(s- i, 3-2) x r (s, s- i) Z |8(s s- 2), 
 or x r (s, sT~7) j8(s, s + i) Z P(s~+i s- i). 
 
 Consequently the sorites must be of the form: 
 0(2,3) a(s, s - i) /3(s + i s) a(s + i, 5+2) a(n- i, n) 
 
A PRIMER OF LOGIC 31 
 
 Z |8(n i), which can be constructed from the chain of 
 syllogisms : 
 
 0(1,2) 0(2,3) Z a(ai) 
 a(ai) 0(3,4) Z 0(41) 
 
 a(s - i i) a(s - i, s) Z a(s i ) 
 a(si) j8(s, 
 
 |8(sTi i) o(sTii s+2 ) Z 0(s+2 i) 
 
 0(n-i i) a(n-i, n) Z #(n i). 
 
 Conclusion in the y-form. 
 
 At least one of the premises is in the 7- form (theorem i). 
 Each 7-form in the antecedent must present its terms in 
 the order (s s - 1). For suppose that y(s - 1 s) should 
 appear as one of the premises. Put each one of the 
 remaining premises in the a-form (principle i). Then by 
 identifying terms (principle ii) the sorites will reduce to 
 an invalid syllogism (principle iii) of the form: 
 
 T(S - i s) a(s, s +1) Z y(s +1 s - 
 
 or a(s - i, s - 2) T(S - 1 s) Z y(s s - 2). 
 
 Pursuing the same reasoning as before it can be shown 
 that no |8- or e- premises can occur. One form of this 
 sorites may consequently be 7(21) 7(11 n- 1) Z 7(111), which 
 can, in fact, be constructed from the chain of valid syl- 
 logisms: 
 
 7(21) 7(32) Z 7(31) 
 
 7(31) 7(43) Z 7(41) 
 
 7(n~M~ 7(n iiTi) Z 7(11 i). 
 
 All the other forms of valid sorites with a 7-conclusion 
 are obtained from the above type by transforming one or 
 
32 A PRIMER OF LOGIC 
 
 more of the premises into the a-form in every possible 
 way under the restrictions of theorem i. Each one of 
 these types can be built up from a chain of valid syllo- 
 gisms each member of which has one of the forms : 
 
 (77T)i, (a77)i,2, or (7017) i, 3- 
 
 Conclusion in the e-form. 
 
 At least one of the premises is in the e-form (theorem i), 
 and there is not more than one e-premise. For, if there 
 are two or more e-premises, put all the premises but two 
 of the e-premises in the a-form (principle i). Then by 
 identifying terms (principle ii) we will come upon an invalid 
 syllogism (principle iii) of the form: 
 
 e(s - i, s) e(s, s+i)Z e(s + i s - i). 
 
 There can be present no /3-premise. For suppose 
 x r (s, s - l) to be a /3-premise. Put all the premises except 
 x r and the e-premise in the a-form (principle i). By identi- 
 fying terms (principle ii) we will come upon an invalid 
 syllogism (principle iii) of the form: 
 
 /3(s, s- i) e(s, s+i) Ze(s + i s- i), 
 or e(s - i, s - 2) /3(s, s - i) / e (s s - 2). 
 
 Any 7-premise coming after the e-premise must pre- 
 sent its terms in the order (s s - i). For suppose 7(5, s - i) 
 coming after the e-premise to present the term order 
 (s- i s). Put all the premises except y(s - i s) and the 
 e-premise in the a-form (principle i). Then by identifying 
 terms (principle ii) we will come upon an invalid syllogism 
 (principle iii) of the form: 
 
 e(s -1,5-2) 7(s - i s) Z e(s s - 2). 
 
