BC 135 
 .T3 
 1920a 
 Copy 1 
 
 
 A SET OF, FIVE POSTULATES FOR BOOLEAN 
 
 ALGEBRAS IN TERMS OF THE 
 
 OPERATION "EXCEPTION" 
 
 A THESIS ACCEPTED IN PAETIAL SATISFACTION OF 
 
 THE BEQUIEEMENTS FOE THE DEGEEE OF 
 
 DOCTOE OF PHILOSOPHY 
 
 AT THE UNIVEESITY OF CALIFOENIA 
 
 JAMES STURDEVANT TAYLOR 
 
 1918 
 
UNIVERSITY OF CALIFORNIA PUBLICATIONS 
 
 IN 
 
 MATHEMATICS 
 
 Vol, 1, No. 12, pp. 241-248 April 12, 1920 
 
 A SET OF FIVE POSTULATES FOR BOOLEAN 
 
 ALGEBRAS IN TERMS OF THE OPERATION 
 
 "EXCEPTION" 
 
 BY 
 J. S. TAYLOR 
 
 UNIVERSITY OF CALIFORNIA PRESS 
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 UNIVERSITY OF CALIFORNIA PUBLICATIONS 
 
 IN 
 
 MATHEMATICS 
 
 Vol. l,No. 12, pp. 241-248 April 12, 1920 
 
 A SET OF FIVE POSTULATES FOR BOOLEAN ALGEBRAS 
 IN TERMS OF THE OPERATION "EXCEPTION" 
 
 BY 
 
 J. S. Taylor 
 
 Introduction 
 
 There are three binary operations between classes which have come into general 
 use in Boolean algebras. These three are "logical addition," "rejection," and 
 "exception," and are expressed respectively by the symbols "+," " |," and " —." 
 Simple and elegant sets of postulates already exist for the logic of classes in terms 
 of "logical addition," 1 and in terms of "rejection." 2 The set of postulates by 
 B. A. Bernstein, 3 however, to whom the third operation is due, is somewhat involved 
 and it is therefore the purpose of this paper to present a comparatively simp'e set 
 in terms of "exception." 
 
 I 
 
 A SET OF FIVE POSTULATES FOR BOOLEAN ALGEBRAS 
 
 IN TERMS OF THE OPERATION "EXCEPTION" 
 
 Let us take as undefined ideas a class K of elements a, b, c, and an 
 
 operation "— ," a — b reading "a except b." Then the logic of classes may be 
 defined as a system 2 (K, — ) which satisfies the following five postulates : 
 
 Postulates 
 
 I. K contains at least two distinct elements. 
 
 II. If a and b are elements of K, a — b is an element of K. 
 
 III. If a, b, and the combinations indicated are elements of K, 
 
 a—(b — b) =a. 
 
 J E. V. Huntington, "Sets of Independent Postulates for the Algebra of Logic," Transactions 
 of the American Mathematical Society, Vol. V (1904), pp. 288-309. 
 
 2 B. A. Bernstein, "A Set of Four Independent Postulates for Boolean Algebras," Transac- 
 tions of the American Mathematical Society, vol. XVII, pp. 50-52. 
 
 3 B. A. Bernstein, "A Complete Set of Postulates for the Logic of Classes in Terms of the 
 Operation 'Exception', and a Proof of the Independence of a Set of Postulates Due to Del Re," 
 Univ. Calif. Publ. Math., vol. I, pp. 87-96 (May 15, 1914). 
 
242 (Ininrsitij of California Publications in Mathematics [Vol.1 
 
 IV. There exists a unique element 1 in K such that, if a, b, and the combina- 
 tions indicated are elements of K, 
 
 a-(l-b)=b-(l-a). 
 
 Definition 1. a'=l — a. 
 
 V. If IV holds, and if a, b, c, and the combinations indicated are elements of K, 
 
 a—(b — c) = [(a — b)'—(a — c'))'. 
 Definition 2. a'=(a')'. 
 
