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12 ( Boc hens ki 1 Lei bni z GB ool e ( E S chr der, G Fr ege A W hi t ehead ( B. Rus s el l ) Ar i s t ot l e ] Bochens ki : A Hi s t or y of For mal Logi c, Not r e Dame, 1961, p. 270 HFL

13 A Dumi t r i u) 1 R Lul l us G Bool e ), W J evons ) A d e Mor gan ) H McCol l ) ; C S Pei r ce ) ( RGr as s ma nn ( E. Sc hr der, ) 3 G. Peano, G. Fr ege, ) 4 B. Rus s el l, A. N. Whi t e head ( C Bur al i - For t i ) G. Cant or, ( 1) ( LBr ouwe r A H eyt i ng ) ( 2 F P Ramsey ), L Wi t t gens t ei n R. Car nap, ), ( 3 D. Hi l ber t, A Dumi t r i u: Hi st or y of Logi c, Vo l. 1V, Abacus Pr ess, 1977, pp, 6-7

14 ) ( W. Acker mann, P Ber nays ), ( 4 J. ukas i ewi cz C. l. Lewi s, ), ( 5 K. G del, ) 6 A Ta r ski P 7

15 ( 1 ) ( 2), ( 3 ( ) o

16 Handbook of Mat hemat i cal Logi c Edi t ed by J. Bar wi s e, Nor t h - Hol - l and Publ i s hi ng Company, Handbook of Phi l os ophi cal Logi c Edi t ed by Gabbay and Guent hner, 4 vol s. Rei del, Dor dr echt,

17

18 ( 1) (

19 1 08

20

21 Ar i s t ot l e Bar bar a, Cel ar ent, Dar i i, Fer l o, Bar b ar l, Cel ar ont ; Bar al i pt on Br amant i p), Cel ant es Camenes ), Dabi t i s Di mar i s ), Fr i s esomor um Fr e s i s on), Fa pes mo Fes apo), Cel ant op Came- no p). Ces ar e, Cames t r es, Fes t i no, Bar oco, * Ces ar o, * Cames t r op I EO, EAE, AEE, OAO,

22 E AO, AEO Di s ami s, Dar apt i, Dat i s i, Fer i s on, Fel apt on, Bocar do A AI, I AI, AI I, I EO, AEO, AOO Bar bar a, Cel ar ent, Dar i i Fer i o) I. 7I. 23 Bar bar a Cel ar ent Bar bar a Cel ar ent Fes t i n M N O N O N M M O N O O N M M M N F er i o O N Fes t i n Fer i o N M M N M N O M O N N M O M O N Fer i o Ces ar eo The Wor ks of Ar i s t ot l e t r a ns l a t e d i nt o Engl i s h, Vol. 1 Oxf or d; 1928,

23 A B, B C, A C o A C B B C B A, C A B A C B C B C A C A C A C B C B p q r p r q ; Da r i i. 4, 7, 23) : ( I ) Bar bar a ( ) Cel a r ent : ( ) A. 2) : SAP PI S ( ) E. 2) : SEP PES

24 ( V , 12, ) SAP SEP S OP, SOP S I P, SI P SAP SEP 37. 8, 59 3, ) : ( VI I. 5, 27, / / ( VI I. 2, ,. 57 1) : / ( 28 7, 59 17; 27 10, ; ( I X. 6;. 8, 11) : ( X). 5, 6; I I. 4) :, / /, ; / / Ces ar e Cel ar ent

25 M N O N M M O N O Fe r i o Ce s ar e ( 29 10) Fes t i no Fer i oo 27 32) H. Schol z,

26 Eu bul i des (Phi l o C hr ys i ppus

27 /,,,, / / W. Kneal e B. Mat es Sext u s Empi r i cus p p q Ma t es. B. St oi c Logi c, Ber kel ey and Los Angel es, 1953

28 p p q q. p q r r p q p q p q r p q r q r

29

30 12 Pet er A be l ar d, ) Por phyr y ( Boet hi us, ) ) 13 Al ber t t he Gr eat, ), R ober t Ki l war dby 1279) ( Wi l l i am of Shyr es wo od 1249) Pet er of Spai n, ) ( Ra i mundus Lul l us, )

31 14 Wi l l i an of Ockh am, Bur i dan 1 358) Al ber t of Saxong, Paul of Veni ce 1429

32 p p, r s ( p q) r p ( s r )

33 ( se ( ment i on) S P S P, S P, ( S P) x) ( S( x - P( x) ) x x S x P a b x) ( S( x - P( x) [( S( a P( a) ) (S( b - P( b) )

34 ( x) ( S( x P ( x) [( S( a - P( a) (S( b) - P( b ) ) ] Ever y man i s r unni ng and t he s ame i s eat i ng) ( x) ( F ( x) G ( x) x) F( x) ( x) G( x) x x F x G x F x G ( A donkey whi ch bel ongs t o ever y man i s r unni ng) ( Of ever y man a donkey i s r unni ng)

35 y) [ D( y x) ( M( x) P( y x) R( y ) ) ] D M p R y y x x y x y ( x) [ M( x y) ( D( y P( y, x R( y) ) ] x x y y y x y

36 p qp p q p q, p p q p q p q 60

37 A B B C A C

38 1 p, q, r 2 3 ) ; 4 ( 0. 1 ( p q) =df. ( p ) q) (0. 2) ( p q) =df. ( p q) (0. 3 p=d f. p p p 1. 1 p q p 1. 2) p p q (1. 3) p ( p) (1. 4) p p (1 5 ( p p) (1. 3) (1. 5) ( 1. 3 ( 1. 4) ( 1. 5) 1. 6) p p q) q 1. 7) ( p q) ( q r ) ( p r ) E. A. Moody: Tr ut h and Cons equence i n Medi aeval Logi c, Ams t er. dam, 1953

39 ) (1. 8) ( p q p q (1. 9) p p (1. 10) ( p q) p q ) ( R. St r ode 1. 11) p ( p q) A A B B) A B A B) (2. 1 p (p q) (i ) p ( i i ) p ( p q) ( q q (i i i ) (p q) (i ) ( i i ) (i v) p q (i i i ) HFL p. 200

40 ( v) p ( p q) ( i ) ( i v), ( 2. 2 p (q p) ( i ) p ( i i ) p ( i ) (i i i ) p ( q p ) ( ) ( i v) ( q p) ( i i ) ( i i ( v) q p ( i v) i ) (vi p ( q p) ( i ) ( v), (2. 1) 2. 2 ) S p, q p q =df. p q) HFL, p. 200

41 p q=df. (p q ) ( 2. 1) 2. 2 (2. l p ( p q) (2. 2) p ( q p) (pq) p q ( p q) p q (p q p q (p q p q )

42 4

43 p p p p p p p p p p p p p ( p p) ( p p ) p ( p ) p p ( p p ) p p p p A A A p, A p A p A p A p A [ ( i ) A i i ) p ] A p A [ ( i ) A i i ) p] 1. A A 2 A p A p 3 A p A p 4. A A A p 1 St yaz hki n: Hi s t or y of Mat he mat i cal Logi c f r om Lei bni n t o Peano, The M. I. T. Pr es s, 1969, p. 46 HML HFL pp

44 5. A A 4 6. A A 7. A 4 8. A 3 A p 9. A A 10. A A A p A [ ( i ) A (i i ) p] 3. A p A [ ( i ) A i i ) p] ] 8. A 3 A p 9. A [ ( i ) A i i ) Ai ) Ai i ) A A A i ) A ( i i ) A 12. A 12 8

45 (

46 17 ( Lei bni z, )

47 Des car t es H obbes

48 17 J oachi n J ungi - us )

49 3)

50 ), 48

51 p q r r q p A A A A Bar bar a Bocar do Bar ok o Bar bar a B C A B A C Bocar do: A C A B B C A C A C B

52 C B C Ba r oko: B C A C A B A B A B A C A C A B B B B A C es ar e A B A A B A Dat i s i A B A A B ADar apt i A A E O A C A A A C D ar i i EO A A A A 16 79

53 , a a b b b a b a b a b a b a b a b a b a b a b a b ukas i ewi cz, A I C KA A A (Bar bar a) CK A I I (D at i s i ) A I C K A A A I A I * 59. CKA A I A A I RS. C C C C C C C

54 Bar bar a Dat i s i A 1 a a a Bar bar a CK A D at i s i A A CK A I I b c b c A b a b a I a c a c a c 1 a b b ; a b

55 a c =15 b 3 c = 12 a = a =14 b =7 c =35 59 CKA A I A c b c b A a b a b I a c RS 100

56 HF L P. 275,

57 1 ) 2 3 T T T T AB=BA AA=A 52 52

58 ( 1) A A; ( 2) AB A; ( 3) A A; ( 4 A A; ( 5 A A; ( 6 ) A A A B B C A C ( 1 ) AB A BC B ( 2 AB BA ( 3 B A ( 4 A B C D E F ACE BDF 5 A BC D A B A C A D A B A B AB= A A B A B AB A A B ABA B AB A B AB A B=A B A B