 Any 7-premise coming before the e-premise must 
 present its terms in the order (s - 1 s). For suppose 7(3, s - i) 
 
A PRIMER OF LOGIC 33 
 
 coming before the e-premise to present the term order 
 (s s - i). Put all the premises except 7(5 s - i) and the 
 c-premise in the a-form (principle i). Then by identifying 
 terms (principle ii) we will come upon an invalid syllogism 
 (principle iii) of the form : 
 
 T(S s- i) c(s, s + i) Z e( 
 
 One form of this sorites may be, consequently, 
 
 7(12) 7(23) 7(3 - 2 T~i) e(s, s - i) 7(s+i s) 7(n n~^7) Z 
 e(n i), which can, in fact, be constructed from the chain 
 of valid syllogisms: 
 
 7(12) 7(23) Z 7(13) 
 7(13) 7(34) Z 7(14) 
 
 7(l S - 2) 7(s - 2 S - l) Z 7(1 S - l) 
 
 7(1 s - i) e(s, s- i) Z c(s i) 
 c(s i) 7(s+7 s) Z e(s~+T 
 
 e(n- i i) 7(n n - 1) Z e(n i) 
 
 All the other forms of valid sorites with an c-conclusion 
 are obtained from the above type by replacing one or more 
 of the 7-forms by a-forms in every possible way. Each 
 new type can be constructed from a chain of valid syllo- 
 gisms, each member of which has one of the forms: 
 
 There exist, consequently, no valid moods of the sorites 
 which can not be constructed from chains of valid syllo- 
 
 gisms.* 
 
 *If (7c) 2 , 4 and (eye) If2 are to be regarded as invalid moods, 
 (see the concluding remarks of chapter III), then it can be shown 
 at once that no 7-premise can occur when the conclusion is in 
 the e-form. The general form of such a sorites will be, 
 
34 A PRIMER OF LOGIC 
 
 EXERCISES 
 
 (1) Construct a valid sorites from the chain of valid syllo- 
 gisms : 
 
 a 2 i7 3 2^ 7 3 i 
 
 7 3 i<*43^ 7 4 i, 
 
 7 4 x ?54 ^ 7si- 
 
 (2) By the aid of the principles of chapter IV, reduce the 
 valid sorites, a 21 y 32 a 43 y 54 Z 7 SI , successively to each one of the 
 three valid syllogisms of example 1. 
 
 (3) Prove the invalidity of the sorites, 
 
 7zi 73* 734 754 ^ 7 S i- 
 
 (4) From what chain of valid syllogisms can the sorites, 
 i,2723 3 , 4 7s46 f 5^ 6i be constructed? 
 
 (5) If (7e)i. 2 and (7e) 2 , 4 be regarded as invalid moods of 
 the syllogism, (see the concluding remarks of chap. Ill), prove 
 the invalidity of the sorites, 
 
 7i2 7a3"7s-a s-i s, s-i 7s+i s~7n n-i Z nr 
 
 a(i,2)a(2 t 3) a(s - i.s)e (s, s + i)a(s+i, 8+2) a(n- i, n)/ e(ni) 
 which can be built up from the chain of syllogisms, 
 
 a(i,2)a(2,3)Z a(si) 
 
 a(3i)a(s,4)Z a(4i) 
 
 a(s - i i)a(s - i, s)Z a(si) 
 a(si) (s, sTi) Z e(s+i i) 
 c(s + i i) a (s-H, s+2)Z e(s+2 
 
 c(n - i i )a (n - 1 1 n) Z c (n i) 
 
APPENDIX I 
 
 On the Simplification of Categorical Expression and the 
 Reduction of the Syllogistic Figures 
 
 If a and b represent classes, there are four ways in 
 which they may be related categorically, the one standing 
 for subject, the other for predicate. These four forms of 
 relationship are always represented by the letters, A, 
 E, I, O, i. e. 
 
 A a b=all a is b, 
 E a b=no a is b, 
 I ab = some a is b, 
 Oab=not all a is b. 
 
 Historical efforts have, been made to reduce the num- 
 ber of these relationships. If symbols be invented to 
 denote some a (a) and not-a (a x ), the last three may be 
 represented by means of the first, for: 
 
 E a b=Aabi> Iab=A a b> O a b=A a bi- 
 
 But an essential difference is here left undistinguished 
 and the number of necessary forms will not have been 
 reduced by this device. If a new symbol be employed for 
 all a (a) and another for the copula is ( Z ), we shall have: 
 
 A a b=aZ b, 
 E ab =aZ bi, 
 lab = aZb, 
 O ab =aZ bx. 
 