 Theorems 
 Theorem 1. a' ' =a 
 
 Proof. a-(l-l) = l-(l-a) by IV. 
 
 but a-(l-l)=a by III. 
 
 and l-(l-a) = l-a" by Def. 1. 
 
 = (a')" by Def. 1. 
 
 = a/> by Def. 2. 
 
 Theorem 2. a' is unique (for any a in K). by IV, II. 
 
 Theorem 3. a — b'=b — a' by IV, Def. 1. 
 
 Corollary 1. a ' — 6 = 6 ' — a 
 
 Corollary 2. a — b = b' —a' 
 
 Theorem 4. (a — a) ' = (b — b) ' 
 
 Proof. (a-a)'-(b-b) = (a-a)' by III. 
 
 and (a — a) ' — (b — b) = (b — b) ' — (a — a) by Th. 3, Cor. 1. 
 
 = (6-6)' by III. 
 Theorem 5. 1 = (e — e) ' . 
 
 Proof. First, (e — e) ' is unique. by II; Th's 2 and 4. 
 
 Secondly, (e — e) " satisfies the equation of IV, in other words, 
 a — [(e — e)'—b] = b — [(e — e)'—a] 
 
 for, a-[(e-e)--b] = a-[b'-(e-e)] by Th. 3, Cor. 1. 
 
 = o-6" by III. 
 
 and 6-[(e-e)'-a] = 6-[a - -(g-e)] by Th. 3, Cor. 1. 
 
 = b-a by III. 
 
 but a — b'=b — a' by Th. 3. 
 
 Theorem 6. a — a'=a 
 
 Proof. Set b = c = a' and a = a in V. 
 
 The left then becomes a—(a'—a')=a by III. 
 
 The right becomes [(a — a')' —(a — a' ')] ' =[(a — a') ' — (a — a)] ' by Th. 1. 
 
 = [(a-a-)T by III. 
 
 = a-a - byTh. 1. 
 Corollary. a ' — a = a " 
 
1920] Taylor: A Set of Five Postulates for Boolean Algebras 243 
 
 Theorem 7. o-(6-c) = [(6' -a) ' -(c-o/)] - by V; Th. 3; Th. 3, Cor. 2. 
 
 Corollary. a - • -(6 - - -c) = [(&' -a') ' -(c* " -a")] '. by Th. 1. 
 
 Theorem^. (b' —a)' — (b — a)=a 
 
 Proof. Set a = a' , c = b in Th. 7, which then becomes 
 
 a--(&-&) = [(6--o")"-(&-a'")r 
 
 but 
 
 
 a' — (6 — 6) —a' 
 
 
 by III. 
 
 and [(&' 
 
 — a' 
 
 y-(&-a- •)]■=[(&• -a)" -(6- 
 
 -a)]" 
 
 by Th. 1. 
 
 hence 
 
 
 [(&" — a) ' — (& — a)] ' =a' 
 
 
 
 hence 
 
 
 [(6' — a) ' — (6 — a)] ' ' =a' ' 
 
 
 by Th. 2. 
 
 and 
 
 
 (6 ' — a) " — (6 — a) = a 
 
 
 by Th. 1. 
 
 Corollary. 
 
 
 (&' —a) " — (&' ' — a) =a 
 
 
 
 Definition 3. 
 
 
 a | 6 = a' — & 
 
 
 
 Theorem 9. 
 
 
 a a = a' 
 
 
 by Th. 6, Cor. 
 and Def. 3. 
 
 Definition 4. 
 
 
 a' = a\ a 
 
 
 
 Theorem 10. 
 
 
 a' = a' 
 
 
 by Def. 4, Th. 9. 
 
 Theorem 11. 
 
 
 (b\a)\ (b'\a)=a 
 
 
 
 Proof. 
 