59 A B A B A B AB=AB 1 A B AB 1 2 A B A B 1 A=B B= A A=B 1 ) A B B A A B ; B= A 2 A B B A B = A A B 3 A=B B C A=C A=B C B 1 B C ) 4 A B B C A C B C A B 1 A= B A C 3 A L L A L C. I Lewi s : Sur vey of Symbol i c Logi c, Ber kel ey. 1918, p. 379;

60 A B N L B L B N L 1 B N=N B A B A B 2 A A=A 2 5 A B A C C B A B C A( 1 A=C C B 6 C B A B C A C B A B A = B C A 7 A A A A A 3), 6) A A A= B A B ) A A ) A B 9 A = B, A C B C A C=A C B A, A C=B C 1 0 A = L B = M A B L M B M, A B A M L A A L ) A B L M C=L 11 A=L B=M C=N A B M N

61 12 B L A B A I 13 L B= L B L 14 B L L B= L 15 A B B C A C A N B N B 1 6 A B B C C D A D 1 7 A B B A A 1 8 A L B L A B 19 A L B L C L A C L 2 0 A M B N A 21 A M B N C P A C M N P BL B M B N A B AB A B B ar ba r a 15 B

62 / 2

63 17 18 Ger oni mo Sac- cher i AEE AE E AEE AEE AEE

64 AEE J. H. Lamber t, G. Pl oucquet, L. Eul er, ) 18 a, b, c, d n m l, x, z -, a b a- b a/ b a b a b a b a =ax a/ x a b= a / b b a+ab ba a x/ a= x a a+a/ b=ab xx=x HML pp ,

65 A B A=B A>B B A B A) A B A A A B A mb, m Bar bar a B=mA C=l B C=l ma x R z R x z R es t a>b a b a b) a b ab c>d eb a bde c a b) c d e b) (a b d e) c c a b c) (d e a d, a e, b d, b e, c d, c e ( ab ) ( c d) ( a b) ( cd) A B A B A=b) ( C D) A c (A= b=c) D( c C )

66 a a=a a a=a J. D. Ger gonne, a b a b acb a b a b a b axb a b ahb a b a 0 b 0 ) aab a b (a=b) ( ( a b) ( b a) ) a b ab b a) 0 (a b) ( (( a b 0) (a b) ( b a ) ) aob (a 0 0 a b a b ) ( b ) ( (a ) (b a) ) aeb ab=0 a b a b acb a b ahb a b axb a bac ba b a b ahb a Xb a b 19

67 ( GBen t ham, ) t X=pY; t p = t X py; t X= t Y t X Y px=py px px= t Y py px t Y W Hami l t on, X Y X Y X Y X Y X Y X Y X Y X Y 8 8 t M=t p t p= t M ps M t S=t M S PS t P t t P

68 ( Gr as s - mann) 1844 G P eacock) G F Gr ego r y) (DeMor gan W R H a- mi l t on

69 G. Bool e, ( 1847), ( 1854 R. Rhees 1952

70 X Y Z X X x y X Y Z X YZ x+y x y x y x x + xy x y 1 0 x x x G. Bool e, The Mat hcma t i cal Anal ys i s of Logi c, Oxf or d, 1951, pp. 3-4

71 x x- =df x( 1- ) = xy=yx, x+y=y x, x y z) = xy x z, x (y- z) =xy- xz, x=y xz= yz, x= xz = z x=y x- z=y- z x( 1- x) = 0, x=x =x = x x =x x( 1- x) 0 x = x, x- x = 0 x( 1- x) =0 x y 1 0 x=1 x =0

72 X Y x ( 1- y ) =0 A X Y xy= 0 E X Y xy= l X Y x( 1- ) = vo v E I E I x x x E I Y X, Y X A 1- ) x=0 x vy v (1- y vy= 0 x=vy Y X A E xy=0 x[ 1- ( 1- ) ] =0, A X Yo A x( 1- y) = 0 X Yo Anal ys t s of L The Mat hemat i cal ogi c, p. 26. MAL

73 A E I O X Y v =x X Y v=x{1-1- ) } X Y v=x( 1- ) X Y v=x( 1- y) v X Y..... x( 1- ) =0 X Y xy=0 vx( 1- ) =0 vxy=0 X Y , X Y O v AEA AE, EA AA AE AA A : Y X y 1- x =0 1- x) y=0 Z Y z( 1- ) = 0 zy- z= 0 1- ) y z( 1- x) 0 Z X MAL p 29

74 v A E AA, AE E A EA AE EA EI Y X y( 1- x) =0 y=vx Z Y z y=0 0=z 0= z x X Z Y X = vx X Z vzx =0 AE EA v AI, EI AO EI OA I E AI, AO, EI, EO, I A, I E, OA, OE I A, I E A I I A: Y X y( 1- x) = 0 Z Y vz= y vz( 1- x) =0 Z X Z Y v z vy, Z X vz( 1- x = 0 v yvx y- vx 0 Y X y=v( 1- x) y+vx- v=0 Y X vy=vx vy- vx=0 Y X vy=v( 1- x) vyv x- v=0 Y X

75 f ( x), f ( x, y) x, y 0 1 a b, f x =ax b( 1 - x) x 1 0 a b f ( 1 = a 1 b( 1-1 = a, f ( 0 = a 0 b( 1-0 = b f x = f ( 1 x+f ( 0) (1- x f ( x) x f ( x) =0 x f ( 1) x f ( 0) ( 1- x) =0 f ( 1) x f ( 0 - f ( 0) x=0, (f ( 1 - f ( 0) ) x f ( 0) =0 x f( 0) 1 f ( 0 - f ( 1 f ( 0) f ( 1 1 x 1 (2), f ( 0 - f ( 1 f( 0 - f ( 1 (1) ( 2 f( 0) f ( 1) x( 1 x) f ( 0 - f ( 1 f ( 0 - f ( 1 x( 1- x) =0 f ( 0) f ( 1 f ( 0 - f ( 1 f ( 0) f ( 1 =0 f ( 0 - f ( 1 f ( 0) f ( 1 =0

76 f ( x) =0 x f ( 1) f ( 0) =0 x x f ( x) =0 x x 1 x 0, M P m( 1- p) =0 s M s ( 1- m =0 m- mp s - s m= 0 m [ f ( m) =f ( 1) m f ( 0) ( 1- m ] (1-1 p +s - s 1) m ( p+s - s 0) ( 1- m) =0 ( 1- p) m s ( 1- m =0 f ( 0) =s, f ( 1) =( 1- p) s( 1- p) =0 S P f ( x. ) x y x f( x, y) =f ( 1, y) x f ( 0, y) ( 1- x), f ( x, y =f ( 1, 1) xy f ( 1, 0 x 1- ) f ( 0 1) ( 1- x) y f ( 0 0) ( 1- x) ( 1- ) f ( x, y x, y z 8 xyz, Bool e, An I nvest i gat i on of The Laws of Thought, London, 1854 p. 101.

77 xy( 1 z ) x( 1 y z x( 1 y) ( 1 z) ( 1 x) yz ( 1 x) y 1 z), ( 1 x ) ( 1 y ) z, ( 1 x) ( 1 y) ( 1 z ) x y z x=yz x- z =0 x - z 8 xyz ( 1- x) ( 1- z) ( 1 - x) ( 1- ) z ( 1- x ( 1- y) ( 1- z f ( 1, 1, 1), f ( 0, 1, 0), f ( 0, 0, 1 f ( 0, 0, 0) 0 1 xy( 1- z) x( 1- y) z+x( 1- y) ( 1- z) ( 1- x) yz=0 0: xy( 1- z =0 x( 1- y) z=0 x( 1- y) ( 1- z =0 1- x) yz=0

78 A B C D A B C D A B C D C D A B A B C D A B C D X Y X Y X Y x z X Y Z 1 x( X Y Z X, x 1- x X Y 1 X Y 2 X Y x x ( 1- )

79 3 X Y 1- x) y 4 X Y (1- x) ( 1 - ) X Y Z 1 xyz 2 xy( 1- z) 3 xz( 1- ) 4 yz( 1- x) 5 ( 1- y) ( 1- z 6 ( 1- x) ( 1- z) 7 z( 1- x) ( 1- ) 8 ( 1- x) ( 1- y) ( 1- z) X Y x y X Y x + x 1 X x 0 X X 1- x x( 1- ) =0 x=1 x=0 X K MAL pp

80 X Y K P (X Y) =xy; X Y P ( X Y ) x+ x 1 x=0 1 0 a b, c d (1) a, b c d (2 b c a, d (3 a, b c d (4 c d a, b b, c a A A = t A d f d t A A A t d B= t B a f B a t B B= t a ) ( B B a a B B B a) =t b c, B b c b c b c a a b c )

81 ( W. S. Jevons ) ( 1864), ( 1874) Aa=0 0 A=A( B b) ( C c ) ( 1) A = B ( 2 A AB ( 3 AB=AC B ( A ( C ) f ( x, z f (, x, z

82 1869 ( J. Ven n, AB, A, B 2 B A A 0 A B 3 A B A B=0 4 A B AB A B A 0

83 B C A B A C B C B C A B A B A C A C C. S. Pei r ce, a b a + b 1. a b a b b a a b a b 2 a b a b a b a b, b a a b 3. a, b a b 4. 0 a a 0 a 5 1 a a a 1 6 a b a, b 0 a b a b a b a + b 7. ab a b ab a, b a b a A, A, A b, B2s B Col l e ct ed Pa per s of C. S. Pe i r ce, vol. 3, Har va r d Uni ver si t y Pr es s, 1933, pp C. P.