 The four separate categorical forms have, accordingly, 
 been gotten rid of at the cost of introducing four new unde- 
 fined symbols, so that no economy of our indefinables has 
 been effected. 
 
36 A PRIMER OF LOGIC 
 
 It is to be observed that the word some, which is im- 
 plicit or explicit in the meaning of part of each proposition, 
 means some at least, possibly all. Another set of proposi- 
 tions, in which some is to mean some at least, not all, may 
 be used to replace the traditional ones. These other forms 
 are: 
 
 a ab = all a is all b, 
 
 /3 a b=some a is some b, 
 
 Tab = all a is some b, 
 
 ab =no a is b. 
 
 In addition we shall have to employ 
 the hypothetical form, 
 
 x ab / y & b =x a b implies y ab , 
 { x ab ^ y a b} r = x ab does not imply y ab , 
 
 the conjunctive form, 
 
 Xab ' y a b = x ab and y ab , 
 
 the disjunctive form, 
 
 X ab 
 
 Each member of the set, A, E, I, O, may be expressed 
 in the members of the set, a, /3, 7, c, and conversely, so 
 that the two are, in fact, logically equivalent, although 
 each one has certain advantages peculiar to itself. 
 
 The members of the second set have this property, 
 that, if one is true, then all the others are false.* We 
 assume, accordingly, the 
 
 Postulates: a &b L /3' ab 0ab ^ T'ab Tab / T'ba 
 dab T'ab jSab c'ab 
 ab ^ c'ab Tab ^ c'ab 
 
 *Provided we exclude the limiting values and 1 for a and b. 
 The ordinary definitions of these limits allow 7 i and c i be true 
 together. 
 
A PRIMER OF LOGIC 37 
 
 from which follow, by the principle of the denial of the 
 consequent, the 
 
 Theorems: e a b / a' ab Tab ^ o/ ab 
 
 Cab / 0'ab Tab / 0'ab 
 
 Cab / T'ab /3ab ^ a'ab 
 
 Consequently* 
 
 a a b'/3ab=0 /?ab'Tab=0 Tab*Tba=0 
 
 tt ab *7ab=0 0ab * C ab =0 
 
 a ab * c a b=0 Tab ' ab=0 I 
 
 From the Definitions:** 
 
 A a b=a ab + Tab 
 
 E a b = Cab 
 
 lab = tt ab + )3ab + Tab + Tba II 
 
 Oab = ab + /3ab + Tba 
 
 we obtain immediately*** 
 
 dab = Aab " A ba 
 /3 ab = lab * Oab ' Oba 
 Tab =A a b " Oba 
 Cab = Eab 
 
 It is an advantage of the forms of the original set, 
 an advantage which the set, a, /3, T> c, does not possess, 
 that the contradictory of any letter is represented by a 
 single other letter of the set. Suppose that we were to 
 
 * a /3 = reads: a(is true) and fi(is true ) is impossible. 
 
 **A, E, I, O are simply the sums given in equations II. 
 That they are the traditional Aristotelian forms, is only an 
 accident of the reader's application. Hence equations II are 
 definitions and not postulates. 
 
 ***Multiplying out the sums in II as if they were ordinary 
 polynomials, applying the results of I, and assuming that a, J3 
 and c are simply convertible. 
 
38 
 
 A PRIMER OF LOGIC 
 
 combine this advantage with that of simple convertibility 
 in a new set of forms. 
 
 To do this it would seem to be enough to subtract 
 from the meaning of A a b the part 7 & b, (equations II), and 
 to add this part to the meaning of O a b.* Our new set of 
 forms becomes: 
 
 HI 
 
 = Cab 
 liab =Ct a b + 
 
 + Tab + 7ba 
 + Tab + 7ba 
 
 the analogues of the old letters being represented by the 
 corresponding Greek vowels. 
 