 
 (&' — a) " — (6' ' — a) =a 
 
 
 by Th. 8, Cor. 
 
 but 
 
 (& 
 
 ■ -o) •-(&■' -a) = (&| a)' -(&■ 
 
 = (6 | a) | (6' | a) 
 
 |a) 
 
 by Def. 3. 
 by Def. 3;Th. 10. 
 
 Theorem 12. 
 
 
 a'| (6'|c) = [(6|a')| (c r | a')]' 
 
 
 
 Proof, 
 but 
 
 a' ' 
 
 _(&--_ c ) = [(6-_ a -)-_( c --_ ( 
 
 a ' ' — (6 ' ' — c) = a ' ' — (6 ' | c) 
 = a ' | (6 ' c) 
 = a'| (6'|c) 
 
 or 
 
 by Th. 7, Cor. 
 by Def. 3. 
 by Def. 3. 
 by Th. 10. 
 
 and [(&'— a")'— (c' ' — a')]' =[(& | a')' — (c' | a - )]' by Def. 3. 
 
 = [(6|o')|"(c" \a)Y by Def. 3. 
 = [(b j a') | (c' | a')]' byTh. 10. 
 
 That postulates I — V are sufficient is now evident. In the light of Definition 3 
 postulates I and II give Pi and P 2 of B. A. Bernstein's set of four postulates in 
 terms of "rejection" referred to in an earlier part of this paper; while Pz and P 4 of 
 that set are here exhibited as theorems 11 and 12. That postulates I — V may 
 likewise be derived from Pi — P of Bernstein's set is also easily shown. Thus the 
 two sets of postulates are equivalent. 
 
 Consistency 
 
 The consistency of the set of postulates I — V is demonstrated by the following 
 system composed of two distinct elements ei and e% which satisfies all five postulates. 
 As in succeeding examples, ei — e, will be given by means of a table; so that if, as 
 
I'll 
 
 Universittj of California Publications in Mathenuilirs 
 
 [Vol. 1 
 
 in the present instance, e x — ei = e 2 , e x — e 2 = e u e 2 — ei = e 2 , and e 2 — e 2 = e 2 , this will he 
 stated in the form: 
 
 — 
 
 ei 
 
 ^ 
 
 ei 
 
 <h 
 
 ei 
 
 e 2 
 
 e 2 
 
 e 2 
 
 That this system 2 satisfies all the postulates the reader may verify without 
 difficulty. 
 
 Independence 
 
 The independence of the five postulates is demonstrated by exhibiting five 
 systems each satisfying all but one of the postulates, the unsatisfied postulate 
 being I — V in turn. The system 2 failing to satisfy the i postulate will be desig- 
 nated 2"\ e{ — e } = x means that ei — e, does not give an element belonging to K. 
 
 2" 1 ; K a class of one element e h with ey — e\. = ei. 
 
 2~ 2 ; K a class of two distinct elements ei and e 2 , with e; — e,- defined by the 
 accompanying table: 
 
 ei 
 
 e 2 
 
 ei ei, 
 
 e 2 at- 
 
 2~ 3 ; Xa class of three distinct elements, with e » — e,- defined by the accompanying 
 table : 
 
 — 
 
 ei 
 
 e 2 
 
 e 3 
 
 ei 
 
 ei 
 
 ei 
 
 ei 
 
 e 2 
 
 e 2 
 
 ej 
 
 e 2 
 
 e 3 
 
 ei 
 
 ei 
 
 ei 
 
 III fails for a = e 3 . 
 
 IV holds, for e 2 , and e 2 only, satisfies the conditions imposed on 1. 
 