84 a b a b x (A, B), x A, B a b a b a=b a b a b a=b ; a b=( a b) ( a, b) a+a a a a a a b a, b=0) a b=a+ a b, b a (a b) c, a (b c, b a a a a, b b, a (a, b) c, a, ( b c ) ( a b), c a b c c c a, b a c) ( b, c) b x, a, x a b x x a b a - b=a- a, b) x a b=a b+a - b a b a a b

85 a, b v, a, b a, [ 0 1],, b v ( 0, 1 ) a b a b a b b a, b=0 a b a; b b, x, a x= a ; b x x a; b = a,b a x a ; b + a, a b a; b 1; 0) a; b a; b a, b v, ( 1; 0), a, a a ; b a b a b a b 1870 x y z z x = = x= x x < y x x x> x x C. P. vol. 3, p. 28,

86 (1 x y z x z (2 x x ( 3 a b x a x=b (4 a b x b, x=a (5 a b c a c b (6) a, b a a, b b (7) a a b b a b (8 a b c c c a b b a x x f y f..., f 1913 ( Shef f er 1. 2 a b a b C. P. v ol. 3, C. P. vol

87 3 a a (a a) (a a) = a 4 a b a { b ( b b) }=a a 5 a, b c {a (b c) } {a b c) }={( b b) a} {( c c) a} a b a a 3 a = a + ) =a a a a) 4 =a a b = a ( ) [ a ( b ) ] =a b) [ a b) ] 5 = =a 0[ a 1 ] = [ = 1 [ 0] = a c ) ) = ) =a =( a ( a c) 5 ) = a a ( a ) A B A B, A B A B,

88 A B A B A. a b a b E. a a b a b b ) b ) I. a O. a a b A B A B C. P. vo1. 3, C. P. vo1. 3,

89 p c P.. C P C P C P C (1) x (2) x z y z (3) { x ( z) } = { (x z) } 2) (4 1 x y) ( x ) 2 C. P. v ol x y x

90 5 4 2) x Bar x y z y x y y z bar a x z z (6 5 2 (7) S S x S) x P) P P (P x (S x) (8) 6 M (S M (S P) (S P x ) (S M x P S M) (S M P (S P x P) ( S M) x

91 {( x y x} x E. Schr der, , ) a b b a b b a b 1. a a 2 a b b c a c 3. ( a+ c a c b 4. c ab c a c b 5. a( b c ) =ab ac 6. a a a a a a 1 9. aa 0 a b a b a b 0 1 c S yaz hki n, H M L, p. 209,

92 H Mc Col l A, B, C, A = 0 A A=1 A A =B A B 2 A B C AB C A, B C ABC=0 3 A B + C A, B C A B+C = 0 A B C=1 4. A A A A+A =1 AA=0 5 A + B C 6 A: B A B A B A: B A=AB Boch enski H FL pp p p

93 A B A +B ) ( A: B A=AB) A, B A, B 1 A( B+C) =AB+AC; (A B) ( C D =AC AD BC B D 2 A AB; A AAA A A= A( B B ) =A( B+B ) ( C C B B = 1=C C ) = 3 ( AB ) =AB +A B A B = AB + A ( BB) =AB A =A B B ( A+A ) =AB B A A = 1 B B = 1 ABC) 4 ( A B =A B ( A B C = A B C 5 A B= {( A B } = A B ) =A B A= AB B 1 1 A: B B : A A: B B : A 1 2

94 A: B AC: BC C 13 A: a, B:, C: r A B C: a r 14 AB=0 A: B B: A 15 A: B, B : C A: C A B A: B A B 16 A B B A 17 A: B A C B C = 1 (2) 1 1 a=1 a b=1+a b c (3) ( ab a b ) = a b ab (a b ab ) =ab a b (4) a: a b: a b c (5) a A) ( a B) ( a C aab C (6) a: b) : a b; (7 ( a=b) =( a: b) ( b: a) (8) a=b) : ab a b (9) ( A: a) ( B: b) ( C c ( ABC abc) ; ( 10) ( A: a) ( B: b) ( C: c) :( A B C : a + b c (11) ( Ax ( B: x) ( C: x A+ B C : ) ; 1 2) ( x: A) ( x: B) ( x: C) = x: ABC ( 13) ( A x ( B: x +( C : x ABC :x) ; 1 4) ( x: A) ( x : B ( x : C z :, AB C )

95 A=B) A= B A, B,

96 A. De Mor g an, ( V

97 X Y X x X 8 A X) YX. ) x X Y = X Y= Y X E X. Y=X) y=y) x X X I Y = X Y = Y XY=X: y=y: x X Y= X Y= Y X O X: Y Xy y: x X Y = X Y= Y X A x) y = x Y=Y) X X Y = X Y= Y X E x.y=x) Y=y) X X Y = X Y= Y X l xy=x: Y=y: X X Y= X Y= De Mor gan For ma l Logi c ( Lon don p. 63

98 Y X Ox: y xy Y: X X Y = X Y= Y X 8 X Y 8 P=O O D=A A D =A O D =A O C=E E C = E I I + I C =E I P X Y X Y X Y X Y X Y X X Y X Y X Y Y D X Y D X Y X Y X Y X Y X C X Y C X Y X Y X Y C X Y X Y X Y X Y

99 A E E X) Y+Y. Z X. Z D D D X Y Y Z X Z A +O A +O A A A O OA O O O D DD A A A A X) Y Y) Z X) Z 0 O Y X Y) Z Z X A O O X ) Y+Z Y = Z X 1850 X X X X X X A X) ) Y E X ) ) y X ). ( Y X( ) Y O X y X ( Y [ ] ) 1 8 V

100 X LY X LY X Y L X Y L X. LY X Y L X Y Y LY. X X. LY 1 X L( MY ) X Y M L X Y M L ( L of Mof Y) X L M) Y X LMY LM x. LM y z ( xlz z My ) z x z L z y M L M LM z xl z zmy LM M L ; L M M L M L 2 L X LY Y.. L X X Y L Y X L L X L X Y L XL L LY Boc henc ki HFL p. 375 HFL p. 375 p. 376

101 3 X Y L X Y L l X. LY X l Y X X X Ll X X L L X XLy Ly x y L x y L X L M) Y X Y L Y M 1. L L X LY Y L X X. L ) Y Y L ) X L L L L L) X.. LY Y.. L X Y L X Y L X Y L X X L ) Y X L ) Y Y L) X Y L X Y L X 1 2. L L X LY Y.. L X X LY Y.. L ) X HF L, p. 376

102 L L L L X X ) Y LY Y L X L Y L X 1, L L X L) YY L X L L 3. L L ) X.. LY X LY Y L X Y.. L X L L L L X LY X L Y Y.. L X Y.. L X 1 L I Y.. L X Y.. L) X L L 4. 1 ) X L Y X L Y Y L X Y L X 2 X L Y X L Y X L Y HF L, p. 376 HFL, p. 376

103 X L Y A B B A ), 5. X (LM Y X M L ) Y X L M) Y Y L M) X Z Y.. LZ Z. MX Z Z.. L Y X M Z, X M L ) Y L M L M ( LM L M M L ( LM M L LL) ) L LLL) ) LL) ) L; x Ly, y Lz xlz L y x z y z x x 6. L L LL LL) ) L L L ) L p. 377 HF L p 377,

104 L xl y yl z xl z yl x zl y zl x L x y y z x z 1870 Rus s el l : Pr i nci pl e s of Mat he ma t i c s, Nor t on, Ne w Yor k, p. 23 Not a t i on f or Logi c of Rel at i es, C. P. Vol. 3 C. P. Vol. 3, 45-46

105 a h, w m l b g b ( A, B, C A, C, B) A, B A B C. P. V ol. 3, 2 18

106 A, B, C, D A: A A B B A A C A: D B: B B C B: D C: A C: B C: C C: D DA DB D C D D l l = i j (l ) i j ( I J ) l ) I J l ) 1 0 l 1 l b, l 41 6 C. P. Vol. 3, 329 l b b

107 l b ( l b) = l ( b) 2 ( l, b), ( b) = ( b) ( l ) i i l b ( l + b) ( l b ) 3 = (l b) ( b} x x 4 l b ( l b) = l ) b 5 l o ( ) = 6 C. P. Vol

108 ( ) = l ) 7 l 8 l x m l x m l = =l =l = (l b) =( b l ) ( l b) = ( ) 1. l s b s l + b s. s l s b s l b. l b s l s b s. 2. s b s l s b. (l +b) x l s b x 3 (l +s b s ) l b+s

109 4 (1 ). l b=l b (2 l ) b= 1 1 l (b s) =( l b) (2) l ( bs = l b) s s 2 (1) (l +b) s =l s bs (2 l b s) s (l ( b s) 3 (1 l ( b s) lb l bs s (l b) s (2 lb= b=