 From equations I and III, and remembering that the 
 sum of dab, /3 ab , 7ab> c a b, Tba makes up the propositional 
 "universe," the results of the following table, yielding all 
 the moods of immediate inference, will easily be seen to 
 hold. 
 
 True 
 
 Implies the 
 truth of 
 only 
 
 Implies the 
 falsity of 
 only 
 
 False 
 
 Implies the 
 truth of 
 only 
 
 Implies the 
 falsity of 
 only 
 
 a 
 
 a, i 
 
 e, 
 
 a 
 
 
 
 a 
 
 e 
 
 e, 
 
 a, i 
 
 
 
 L 
 
 e 
 
 i 
 
 i 
 
 
 
 L 
 
 , 
 
 a, t 
 
 o 
 
 o 
 
 a 
 
 
 
 a, i 
 
 , 
 
 An induction of these results shows that a = o' t 
 o = a f , e = i f , i= e', and that, consequently, contradictory 
 
 *Here would seem to be another instance of the manner in 
 which the language of symbols may free a science from the 
 accidents imposed upon its development by the language of 
 speech. The last two members of the new set have apparently 
 no simple verbal expression. 
 
A PRIMER OF LOGIC 
 
 a, 
 
 pairs are a, o and c, t. Likewise contraries are 
 subcontraries are i, ; subalterns are a, t and , o. 
 
 If we define an affirmative form as one that becomes 
 unity when subject and predicate have been identified 
 and a negative form as one that becomes unity when sub- 
 ject and predicate have been made contradictory, then it 
 is a result of the following 
 
 Postulates:* a aa is a true proposition, 
 
 Theorems: 
 
 18', 
 
 > 
 
 > 
 
 M 
 
 M > 
 
 > M 
 
 Qt 
 
 P aa 
 
 > 
 
 / > M 
 
 T aa 
 
 and equations III, that a and t are affirmative and that e 
 and o are negative forms. 
 
 If a distributed term be one modified by the quanti- 
 tative adjective all, it will be seen that a and c distribute 
 both subject and predicate, while i and o distribute neither. 
 These results are summarized in the table below, the 
 distributed terms being underlined. 
 
 
 Affirmative 
 
 Negative 
 
 ,f 
 
 Universal 
 
 dab 
 
 ab 
 
 Particular 
 
 tab 
 
 ab 
 
 *x is true is to be represented by x=l, x is false by x = 0. 
 (See Boole, Investigation of the Laws of Thought, ch. XI, p. 
 169). The theorems follow by equations I, and equations III 
 become as a result of them, a a a=l, c aa = 0, i aa = l, aa = 0, 
 aa = 0, 6 aa = l, i a a = 0, 0aa=L Employing the usual notation, 
 a^not-a. 
 
40 A PRIMER OF LOGIC 
 
 The traditional rules for detecting the invalid moods 
 of the old syllogism, constructed from the set A, E, J, O, 
 hold for the new syllogism, built up out of the forms, 
 a, e, i, o. These rules are: 
 
 1. Two negative premises do not imply a conclusion. 
 
 EX. bn c b Z Cca 
 
 2. Two affirmative premises do not imply a nega- 
 tive conclusion. Ex. a ba a cb Z e ca . 
 
 3. An affirmative and a negative premise do not 
 imply an affirmative conclusion. Ex. aba cb ^ a c& . 
 
 4. Two premises, in neither of which the middle 
 term is distributed, do not imply a conclusion. 
 
 EX. t ba tcb ^ t ea . 
 
 5. Two premises, in which a given term occurs un- 
 distributed, do not imply a conclusion, in which that same 
 term is distributed. Ex. a ba t c b Z a ca . 
 