 V holds. It holds obviously for a, b, and c limited to e\ and e 2 , for that part of 
 K is identical with the system used to demonstrate the consistency of the postulates 
 with 6i and e 2 simply interchanged. The other possibilities may be disposed of 
 as follows: 
 
 (1) a = ei or e 3 , b = e i} c = e,- (i,j=l, 2, 3). 
 
 e\ oi3—(e i —e ] )=ei, [(e x or3 — e t ) ' — (e x or3 ,— e/)]" =[e 2 — ei] ' =e x 
 
 (2) For a = e 2 we have the following as yet undisposed of cases : 
 
 e 2 — (e 3 — ei) = e 2 , [(e 2 — e z ) ' — {e 2 — e { ' )] ' = [e Y — e,] ' = e 2 
 
 e 2 -(ei-e 3 )=e 2 , [(e 2 — e x ) ' -(e 2 -e 3 ')]' =[ei— ej' =e 2 
 
 e 2 — (e 2 — e 3 )=e u [(e 2 -e 2 ) ' — (e 2 -e 3 ')]' =[&— ej' =ei 
 
 2" 4 ; Ka class of two distinct elements d and e 2 with d — e ; - defined by the table : 
 
 — 
 
 ei 
 
 e 2 
 
 ei 
 
 ei 
 
 e x 
 
 62 
 
 e 2 
 
 e 2 
 
 Neither ei nor e 2 satisfies the conditions imposed on 1 by IV. 
 V is satisfied vacuously. 
 
1920] Taylor: A Set of Five Postulates for Boolean Algebras 245 
 
 2~ 5 ; K a class of three distinct elements with e — e,- defined by the table: 
 
 ei e 2 e 3 
 
 ei ei ei e\ 
 e 2 e 2 ei e 3 
 e-s e 3 ei ei 
 
 III is obviously satisfied. 
 
 IV is satisfied, for e 2 satisfies the conditions imposed on 1. 
 
 V fails for a = b = c = e 3 . 
 
 II 
 COMPLETE EXISTENTIAL THEORY 
 
 As has already been shown, postulates I — V are independent in the ordinary 
 sense that no one of the postulates is implied by the other four. Professor E. H. 
 Moore, 4 however, has suggested the question in connection with sets of postulates 
 of determining not only the implicational relations existing among the postulates 
 as they stand, but also all the implicational relations which exist among properties 
 defined either by the postulates themselves or by the negatives of the postulates. 
 A set of postulates is said to be completely independent if, and only if, no such im- 
 plicational relations exist. For example, I have shown in an earlier paper that 
 while Bernstein's set of four postulates in terms of "rejection" already referred to 
 are independent in the ordinary sense, they are not completely independent, since 
 the negative of the first postulate implies the third and fourth. 
 
 Any system 2 (K, — ) of the type prescribed earlier in this paper has with 
 respect to the five postulates there stated one of the 2 d = 32 characters: 
 
 (1) ( + + + ++),(+ + + + _-),( + + + -+), (+ ),(-- ); 
 
 the i th sign of the character being plus or minus according as 2 does or does not 
 satisfy the i th postulate. The body of thirty-two propositions stating for the 
 various characters represented in (1) that there exists or does not exist a system 
 having the character in question constitutes what Professor Moore has called 
 "the complete existential theory" of the five postulates. 
 
 For the five postulates in question the complete existential theory consists of 
 fourteen propositions of existence and eighteen propositions of non-existence. 
 The eighteen non-existencies arise from the fact that the negative of I implies 
 III, IV, and V, and that also the negative of IV implies V. 6 
 
 4 E. H. Moore, "Introduction to a Form of General Analysis," New Haven Mathematical 
 Colloquium, Yale University Press, p. 82. 
 
 5 J. S. Taylor, "Complete Existential Theory of Bernstein's Set of Four Postulates for Boolean 
 Algebras," Annals of Mathematics, Second Series, vol. XIX, No. 1, pp. 64-69 (September, 1917). 
 
 6 The question naturally arises as to whether it might not be possible to modify the postulates 
 in a way such that they would become completely independent. The writer has investigated 
 this possibility in considerable detail but has been unable to make such a modification without a 
 considerable loss of simplicity. The simplest change found to bring about the desired results is 
 as follows: — 
 
 I'. K contains at least four distinct elements. 
 