110 4 (1) l b = ( 2) l b ( 3) l b (4) l b= 4 ), 5 1 ( l b) s = { l ( s p) b( s ) } ( 2 ) l ( b s ) = {( l p) b ( l ) s } ( 3) ( l b) s= {[ l (s p) ] [ b (s + ) ] } ( 4) l ( b s = {[ ( l p ) b] [ ( l p s ] } 5 1 w ( 1) s =s x y x s w) w ( 2) s sw sw s w = s... ) = s w s w1 s w s w W W s w = W s s w =s w w 1 + s w w=s ( w C. P. Vol. 3, ,

111 s w s w s w s w + sw s s w s z ( 3 x z 2 l l s 3 w w ( 3 b, m a bam b a ba b b b m

112 ) l =( l l l ( l l l l =l l l l i j l l = (l ) ( J ) C. P. Vol. 3,

113 ( l b) {( l = ) ( b} ( b) = {( l ) l ( ( b i k i k k i i i i k a (a ) i k i i k i k i 1. i i i i 2. { (i ) }{ (j ) } = { (i ) i ) } ( ) ) } { { (i }{ j (i ) ( i ) } 2 1 (a s l ) 1 ( + 2), b ) (

114 b ( ( 3 1 ) 2 b l s 3 ( ( } ( 4) (4 l b ( x x x x=x x + + x x =x x x x x i j i i x x i i x i i x x x x x x x = x x = x x = x C. P., Vol. 3393

115 x x x = = x x x = x x y y i j s a a n n n ) n w waw (a l l l x f x+ l

116 (b c b a b (b c a b c (b ) a b a ( b c ) i i a ( b c ) =f x y+z= z a c a c C. P. Vol. 3,

117 i j i j i { x x i j i j q q ) = ( q q + ) q q i i k q = x x x x ) x } q q = (q q q ) q: I x x + x x ) =

118 x

119 G. Fr ege, (1879), ( 1884), ) ( 1891), ( 1892), (1892)

120 , 19 A. Cauchy We i er s t r as s

121 17 A " A A A A B

122 ( 1) A B ( 2) A B ( 3) A B 4 A B BA B A) A A A B A A B B A B A), A B A B A B A B

123 ( B A) A( B A) B A B A B A B A B A B B A - B A) B A ( B A) (A B A B B A 8 J. van Hei j enoor t Fr om Fr ege t o G del ( Har var duni r r s i t ypr es s, 1977) F G.

124 , (x) x ( 1891) 1893 (1904 s i n( ) ) + 3 s i n 3 (A ( A A (A, B A B F. G. P. 22. Tr an s l at i ons f r om t he Phi l os ophi c al Wr i t i ngs of Got t l ob Fr ege ( New Yor k, 1952) p. 24. P. W. F., p F. G. p. 23.

125 =1 = 1 ( A) A ( ( A, B) B A B A [ (, ) ], (A, B, P. W. F. p. 30.

126 x x x) (x) x x ( x) x) x 9 ( 1), ( 2), ( 8), ( 28), ( 31), ( 41), ( 52), ( 54 58) [1] p ( q p) p ( q [2] ( p ( q r ) ) ( ( p q p r p (q r p p r ) [3] ( p ( q r ) ) ( q ( p r ) ) p (q. r q (p r [4] p q q p p q q p [5] p p [6] p p, [5] [ 6 ] [7] ( a=b) ( F ( a F( b) ) a=b F( a) F ( b)

127 [8] a=a [9] ( x) F F( a) x F a F B A B A a A a ( a ) A ( a A x) (x), a A (a B ( A ( a) B ( A x) ( x) ) a A B a a B ( A ( a) B A ) (a) B A) ( x) ( x ) B A ( x) (x ) ) [ 1] - [ 6] 1 2 F. G. pp p. 26

128 ( 2 ) ( 1 ) : ( 3 ) 2 2 ) 2 1) 3 1 ) 1 ) 2 a b 1 ) 1) 2 ) 2 ) 3) 1 ) ) 2 ) ( ukas i ewi cz ) [ 28) ] q) q p) p

129 A, B, T,, X 91) M F M F F. G. p. 28.

130 A A A A A A A A= B A B A B A B F( G ) F( ) F ( F( 1. F ( 2. F ( ) F ( F F x F( F( ) ( 1) = ; ( 2

131 = ) = = ) ( ( ) 2 \ ( 3=5) \ ( x ) ( x) x). p ( q p p p a. ( x) F( x) F ( a) b. ( F) M (F( ) ) Ms ( F( ) ) F M G( a=b) G[ ( F) ( F( a F( b) ) ], V. (A B) ( A B) A B ) A a V. ( F( ) = G( ) x) ( Fx G x F G x x F x G a V. a=\ (a= ) a (a = e( a= HFL, p. 368.

132 ,

133 f (, f (, 1 f f 2, f f x F x f F x F x f x F F f f F. G. p. 57.

134 x 1 f x F, 2 F f F F f x x f x f x f x F G 1. F G 2 G F ; 3. x, y, z x, x z z 4 x y z x z, z x y 1. F G, F G 2. F F 3. n n 0: 0 0 p. 60. p69. TheFoundat i ons of Ar i t hmet i c ( J. L. Aus t i n Oxf or d, 1950, F. A.

135 [ ] 0 n m F x F n F x m F. A. F, A ,

136 F F F F F F F (F F F F 0 m f ( m) n), f ( m m 0 A [A] A 0=df. =[ ] 1=df. {0}=[ {0}] 2=df. {0 1} =[ {0 1}] 3=df. {0 1 2} [ { 0 1 2}] f x n 0,

137 0 n 0 n n 0 n 0 n 0 ( von = Neumann) 1={0}={ }, 2={0, 1}={, { }} 3={0, 1, 2}={, { }, {, { }}}, 0 0 F. A. 83.

138 , 1893 a, b, c a b b c a b b c P. W. F. p. 61.

139 a= b b a a =b a=a b a a= ba =a = 2 = =4 4 4= P. W. F. p. 63. p. 62, p. 6, p. 65. p. 78

140 =4 P. W. F. P. 29. P. w. F. P 74. pp 58 59,

141 t hat

142 ( 1 ( 2 ( 3 (1) (2 (1) (2

143 a ( b a b a b x 4x x( x 4 x 4 x 4 ( 4) P. W. F.. PP

144 (. ( 4 ) 4 = ( 4 ( L. Wei t zman B. Li ns ky ( Pr i nci pl es of Mat hemat i cs B 1993 F. G. p. 125.

145 ( 1910 V w w w V V 31 V ( F( ) = G( ) x) ( F x G x F G F. G. p P. 127.

146 V) V V W. V. Qui n e) 1955 Gi us eppe Peano, P W F p p. 243.

147 ( 1889) a, b, c, a b a b a, b, c, a b c abc a a [ p ] [ P ] a a a a P 1 1 a a 2. a a a [ P ] a a a A. Dumi t r i u: Hi s t or y of Logi c, vol. i v ( Ab ac us Pr e 1977), PP67 80,

148 3. a=b. a b. b a [ Def ] [Def ] a=b a b b a 4. a =a [ P1. P4] [P1.. P4] 4 1 a 5. ab a [ P 6. a=aa b, a aa aa a 3) P6: a= aa a aa. aa a 3 7. ab ba [PP] 8. ab=ba b a a b a ba b 3 ab=ba 7 8 ba ab 9. abc acb [P ] 10. a b c=acb [P9. p10] a b. ac bc [P ]

149 a b[ P 12. a b. ] 1 2 b b 13. a c[ P c a P p a b. b - a [ p ] a b b a 2. a=b. - a=- b 3 - a) =a [P ] a b. 4. b - a 5. a= b = a - b [P2.. P5] 6. a b =- [ - a) - b) ] [De f ] 7- a b) = a) - b) 8 - ab ) = - a b) 7 8 [P6.. P7] a K a K a x a x a b a b a

150 b a b a b a b a b ab a b x x ( xa) =a a a P P x x 3 a.. a K, a a a x=y ( y=x x x y x y=x y= x b. = x a. xb a. a b x a b x x a.. x b 1903

151 R. Dedeki nd) 1888 S S S S S S K K K K( K K) K K A A A K K A A A A A A 1 A 1 A N 1:. NN. N = 1 r. 1 N. N N N

152 N 1 8 ( ) NN N N [ ( N) ] N n N 1 N=1 0 KS 1 ; S n N n S K K ( i ) 1 K i i (K K 1 N 1 N [ (N) ] N N a b a N 1 2. a N a=a a a= a, 73; 592 F. G. p F. G. pp

153 3. a, b N a= b = b= a a b a =b b=a 4.a, b, cen. a=b b= c:. a = c b a, b, c a= b b = c a= c 5.a= b b N: a N a =b b a 6. a N. a+1 N a a 1 a 7. a, b N. : a=b. =. a+1=b+1. a, b a= b a 1 = b a N a+1 =1 a a K xn. xk. 9.. x+1 k :. N k 1 x x N x x + 1 N K ) 9 1 0) 1 x+1 1, 18) :