 The valid moods which remain, and which of course 
 are valid in all four figures, since each one of the forms 
 is simply convertible, are twelve in number, viz: 
 
 aaa aat aee aeo 
 
 au aoo eae eao 
 
 tat OQ.O eio teo 
 
 It has been previously observed, (note p. 2), that 
 equations I hold generally only when the limits and 1 
 are excluded as possible values of a and b. If these pos- 
 sibilities he included, we shall have to assume: 
 
 {Tab Z e'ab }' and .*. { e ab Z 7' ab }', since T OI e OI ^ 0. 
 Equations I then become: 
 
 Ctab ' 0ab=0 |8 a b '7ab=0 
 
 j8 ab ' C ab =0 IV 
 
A PRIMER OF LOGIC 41 
 
 Under these conditions, the fact which the old logic 
 always took for granted, that E ab is the contradictory of 
 I ab , and Aab the contradictory of O ab , no longer holds 
 true. For, while the sum of each of these two pairs of 
 forms is the prepositional universe, their product is not 
 the prepositional null, (equations II, IV). In order that 
 Eab * lab and A ab * O ab shall vanish for all values of the 
 terms, it will be necessary to exclude Tab e a b from the 
 product. A new set of forms, in which part of the meaning 
 of t a b is subtracted from i ab and added to e ab , will satisfy 
 this requirement. Let this new set be: 
 
 C 2 ab = ab + Tab + Tba 
 
 2 ab = ab + Pab + Tab + Tba 
 
 If a, e, i, o, be replaced by a 2 , c 2 , i 2 , o 2 , respectively 
 in the table, (p. 38), all the results of such a new tabulation 
 will be seen to hold, (equations IV, V). The same defini- 
 tions as given before will make a 2 and i 2 affirmative, e 2 and 
 o 2 negative forms, (equations V, and the postulates and 
 theorems, p. 39), but since e 2 distributes neither subject 
 nor predicate, e 2 i 2 o 2 and i 2 e 2 o 2 will not be found among 
 the valid moods of the syllogism, (see p. 40). The same 
 rules (p. 40) for the detection of the invalid moods will 
 hold for the new syllogism, but rule 1 is now redundant, 
 being a corollary of rule 4. 
 
 It might perhaps appear that our original symmetry, 
 (that of equations I), which was interrupted by the neces- 
 sity of allowing TOI to stand as a true proposition, could 
 be saved by assuming that the null class exhausts no part 
 of the universe, i. e. all of nothing is some of everything, 
 might be regarded as a false proposition, Now A = l, 
 
42 A PRIMER OF LOGIC 
 
 or cioa-f 7oa = l, is Schroder's definition of the null class, 
 and A oa will be a true proposition for all values of a if 
 7oi be true, whereas, 7oi=0 involves A i=0. These con- 
 sequences lead us to the alternatives of either giving up 
 our symmetry, (in equations I), or else of regarding the 
 null class as not essential to our algebra. 
 
 It is finally to be noted what was obvious in the 
 beginning that, while the members of the set, a, e, t, 0, 
 can be expressed in the members of the set, a, 0, 7, e, 
 the latter can not be expressed in the former. Conse- 
 quently, an essential difference has been lost, and the 
 existence of a completed logic of the new forms would not 
 put aside the necessity of working out the logic of the old. 
 
 The attempts of the logician to discover a set of 
 categorical forms, which establish a complete symmetry 
 among the moods and figures of the syllogism, are as old 
 as the science itself. The end would be attained if a new 
 set could be selected so as to satisfy the following con- 
 ditions: 
 
 1. Each form of the set must be simply convertible. 
 
 2. Corresponding to any member of the set, there 
 must occur another which represents its contradictory. 
 
 3. The new set must yield at least one valid mood 
 of the syllogism. 
 
 4. Each member of the new set must be represent- 
 able in the members of a set already proved necessary 
 and sufficient to express all differences, (the set A, E, I, 
 O, say), and conversely. 
 
 If Xab be any categorical form, the simplest functions 
 of x, which are themselves categorical and which are in 
 general simply convertible, are x ab * x ba and x ab + x ba . It 
 will be enough, therefore, in order to satisfy condition 1, 
 
A PRIMER OF LOGIC 43 
 
 to assume as a new set of forms either such a sum or 
 such a product of each one of the old forms (A, E, I, O, 
 say). 
 