 V. There exists an element e in K such that, for each a, b, c choice for which there exists any 
 element in K satisfying the condition imposed upon 1 by the equation of IV for each pair of elements 
 in the group of elements obtained by combining a, b, and c in all possible ways, the element e does so, 
 
L! Hi l iiin rsit // of California Publications in Mathematics | Vol. 1 
 
 Propositions of Non-Existence 
 
 The eighteen propositions of non-existence, as implied above, may be expressed 
 by the two following propositions: 7 
 
 (2) S _1 DS 34B 
 
 (3) S" 4 DS 6 
 
 The truth of these two propositions is readily perceived from the following 
 considerations. First, the hypothesis that I is not satisfied necessitates either 
 K"" a (a class without any elements) or K" ; "" u ' ar (a class with only one element). 
 But if K contains no elements, postulates III, IV, and V are satisfied vacuously. 
 And if K contains only one element, then they are satisfied either evidently or 
 vacuously, according as 2 does or does not satisfy postulate II. 
 
 Secondly, the hypothesis that IV is not satisfied obviously results in the vacuous 
 satisfaction of V. 
 
 Propositions (2) and (3) render impossible the existence of systems with the 
 following eighteen characters. 
 
 (- + + + -), (- + + -+), (- + - + + ), (- + + --), (- + - + -), 
 
 (- + --+), (- + ), (-- + + -X-T-- + -+), ( ++), 
 
 (-- + --), ( + -), ( +), ( ), (+ + + --), 
 
 (+ + ), (+- + --), (+ )• 
 
 Propositions of Existence 
 
 The fourteen propositions of existence are established by the exhibition of 
 fourteen systems having the remaining fourteen characters ; there are two examples 
 for K sinoular , seven for K dual , and five for K trip,e . In each case K contains the least 
 number of elements possible. 
 
 Examples for K singular 
 
 Systems having the characters ( — H + + +) and ( h + +) respectively 
 
 are the following: 
 
 System Ii. Character ( — (- + + +); class composed of single element e\. 
 with e x — ei = ei. 
 
 System I 2 . Character ( (- + + ); class composed of single element d, 
 
 with e. — ei^ei. 
 
 and such that, for such an a, b, c choice, if p.' be defined as e — e, and if a, b, c, and the indicated com- 
 binations are elements of K, 
 
 a — (b—c) =[(a — b) ' —(a — c')] ' 
 Since considerable space would be occupied by a proof of the fact that the postulates thus 
 modified are completely independent and since the reader should meet with no very serious 
 difficulty in establishing this fact for himself, such proof is here omitted. 
 
 7 2"'DSi h ' ' " means, "If a system 2 does not satisfy the i tb postulate, then it does satisfy 
 postulates j, k, . . . ., and n." 
 