154 a, b N.. a+ b+ 1 = a b ) 1 a a b b) +1 a b a ( b+ 1) a + b + 1) a b 4, 1 2) : 1. a N.. a 1=a 2. a, b N ( b 1) =a b a 11) : 2 N P1. 1 N 1 1[ a] ( P6). 1 N N ( 2 1) ( 2). 1 1 N ( 3) P =1+1 ( 4) (4). ( 3). ( 2, 1+1) [ a, b] ( P5) : :2 N 1 ( P1) 6( P6 1 a ; 1[ a] ( P6 P6 1 [ a ] 1 ) ( 2 P10 2 = 1 1; 3=2 1; 4=3 1 ; 5 2 a b 2= N:. 2 N,

155 ( 4 3 ) 2 N 4 ), ( 3 2 N ) a b a b, a b a b 5. x x x

156 ( Ber t r and Rus sel l,

157 ( A. N. W hi t ehead, ( Pr i nci pi a Mat hemat i ca P. M., ) 1919 P. M P. M

158 (1) p, q, r, s ( 2 p p ( 3 P ( 4 ( 5) p p p (6 P q p q P q = q Df. * P. M

159 Df p q p q 1. 1 P ( P 1 2 :p P.. p P p p p Taut ) 1 3 q.. Pq P q p Add) 1 4 :p q. q p P p q q Per m) 1 5 :p (q r q ( p r p p q r q p r As s oc) 1 6 q r : p q.. p r p q r p q p r Sum ) 1. 1 p p P. M. vol. 1, Cambr i dge, s ec nd edi t i n, 1925, p. 94.

160 Ber nays * 1. 5 P. M q.. p q 1) 1 01) q.. p 1 q Add q.. p q p p q.. p ( 1) : p. ( q r,p (r q) P. 98.

161 2. 03 : p q. qp p q q p p q. q p 2 16 p q. q p * q p.. p q p q r : q.. p r p q r q p r ) q r :p q.. p p q. q r. p p p 2 21 p.. p q p. q p q Df = ) p. p q pr q r = p q. q r Df p ) q q r p q r *3 2 p q.. q r r

162 p q p q : p. q. p 3 27 :p. q.. q p. q. :p. q r p q r p q r q p. q r : q : p. p q.. q p p q q p q. p r : p.. q. r p q :p. r q. r p q p r q s. :p. q.. r p. p) p q = p q. q 4. 1 :p q. q p * :p q.. p q p p p Df

163 4 21 q.. q p 4 22 :p q. q r.p r * :p p. p * :p. p p p. q q. p * :p q. q p * :( p. q). r p. ( q. r ) * :( p q ) r p ( qr ) *4. 4. :. p. q r p. q.. p. r * :. p.. q. r p q. p r p q. p p. q p q p p. q q.. p. q q p 5 1 :p. q.. p q p. q r : : p. q.. p. r p q.. r. : p.. q r x x x x ( x P. M. vol. 1, pp

164 . x x x) x). x x x x ), x). x x x ), : x. x x x ) x ). x x) x x. x ( x). x P. M p, q x,, x x). x x) px 9 x x { x. x}. x x * { x x } x x Df Df P. M. vol. 1, P. 11.

165 9. 03 ( x) x.. p x p Df x x p x x p) p 9 04 p ( x) x: ( x ). p x Df x) x.. p x). xp Df p. x). x: x ). p x D f ( x). x. ). = ( x ) : ( y), x Df x) ( x ( x y * ( y). y. ( x). x = x ). x Df 6 x x. ( z) z P p 9 11 x. z z 9. 1 x Pp Pp * p q,.p

166 . q, p q ( *1. 11)... x x x x x y. x * x, x a a pp * a a ; P p *9. 14 *9. 15 x x u v ( 1) ( 2 ( 3) u v ; ( 4) u, v, ( 5) u ( ). (, y v z., x) (, ),, ( 6 ; ( 7) u v u; ( 8 ) u x x, v ). : P. Mvol. 1, p. 1 32

167 . ). 9 3 ( x) x x. x: x x q. x x. q x. x x), x ( x ) x p. x) x. p x p x p y x ) x z z z z z z. z. z z z x x z z z z x 10 3 x: x). x. x). x x) x ) x x) x: x) x x x

168 1 1 1 ( x, ) ( x, ) ( x) x ) x, y ( x, ) y, x) ( x x) : y ( x, y y x x y * x) ( ) ) x) (x, ) x x). ( x, ) y ) : x ). ( x, x) : ( ). ( x, ) G del, )

169 z ) x z x x 1 z) z x 0 < x (x, ) (x, ) x ) x K x, y K (x, K ( x, ) 22 * R S xr y y Sy Df R S x x R x S

170 23 02 R S= xry. xsy R S R S R S= (xr y. S ) Df R S R= { (xr ) } D f * R S=R S Df R S R S R =R R R * R S) = R R S R S ( x, y). x( R R) y. x y R R ( x, ). { x( R R) y} x y R R 1) x, s i n x, l og x S Df

171 x = R y ( x) ( xry) Df R y R x R R y R R ( R y) (2 R xry x R R R R = R R R R R Cnv Cnv R R Cnv ) =Cnv R

172 E! Cnv P P E! = * : P= Cnv C nv P=P P P : y ypx x( Cnv P ) y. P x y P y x P ( 3 R y y x R x y y x x y= (xry x= (xry) R y y x x R y x y { = y) } Df = (xr = (xry) } Df = (4 R R R D R R R R R D R= {( y). x Ry} { x). xry} {( y) : xry R= C R=, R }

173 D R R C R D= [ = {( ). x Ry}] Df * = [ = { x). x Ry}] Df C= [r = { ( y) xry yrx}] Df D R, R C R y= ( xry) D R = { ). x } y DR ) 5 L M (l b). l ) b R S S R S = {( y). x Ry. ysz} Df R S x z x R z S R R R R =R R Df R =R R Df

174 34 2. Cnv ( R S) = * ( P ) R= P R) P R = P ) (P R) P R = P R) ( R) P ( R) ( P ) ( P R) P ) R ( P R) ( R) R P. S R S p R P S R S P D (P ) DP P ) P P P :E P z. P z = P ) z P z P ) z P P z z z P z P ) z E! ( P ) z P ) z P z z z z

175 (6) R y y R R R R R R R R R = {( y ). y. xry } Df R = ( =R ) Df t he t hef at her of x( x t he doubl eof x ( x t he s i ne of x ( x y R x y, x R P S P P

176 P S P ( 1) x x ( 2 x z x z 3) x y x x

177 * (x, ) t he t heaut hor of Wav er l ey t hemor ni ng s t ar ( x) ( x) x x

178 x =x x x x x= x ( x) ( x) =( x) ( x) x ( x) ; x ( x) = ( x) ( x) 1

179 . (1) x ( 2) x x ( 3) x x 1) ( Z c x x c c i ) x x c i i ) c x c ( 1) x x c (i i ) f f {( )( x ) } c) : x x= c f Df i 2 c x x c P. M. Vol. 1, P

180 b x x E! ( x) ( x b ) : x.. x=b Df E! b E! ( x) ( x) a E! a E! a 3. f x) ( x) f ( x) ( x) f x x x x f x x P. M. Vo l. 1. P. 174

181 x x x { c) : x x=c: f c}; ( c) x. x =c f c { [( x) ( x). f ( x) ( x ) } ( x) ( x) ) ]. f x ( x x)( x ( x) ( x ) f ( x) ( x) f ( x ) ( x) 169 P. M. Vol., pp

182 14. 01[ ( x) ( x)].f( x) ( x c). x. x=c: f c Df E! ( x) ( x) x x. ( x) ( x ) x ) ( x) x) (( x). E! ( x) ( x) x) ( x) x) ( E! ( x) ( x ( x ( x x ( x x) ( x x =b: ) ( ) =b x. b= x: : b= ( x) ( x x x x= b ) x=b) x x b=x ) b= :E ( x) ( x) b). ( x) ( x) =b ( x) ( x b b ( x) ( x). b). b= x) ( x). b ) ( x) x :E! ( x) ( x ) ) ( x = x ) ( x) ( x ) ( x)

183 x) ( ) ( x)( x :a= x) ( x) ( x) ( x =a x) ( x) =( x) ( x ). ( x) ( x) =( x) ( D. Hi l ber t, ( x E! ( x) ( x ) 2

184 ( ) ( x) ( x ) x; ( ) ( x) y 0 E! a E! a N. Res cher E! x x) p) [ Px&( P y] ( x) A( x) A A P 1 Re s c her, Topi cs i n Phi l os ophi c al Logi c, D. Rei del Publ i s hi ng Coml. pa ny, 1968, pp

185 2 x) E! x Mei nong c x c x

186 246

187

188 1 p q p 2 [ p ( q r ) ] [ ( p q) ( p r ] 3 p q) ( q p) 1 ) ( p q) [ ( q r ) ( p r ) ] 83

189 ( 2) p p ) p, ( 3) p p q ) p p p p p p) ( J. Ni cod) [ p ( q ) ] [ t ( t t ) ] { s q) [( p s) ( p s) ] } A A C C ( p p p ( 2) q p q) ( 3) ( p q ) ( q p) ( 4) ( q r [ ( p q) ( p r ) ] 1928 D. Hi l ber t, ( A cker mann 2 ) p p q 1 934