 The equation, {x ab * x ba }' =x' ab + x' ba , suggests at 
 once what our manner of satisfying condition 2 must be, 
 for since the product, x a b ' Xba is the contradictory of the 
 sum x'ab + x'ba and x the contradictory of x', if x ab + x ba 
 [respectively x ab ' x ba ] be chosen as one of our new forms, 
 x'ab ' x'ba [respectively x' a b + x' ba ] must be chosen as one 
 of the others. 
 
 Remembering that 
 
 lab =Iab * Iba =Iab + Iba, 
 
 and that E a b = r a b and I a b =E' ab , 
 
 it follows that our choice of a new set of forms is limited 
 to the following two: 
 
 A I =A ab 'A ba , Ex=E ab , (1) 
 
 O ba , I x = Iab, 
 
 A 2 =Aab + A b a, E 2 =E ab , (2) 
 
 O 2 =Oab 'Oba, I 2 = lab- 
 
 It will be found, however, that the set (2) yields no 
 valid moods of the syllogism. Consequently, applying 
 condition 3, our choice is seen to be unambiguously re- 
 stricted to set (1), which yields twelve moods, valid each 
 one in each of the four figures. These will be found to 
 be, (dropping the subscripts): 
 
 A A A, A A I, AEE, A E O, 
 All, AGO, EAE, EAO, 
 I A I, OAO, EIO, IEO. 
 
 It will be impossible, however, to satisfy condition 4, 
 since every expression involving the new forms will be 
 
44 A PRIMER OF LOGIC 
 
 simply convertible. Consequently, an essential difference 
 has been left undistinguished, and it will not be possible 
 to substitute the new forms for the old. The new forms 
 are, in fact, identical with a r , e x , t x , o lt considered above. 
 From this latter discussion and from the discussion 
 that has gone before, we conclude, that, if it be necessary 
 to retain in our set of forms at least one that is not simply 
 convertible, it will be impossible to satisfy the condition 
 2 above, unless the null-and one-class be excluded or de- 
 nned in some way other than ordinary. 
 
APPENDIX II 
 
 Historical Note on De Morgan's New Prepositional Forms 
 
 In introducing the notion of contradictory terms into 
 logic De Morgan discovered two new prepositional forms, 
 which cannot be directly expressed by means of the A, E, 
 I, O relations of traditional logic.* Suppose that we de- 
 note these two forms by U and V, i. e. 
 
 U (ab) =A11 not a is b, 
 
 V (ab) =Some not a is not b, 
 
 A (ab) =A11 a is b, 
 
 E (ab) =No a is b, 
 
 I (ab) =Some a is b, 
 
 O (ab) =Some a is not b. 
 
 U and V are simply convertible, for (if a = not-a) 
 U(ab) =A (ab) =A (ba) =U (ba), (converting in A by the 
 principle of contraposition), and 
 
 V (ab) = I (ab) = I (ba) = V (ba), (converting simply in I). 
 
 V distributes both subject and predicate while U 
 distributes neither, for V(ab) =O(ab) =O(ba) (converting 
 in O by contraposition) and, since O distributes its 
 predicate, both a and b are distributed terms; similarly 
 U(ab) =A(ab) =A(ba) (converting in A by contraposition) 
 and, since A does not distribute its predicate, a and 
 b are undistributed terms. 
 
 If an affirmative form be defined as one that becomes 
 the subcontrary of itself when the subject and predicate 
 
 *Formal Logic, p. 61. 
 
46 
 
 A PRIMER OF LOGIC 
 
 have been identified, and a negative form as one that 
 becomes the contrary of itself under the same conditions, 
 it will be seen that A, I and V are affirmative, that E, O 
 and U are negative forms. 
 
 These results are summarized in the following table, 
 the distributed terms being underlined. 
 