!920] Taylor: A Set of Five Postulates for Boolean Algebras 
 
 247 
 
 Examples for K dual 
 
 System Hi 
 
 (+ + + + +), e l e 2 e x 
 e 2 e 2 e 2 
 
 System II 3 
 
 ei e 2 
 
 ei e 2 
 
 (+- + ++), e x | ei 
 
 ei 
 
 e 2 \ e 2 
 
 X 
 
 System lis 
 
 
 — 
 
 ei 
 
 62 
 
 (+ + --+), ei 
 
 e x 
 
 ei 
 
 e 2 
 
 ei 
 
 ei 
 
 System II 2 
 
 e x e 2 
 
 ( + + H h), e x I ei ei 
 
 62 I e 2 e 2 
 
 System II 4 
 
 — I e t e 2 
 
 (+H h — ), ei I ei e 2 
 
 e 2 e x e x 
 
 System 116 
 
 e 2 
 
 e\ e 2 
 
 x x 
 
 System II 7 
 
 ei e 2 
 
 (H h), ei \ x e x 
 
 e 2 I e x ei 
 
 Examples for K triple 
 
 System IIIi 
 
 e e 2 e s 
 
 ( + + + + -), ex 
 e 2 
 e s 
 
 ei ei ei 
 e 2 e x e 3 
 e 3 e x e x 
 
 System III 3 
 
 - 
 
 e x 
 
 e 2 
 
 e 3 
 
 e x 
 
 e x 
 
 e x 
 
 X 
 
 e 2 
 
 e 2 
 
 e x 
 
 e% 
 
 e 3 
 
 e 3 
 
 e x 
 
 e x 
 
 System II I 2 
 
 — I e x e 2 e 3 
 
 (++-++; 
 
 e x 
 e 2 
 ez 
 
 e x e e x 
 e : e x e 2 
 e x e x e x 
 
 System III 4 
 
 — I e x e 2 e 3 
 
 (+ - + + -), e x e x e x x (+-- + +; 
 
 System Ills 
 
 — I e e 2 e% 
 
 (H V — ), e x I e x e 2 x 
 
 e 2 J e x e x x 
 
 e 3 I e x e 2 x 
 
 e x 
 e 2 
 e 3 
 
 e x e x e x 
 e 2 e e 2 
 e x e x x 
 
248 University of California Publications in Mathematics [Vol.1 
 
 III 
 
 THE ELEMENT 1 AND NEGATION 
 
 It is interesting to note that it is impossible to express either the element 1 or 
 negation, "not — a," directly in terms of the operation of " + " or "— ," although 
 this can be done in terms of "rejection." It has been found necessary in all sets 
 of postulates in terms of "logical addition" or "exception," therefore, to postulate 
 one or both of these two ideas. Curiously enough, although there are several sets 
 in which only 1 is postulated and "not — a" then defined, there has been no set 
 formulated in which "not — a" is postulated and the element 1 defined. This 
 might lead one to believe that the element 1 plays a more important role than 
 negation, but that this does not follow in the case of "exception," at least, is demon- 
 strated by the following set of five postulates in which only "not — a" is postulated. 
 
 I. K contains at least two distinct elements. 
 
 II. If a and 6 are elements oi K, a — b is an element of K. 
 
 III. If a, b, and the combinations indicated are elements of K, a—(b — b) = a 
 
 IV. For every element e in K there exists another element e ' in K, unique for 
 each e, such that, if the combinations indicated are elements of K, 
 
 (1) e — e'=e 
 and (2) a-(b-c) = [(c-a) ' -(6" -a")] ' , 
 where each dotted element is an element satisfying (1). 
 Definition 1. a' ' =(a')' 
 
 Theorem 1. a' ' =a 
 
 Proof. Set b = c = a in IV (2) 
 
 the left becomes a—(a — a)=a by III, 
 
 while the right becomes [(a — a') ' — (a' — a')]" =[(a — a') ']' by III, 
 
 = [(a)T byIV(l), 
 
 =a" byDef. 1. 
 
 Theorem 2. a -a = a by IV (1), Th. 1. 
 
 Theorem 3. b' — a' =a — b 
 
 Proof. Set a = a, 6 = 6, c=6 - in IV (2) 
 
 a- (&-&•) = [(&' -a") '-(ft" -a')] " 
 whence a-6 = [(6 - -a')']" by IV (1), Th. 2. 
 
 = 6' —a' 
 Corollary. a ' — 6 = 6 ' — a 
 
 Theorem 4. a-(b-c) =[(b ' -a) ' -(c-a)] " by Th. 3, Cor. 
 
 Theorem 5. (6' — a) ' — (6 — a) =a 
 
 Proof. Set a = a' , c = b in Th. 5. 
 
 Definition 2. 1 = (a — a) ' 
 
 The rest of the development and the proof of the sufficiency of the set of pos- 
 tulates follow so closely those of the set for which it has already been explained in 
 detail that the reader is left to complete the work for himself. 
 
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