190 . 1 p ( q p ) 2 [ p ( p q) ] ( p q) 3 ( p [ ( q r p r ]. 1 p q p 2 p q q 3 ( p q) r q r ) ]. 1 p p q 2 qp q 3 ( p r ) [ ( q r ) ( p q r ) ] V. 1 ( p q) ( p q) 2 ( p q) ( q p) 3 ( p q) [ q p p q) ] V. 1 ( p q) ( q p) 2 p p 3 p p ( V V ( J. vonneu mann, P ( P 2 [ P R ) ] [ P P R ) ] 3 ( P ) ( P)

191 ( 4) ( )(P ) [ P ( ) ] P (5) ( ) P( P P( P( 1928 ( 5) ( x) F x F ( z) ( 6) F z ( x) F ) 1934 G. G ent z en UE UB OE OB [A] [ B] AE AB

192 E EB [F( a) ] FE FB NE NB [ A] OB A B A C B C C a AE EB a AE x) F ( x) EB x) F( x) C F( a

193 1 2 F( a) EE x) F ( x) x) F ( x) x) F x NB NE ( F( a) F ( a) 2 NE NE2 F a AE y) F( y) FE x) F ( x) ( y ) F 1 x) Fx x) F ( x) F( y) (1) ( pp p (2 ) p ( p q) (3) ( p q) ( q p) (4) ( pq) r p r q ) ] FB, OF FE (5) ( x) F F ( y) A B F E

194 ; ( 6) F( y ) x) F( x) E E FE A A A E. L. Pos t, F. G. pp

195 1928 p, q, r, p q, p 0 1, (1 4 0 p p 0 (p p) p[ 1) ] 0 ( 2 p q), p q, p p 0 0 q ) 0 0 A A B B A 0 A 1, A B B AB 0 B 0 p p p ) 0, 1 2 (md4) p 0 1, 1 0, 2 2 ( 2) ( 3) ( 4 0 p p 0 1) 0 p 2 (2 2 ) 2 = 02= 1 2= 2 1 2) ( 3 ) ( 4 ) p p 3

196 A B B p p, p p q B B A B B B C C p p p p... p p A B C p p p ,

197 ) (1) 4 5) 6) p p, p p ( 5) 6 p p 5 ) ( x) Fx F ( y x) A (x), ( ) A( y x) A x p p, ( y) A( ) p p5) 1) 4 6 A x) B( x) A ( x)b p p ( x)f(x) p p F ( y) F ( y), F( y) x) A( x) x) A( x) p p ( 6 86

198 . A ( x) ( ) A ( x) p p ( x) ( F ( x) F( x) x) ( Fx F x p p ( x ) A( x) x) A( x) p x) ( F( x) F( x) p, ( x) A( x) A( ) A ( ) x) Ax ( x) A( ( ) A x ) x ( x) A( x) z x) A x A( z) ) z) ( A( ) x) Ax ( 5 ) 6) x) F x x) F x z z) F x F ( G ( G x) F ( x) F ( z ) x) F( x) F F z z z z z) F F ( x) F F( x)

199 x A (,( ) A( G x) A( x) p p, ( y a p p x) G( x) G( y x) G ( x) p p G p p, G( G( y) A B A B p x) F( x) ( x) F ( x) 0 x F( x, F( 1930

200 (C. I. Lewi s, ) 1912 p q p q ( C. H. Lanf or d) S S,. Lewi s, C. I. A Sur vey of Symbol i c Logi c. Ber kel ey, Lewi s, CJ. and Lanf or d, C. H. Symbol i c Logi c. New Yor k, 1932

201 S p q ( p q) S 7 B1. p q B2. p q p q p B3. p p p p q ) B4. ( p q r B5. p p) B6. [ ( p q q r ] ( p B7. [ p p q) ] q r S =S + ( p q) 3 p S =S S p q) ( q p) S = S p p pp ) p) S =S p p ( p p)

202 , 0, 1 ( 0=0 = 1, ( 1=1) =1, ( 0=i ) =( 1=0 = 0. ( 0<0 = 0<1) =( 1<1 =1, ( 1<0) =0 a =( a 0) a b= a b) b] ab=( a +b ) ( ab a b 0 1 a= 1 ) =a 0 =1) =0 1=1 = 1 a b

203 . a<b) (b<c) <( a<c a a = 0 a+a =1 a=a ) =0, a=a 1921 m m t F. G. pp

204 m m) m m V p q p q m 1. m m- 1) P= ( p. p, p ) 2. r 1 p P t 3. P=( p p,, ),, q,, q ) p =( q q P p p q p 4 p =( p, p,, p p ( ( p p p ( p p ( p p p ( p F. G. p. 283

205 p ) (p, p )

206 G Gal i l eo, Sa gr edo ( Si mpl i ci o Sal vi at i

207

208 1 S = {1, 2, 3 n, } S ={1, 4, 9, n } S S S S S S n S n ) S S B. Bo l zano,

209 G. Cant or, L. Kr one cher, ( 1874),

210 Gaus s, ( Schu macher

211 1885 G. Enes t r om x

212

213

214 ; n N a N N N=1 1 n n , n

215 b a a 3 a, a a a 0. b b n n OC 1 ADBO AD BO P x y 1, x, y x=0. a 1a 2a 3 y=0. b 1b2b3

216 x z z=0. a 1b1a 2b2a 3b3 O C Q x, y O C z n 1879 J. Lur ot h z x, y x, y z ( Br ouwer 2 0 m n m n ) M N M M M M m M M N M M =

217 M N N c co nt i nuum c 1879 M u M M M M um UM M M M um UM a M a a M a a a M a M u M M um H. Rubi n and J. Rubi n, Equi val ent s of t he Axi om of st er dam Choi c e, Am.

218 a M M {0 1} M 2 { 0}, {1 0 1} M u M 2 M a M 2 2 a a a b a b b a ) M N M m N n m n M N M M M

219 1, 2, 1, 2,..., { 1, 2, } { 4, 3, 2, 1} + 5 {1, 2, 3,,, 1 2, 3, 4, 5} +5 Z {1 2 3 } +1{ {1, 2 3 1, 2 } 2 { 1 3 5, 2, 4 6 } 2 1 {1 3 5, } {1 4 7, 2, {1 4 7, } 2 = 2, 5+ = +5

220 2 { k 2 k 3k } Z Z ( E. Zer mel o, Equi val ent s of t he Axi om of Choi cs

221 C c c c P. J. Cohen 1963

222 1812 p p

223 p p p p Bur al i For t i ) {0, 1, 2,..., } +1 3 W W , + 1 F. G. pp

224 ( von Neuman S, S S S S F. G. pp

225 S S S H. Poi ncar e) w x x w x x w x, w w w w R S R S R R S R S R S R R S 17 18

226

227 J. Ri char d u u u s E N p F. G. pp

228 E n n 0 p 8 9 n p N E N E n n n G N G E E N J. Koni g G E E G E G N F. G. p. 14 3

229 N (Rams ey F. G. p. 142 F. G. pp

230 G. Ber r y K. Gr el l i ng 1908

231 Pd ( Pd( Pd( Pd) ( Pd Pd( Pd P Pd Pd P d P Pd Pd( Pd Pd) Pd( 1908 F. G. pp

232 a b a = b a b F. G. p. 200

233 a a b a b a b b a M x N x M x N M N M N M N A( x x a b M N M N M N N MM = N 0 a {a} a a a b { a, b } a b x {a} {a, b} A ( ) M M M M A ( x) x T T

234 (T T V T UT ( T T T 0 UT S T T M, N, R, m n r S 1904 s f x s, f ( x) x F. G. p. 141 F. G. p. 180

235 Z a {a} a {a} 1906 M A M

236 0, { 0}, {{0}}, Z T. Skol em (1 (1 ) (2 (3 (3 a b a=b = F. G. pp F. G. p. 22 9

237 Z, Z, Z } Z Z Z 0 {0} {{0}}, { 0, {0}}, n n 0 M Z n n B, B B Z B Z, Z, Z B { Z, Z Z } B n Z Z, Z Z B Z, Z, Z, U B a b a b U a M b M {Z, Z, Z, } Abr ahama. Fr aenkel, F. G. p. 297 pp

238 (x) x x ={ {{x}, {0}}, + 0}}} x x x x x x M, M M M x p ( ) x ) (x) ( x) 1922 Z 0, {0}, {{0}}, Z = Z (Z Z = Z, Z Z M M M F. G. p. 286 Kneebone, Ma t hemat cal Logt c and t he Founda t i ons of Mat hemai i es, London, 1963, p. 291

239 19 21 ( von Ne uman, A f x A B A x f x f ( x) f ( ) A B 1917 (D. Mi r i manof f a={a}, a = {a } a ={ a }, s t s t s s s ={s F. G. p. 398 F. G. p. 114

240 ZF 10 Z FC C C hoi ce ZF

241 = 1={ } 2={ { } } 3={ { } { { }}} ={ { } { { }} { { } { { } }} } 1={ { } { { }} { { } { { }}, }} F. G. pp