 
 Affirmative 
 
 Negative 
 
 Universal 
 
 A(ab) 
 
 E(ab) 
 
 Particular 
 
 I(ab) 
 
 0(ab) 
 
 Indefinite 
 
 V(ab) 
 
 U(ab) 
 
 Below are tabulated all the forms of immediate im- 
 plication which hold among the six propositions A, E, I, 
 O, U, V. The (144) implications and non-implications 
 necessary to establish unambiguously the results of the 
 table can be derived from a certain number of postulates 
 and the commonly assumed principles of traditional logic. 
 
 True 
 
 Implies 
 falsity of 
 
 Implies 
 truth of 
 
 False 
 
 Implies 
 falsity of 
 
 Implies 
 truth of 
 
 A 
 
 E, 0, U. 
 
 A, I, V. 
 
 A 
 
 A. 
 
 O. 
 
 E 
 
 A, I. 
 
 E, O. 
 
 E 
 
 E. 
 
 I. 
 
 I 
 
 E. 
 
 I. 
 
 I 
 
 A, I. 
 
 E, 0. 
 
 O 
 
 A. 
 
 O. 
 
 
 
 E, 0, U. 
 
 A, I, V. 
 
 U 
 
 A,V. 
 
 0, U. 
 
 U 
 
 U. 
 
 V. 
 
 V 
 
 U. 
 
 V. 
 
 V 
 
 A,V. 
 
 O, U. 
 
 An induction of these results will show that: 
 Contradictory pairs are: A, O; E, I; U, V; 
 Contrary pairs are: A, E; A, U; 
 
A PRIMER o'LtGJe' ' V: f ; , , 47 
 
 Subcontrary pairs are: I, O; O, V; 
 
 Subalternate pairs are: A, I; A, V; I, V; I, U; E, O; 
 
 E, U; E, V; U, O. 
 
 In order to show how the new forms fit the ancient 
 scheme and as an illustration of method let us solve the 
 array of the syllogism. We first observe (see table) that 
 A weakens ambiguously to I or V; that O strengthens am- 
 biguously to E or U.* 
 
 Rules: 
 
 1. In any valid mood interchange either premise 
 and the conclusion and replace each by its contradictory. 
 
 2. In any valid mood strengthen a premise or weaken 
 a conclusion. 
 
 3. In any valid mood convert simply in any form 
 but A or O. 
 
 Postulates: 
 
 A A A (1st figure) is a valid mood. 
 
 AUU ( " " ) " " 
 
 EAE ( " ) 
 
 UEA ( " ) >' 
 
 From these rules and postulates will follow sixteen 
 valid moods in the 1st figure, twenty in each one of the 
 2nd and 3rd figures, and twenty-one in the 4th figure. 
 
 In order to deduce the invalid moods let us assume 
 the 
 
 Rules: 
 
 1. In any invalid mood interchange either premise 
 and the conclusion and replace each by its contradictory. 
 
 *If x implies y but y does not imply x, then x is said to be a 
 strengthened form of y, and y is said to be a weakened form of x. 
 
48 - 'A'PRfMER OF LOGIC 
 
 2. In any invalid mood weaken a premise or 
 strengthen a conclusion. 
 
 3. In any invalid mood convert simply in any form 
 but A or O ; and 
 
 Postulates: 
 
 A A A (4th figure) is an invalid mood. 
 
 AAO ( " " ) " " 
 
 AAV (3rd " ) " " 
 
 AAI (2nd " ) " " 
 
 AAO (1st " ) " " 
 
 EEI ( " " ) " " 
 
 EEO ( " " ) " " 
 
 EU A ( " " ) " " 
 
 EUO ( " " ) " " 
 
 UUO ( " " ) " " 
 
 U U V ( " " ) " " " M 
 
 From these postulates and rules follow the remaining 
 (772) invalid moods. 
 
 It will be seen at once that the rules of the old logic 
 for the immediate detection of the invalid moods of the 
 syllogism no longer hold. To give only one illustration: 
 A term may appear distributed in the conclusion of a valid 
 mood and be undistributed in the premise. 
 
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