242 ( R. M. Robi ns on u (1 x u u x y, x =, x (2 x y y u x u F. G. p F. G. pp

243 x [ x y ] x [x, y A B x, ) x x, ) x = x = ) = ( x, ) 1 1. AB - 2. [ x, y x - y - 3. ( x x -

244 - 4 a b - - x [ a, x] = [ b x] a=b a a, x - x 2 u - - a [a, x] =u 3 - a [ a, ( x, y ] = x 4 - a [ a, ( x, ) ] = 5 - a x - - [a, ( x, ) ] =[ x, y] 6 a b - - c [c, x] = [ a, x] [ b, x] } 7 a b - - c [c, x] =[ a [ b, x] ] 1 - a: x= [ a, ( x, ) ] A 2 a - - b: [ b x] A, [ a, ( x, ) ] =A 3 a - - b, [ a, ( x, ) ] A [ b, x] a - x - - [ a, x] A 2 - a - - -

245 b - x y [ a, y] A [ b, y] = x [ a, y] A - a b - a - a ( A a, x A x [ b, x] A x b - a [ a, x] A B V a - [ a, x ] A x a a b - x a x b a b b a a b a b a b a b a b a b a< b; a b b a>b F, G. p. 43

246 1 - - a - - x xa - - x x a - - y a x y ZF a 2 a b x y y a y - - x b ZF a a b 3. a b x a( x y x y y b Z F a a b - - 2

247 V1) 2. A=, B={ }( { }, { } A B 3. ( u, v) ={{u, v}, {u}} K. Kur at ows ki 1921 u, v = x {{ u v} {u}}={ {x, y} {x} } u = x v= 4 - n [ n 1] [, n] 4 4 ( R. M. Robi ns on NB GBG BG

248 BG a b a b a b a a a a b I A B A B (A B ) ( A x B ) a=b, a b, a P. Ber nays, A s ys t em of axi omat i c s et t heor y, V, J. Symb. Lc gi c, 2( 1937), 65 77; 6( 1941), 1 17; 7( 1942), 65 89; 7( 1942), (1943), ; 13( 1948), 65 79; 19( 1954), Ber nays, p. Fr aenkel, A. A Axi onmat i c Se r Theor y ( Am s t er dam, 1958).

249 E a=b ( a A b A) E ( x) ( x a b) a =b A a 0( 0 A a c a b a = c b c b a( m [ A t ) ) ( m a t ( ) ) ( m t ) ) c c m c m t c 0, 1, 2 3,... 0, {0}, {0, {0}}, {0, {0}, {0, {0}}}, d d e d d v, d A 4 A, A6

250 A 1940 A X Y X 4 x Y x Y} B Y x Y C D E A- D BG, BG BGC,

251 ZF ZFC BG 1 ZF BG 2, BG ZF

252

253

254 / 1 1/ n + 1 n 1 n n 1 m n m n m n - mn n+m m m m m n xn= m x 182

255 m/ 1 x x my m m 1/ 1 1 m n p/ q mq p n m / n p q / 3 1/ 3

256 n n n n+1 n n n n n n +1 n n 1 = 2

257 S S S 1903 x x x x Ks K x ( x ) x x ( x) x B. Rus s el l, The Pr i ncr pi e s of Mat hemat i cs, 2nd ed. London, 1937, 104.

258 ( x ) x r x r 2 3 W. V. Qui ne)

259 ) 0, 0), ( 0, 0, 0) ( 0) ) ; ( 0, 0) ) 3 4 n n n +1 i i i = i 1 i x) { ( x) } x) Po

260 i ncar e w w P. M Vol. 1, pp

261 x, y, x, ( x, y ) x( x y z P. M. Vol. 1, pp

262 x ( x, ). ( ), ( x, ) ) x ) ( z x( x,, z) y ( z, z)!x! x x x.! x! x! x a! a a b! a! b!! (,x) f (! ), g(! ) F! ( ) g!, x) F(, z x) ( ) F(, x) x

263 n n n+1 F(!, x! z x 1 2 p p p n p p p n 1 q, q p p p n q n 1

264 1 2 n n n F(! z ^, x) 1 2 x. F(!, x 2 ). F(!, x) x F n n n 1 (.! x x!x x x! x n+1 n

265 L. Chwi s t ek 1925 Rams ey A B A B B 1937

266 R. Car nap, I nd( x) = (f ) [ ( Her ( f ). f ( 0) ) f x ] x f f 0 f x f 2 (f ) [ ( Her ( f ) f ( 0) ) f( 2) ] 0 2 P. Benac er r af a nd H. Put na m ( ed. ), Chi l os ophy of Ma t he mat i c s (Sel ect ed Readi ngs ), Pr ent i ce - Hal l, I nc., 1964, pp

267 2 2 Her ( f ). f( 0) f( 0) 0 He r ( f = ( n) [ f ( n f ( n+1 ] f ( 0+1) f ( 1 Her ( f ). f ( 0) 1 Her ( f ). f 0 f ( 1+1) f ( 2) 2 NF W. Qui ne NF NF N F 3 e x y z x ( y x x ( x) (x) ( x y NF

268 NF ( ) ( ) NF (. ( ) 3 3. ) 4 4. ( ) 5. ( 5 ( )) 6 6. ) ( ) 7 7. ( ) ) 8 8. = ) ( ( e ) }( e ) ) ( ( ) (( 9. ) = ) ) ( ) ( ) ) ( ( ).( ) ( ( = r ) ) x x x

269 x 11. )( )( ( ) ) x ( x) x ) (( z x) ( zey) ) v (x = x) V x {x} 12. { } = ) x {x, y } 13. { } (( r = ) = ( ; {{ }, {, }} x; y) 15. = ;. 15 (( x y) (( y x) ( x= ) ) ) x x x 5 1. x 2 ( ) 3 x x) ( y) ( ( x 3, x 4 x) x

270 5 ( ) y 3 3 x)( )( ( x )) x x) ( ( xx ( xe x) ) ( NF n n+1 x y x x NF 1-5, V x x x x

271 V 0 1, NF x x) ( ) ( ( x) ) - x x V V NF ( PM PM NF V 3 V, { } 3 NF PM NF PM NF 85

272 NF N F NF

273 NF PM 1970 NF NF

274 1931 Chu r ch)

275 Pr i ncr pl es of Mat hemat i cs, p. xi i

276 P. MVol. 1, pp

277 L. E. J. Br ouwer, ) Heyt i ng) We yl )

278 7 5=

279 Phi l osophy of Mat hemat i c s ( Edi t ed by P. Benac er r a f and H. Put nam), Pr ent i ce- Hal l, I NC. 1964, p. 69

280 n + n 1 1, 2,3 2, 3, 4 n 0 1 pp

281 2 2 = Heyt i ng, A. I ns ui t i oni s m. An I nt r oduct i on. Ams t er dam, 1956, p. 8

282 Bor el )

283 ), 50 I nt ui t i oni s m, p. 15

284 1912 A B B A A B A( =A B A A + B A A A A B C A A B A A + B A A C A A + B A A A + C A +B C A A A A B + C A A A C, C C, B, B, B, D C A A A + B A 0 1 A 2 n- 1) 2n Phi l os ophy of Mat hemat i cs, pp

285 A B A A A r A A - 3) F. G. pp F. G. p. 336

286 1 2 a <b a b a<b a b a=b S S S S S S S S S S S, S, I nt ui t i oni s m, p. 1

287 . k k- 1 k=i. l l 2 l =1 k k= 3) l p p+2 - I nt ui i i oni s m, p. 2

288 ( 1931 (1956 {a } n n n p > 0, a - a {a } k n=n( k), p, a - a < 1/ k k n( k) a { b } l ni ui t i oni s m, p. 16

289 k a k b n k, P 0 { a b a M AM A (1) k, n (2 3 k A

290 (4 a a k a a k M M M r, r, a, a a,, a, a - r 2 - a 1,, a n r M r r c M m c=m M P P S S S S Ini ui t i oni s m, pp p. 37

291 a b a b a b a =b a b a b a b a b a b a=b a b a#b) 55

292 A8 B 4 C 2 8a 4 2a P A B AB) Q B C P Q ( P Q 8a = 2 ( 2 2a) A C A, B, C, P AB, Q B C, P Q A C 2+3=

293 , A B A B A B A B A A A A A A A A I nt ui t i oni s m, p. 6 I nt ui t i oni s m, pp

294 A B C A B p ( p p ). (p q) ( q p). q) ( ( p r ) ( q r ) ). (( p q) (q r ) ) ( p r ). q ( p q ). (p p q) ) q. p p q ). (p q) ( q p). (( p r ) (q r ( ( p q r. p p q ) XI. (( p q) ( p q) ) p I. J ohans s on X, ( p q) ( q p) p p p p X I ( ( p q p p p p XI ' XI XI p pxi p (1 p p p p p

295 p p p q (2) q p q p) p q 2 ( (3 q) p 1) ) q ) (4) p p 2 ) p q 1) p p (4 5 p p p p q 2 (p (p q (6 ) (p q p q (6 7) p q ) ( q (p p; (6 7 ( 8) p q (p q) (p p q ( 9) (p p) ( p p 6) p p ( 9) ),

296 A A A (p q ( p q p (10) (p q ) p q q q (p q) (11 p 12 p (p q) ( q 12) ( ) A( x) x a A( a) A ( a x) A ( ) A( a) a o0

297 11. ( x) F ( x) F ( y). F ( y x) Fx (1) ( x) F x) F x (2) x) F x ( x) F ( x) 1 ) x F( x x x x x x x x ( 3 ) ( x) F x x) F x ( 4 ) x) Fx ( x) F x ( 5) x) F x ( x) F x ( 3 4 ( x) Fx ( x) F x 5 3 (6) ( x) F ( x) x F x 4 (7) x) F x x) F 8 ) ) (8) ( x) F x ( x) F x 8

298 (9) ( x) F x x) 9 7) p x (10) ( x) F x ( x ) F( x) ( 10 x ( 10) x 10 ) 1 1) (10) ( 10) 3 4) 2 (11) x ( x) F x (12) ( x ) F x x) F 5 ) ( 13) ( x) F x ( x) F x (14) x) F x x ( x ( p p q Kol mog or v 1932 a b a b, a b a b, a b a b, a x

299 Gl i venko) T E E T, E T, E 1932 E E E E E E E Davi s The Und eci dabl (New Yor k, 1965), pp

300 AB AB AB A ( x) A( x) x A ( x) ( A B) A B A B ) A ( x) Ax x) ( A( X) E E E E E' T E E T T E E A A E ( x) A (x) ( x) A( x) E E 1 0 E E (( x) A x ( ) A( x) B B ( x) A( x ( x) A( x) A A ( x) A ( A E

301 1 936 E E E E A A E A A Eo r E A. Mos t ows ki

302

303 D avi d Hi l ber t, ) ( 1899), ( 1904), ( 1925), (1927), 1 928), )

304 18 99

305 Thal es Pyt hagor as / Democr i t us ), Ar chi medes

306 Hi P Pi as 460 Pr ocl us Hi PP o cr at es of Chi os ) Ant i phon ( B r ys on 450 ( Pl at o Eud oxus

307 Aut o l ycus 310 ( Zeno of El ea ( Soc r a t es

308 1 (1) ( Pos t er i or i Ana l yi i cs, 71b , ,

309

310 ] 2 ] 3 [ ] A A B B B A

311 C AB C 4 47 ABD FBC, BL= 2 AB D GB = 2 FB C BL G B CL = A K

312 x x a + b ) 17 1 [ ] ] A BC BA C BC BA C BC O O A B A C

313 F E OF = OE; OB ==OC FB =EC AF =AE AB =AC A BC O A B C 1899

314 ( Saccher i, BC A B C D, AD ABC D :

315 B C B C ( 1 ) B C (2 B C ( Gaus s Loba chevs ky, Bo l yai,

316 ( Ri ema nn, ) Bel t r ami,

317 ( Kl ei n, Poi n car e) 1882

318

319 Pas ch

320 H. Wi ener

321 A B a A B A B A B A B C [ ] [ ] A B C 77

322 a A, B B A C A, B, C B C A A C B AC C A B 3 A B a A, B B, A AB A B A B a A, B C a ABC[ A, B, C ] A, B C a A B AC B C. A, B a A a a a A B AB A B A B A B A B A" B" A B A B A" B" AB BC a B

323 A B B C a B A B A B B C BC, AC A C (h, k) a h O k (h, k) ( h, ( h, A BC A B C AB A B AC A C BAC B A C ABC A B C a A a A a A a V V. A B CD A B A A, A A, A A A.., A A CD B A A V, V

324 A, B, C, D ( 1) C A D B A C C B D B A D ( 2 C B D B A C C A D A D O O A, OB, OC, OD O A A O D D C A BD ) [ ] C A A C A CA A CA A CA AA CA O CA O C A C A A C A B D O ( BD A, C, D O B A C CA A, B, O AB ( C A ) B O B O, A, D CA C A O A AD ) O D) O, B, D C A,

325 C A O, B, D OD ), CA BD C C, C B D C B D B C D BD CD O, C, D B D B D OC ), O C C O, A, C B D BD O A) O, A, D BD BD ) A D B D (AD B B A D 1 1. (x, y), v: w u v 0 ) x+vy+w=0 V V

326

327

328 F. G. p. 131 p. 137

329

330 S, ST, T) S A( A) A( a) A A A A a 0 O Hi l ber t : Di e l ogi s chen Gr undl agen der Ma t hemar i k, Mat h. Ann. 88 (1923), F. G. pp

331 , 2, 3, 4, 3. x A( x) A( x) x x a a+ 1 = 1 +a = =1 2

332 A( x) n A( n) a 1 = 1 a a a a, 1934 ( 1) (

333 F. G. P. 475 P 479

334 p 479

335

336 1. a b x

337 P 3 F. G. p. 6 15

338 4. 569

339 F. G. P 465 p 475

340

341 (

342 ( von Neumann A, A A A T. Skol em 1920

343 k n k B B (m<n) n 1 C Cn C C D F. G. pp

344 B F( x ) F( x ) B F ( x ) F( x ) C F( x F x ) ) ( F( x ) F( x ) ), D x ) ( x ) x ) [ ( F( x ) Fx ( F ( x ( ) F( x ) ) ] C 1 n C C D D A 2 n C A A n, C C C C n C, C, C, C C, C C, C K ni g) A

345 F( 0), G( 1, 2), H( 1, 2, 3) A A A A k c B k 1 2 A A A A A A A A A A A A 1928

346 n n C (L. Lowenhei m A A F. G. p. 589 Wang Ha o, Fr om Mot he ma t i c s t o Phi l os ophy, London N ew Yor k, 1974, p. 8.

347 pp. P

348 ) A, A A A A A A A

349 , 2,, n+1 n 1 m 1 l 1 n1 F. C, pp

350 1, 2,, n 1, 2,, n 1 L L L < L R L 0 L 1 R L L 0 1 L L L L L<L n L L v L L L L < L, L,, n L, L, L L ( n v n >n a a e 1, a n

351 1930 I X) ; X ( ) ) A ( ; ( ) A ( ; ),( ) ( ) A ( ; ) r s, x, r,, x, x, x s

352 x B P B n B B A A. Robi ns on 1960 R R R R R R

353 M1, M N k=1 2) B A, A A, M N M,, M N M A A A T h e Unde ci dabl e pp. 4 74

354 1 (1) ( [ ] [ ] = ) (2 ) 0 x, y z (, ) 2 (1) 0 (2 (3 s s (4 s ( s ) ( t 5 t s ) ( t ) (6 3 (1 s t ( s ) =( t (2 X (X (3 X Y X) ( Y ( 4) X ) (X ) ( X ) (5) 4 (1) ( A B ( A B) ) ( A B ( ) ( 2 A

355 (3) ( A B) ( ( A B B A ) ) 5. (1) x=x x= A x A( ) ) (2 0 x ' (x ' = ' x= ) x+0=x (x ' ) =( x+ ) ' x 0= 0 x=( x y) x A( 0) ( x) ( A x A( x ) ) A x x) Ax B B B B

356 1) p 1) p p p ( 1) p1) p p p p ,,,, =, 0, (, ) x 17, 19, z 2 3 k, k k P P i

357 ( z) ( z + x=y 12 13, 7, 23, 14, 13, 23, 11, 9, 7, 8, 19, 14 z + x=y x ) ( x = ), x x=0 ), m n 2 3

358 (1) D A X( D) A d) 15,, 5 Ax ( d) Ax( d d = d= d= (2 D E F CO ( D, E, F) Co( d, e, f ) d, e f d e f (3 Y A PF ( Y, A) Pf ( y, a), y a y a Pf ( y, a) Pr (y ) [ Y ] Gl (n, x) [ x n x = 2 3 P P G l ( n x k ) ] L ( x) [ x ] Pf (, a Pr ( y Gl ( L( y), y) = a

359 a 0=a a+b =( a b) ; 0 b a 2 a+ 1) a +1 ( a+0) ) a 2= a ) V

360 (1) ( a = a S ( ) ( a,, a =q n q C ( ) ( a,, a ) =a (1 i U ( ), ( n, i n,a x), x ) a = ( x (a ( a, a ) ( V) ( a n m, x, x m x, x n ( V),, x (v a) ( 0) =q (b ) =x( b, (b) ) ( q, x ) ( a = (V b 0, ), a a, a (, a,, a ) =x( b ( b, a, a ) a, a n 1 n- 1 x n 1 (V va) x Vb, x (k 1) a+b a 0 a+b =( ab ) a

361 a b a b a s g( a) ( s g( 0 (a b a b 1.

362 2. 3, R 4. Pf ( y, a) [ Y A

363 0, 1, 2, 3, u, v, w n

364 y 1 9 m 19 y m 19 m 1 9 A y Sb( m, m) m n

365 n m 10 n m 19 n m n n m 1 9 Sb( a, b) w b w w Sb( a, b) S( u, v) pf ( y, a) Y A p p w S b( a, b) S b( p, p)

366 1934 A A A A A A A A A n p Sb( p, p)

367 U( z ) U( z U( z U ( z U( z ) U( z ) U( z U( z U( z U( z F F( z ), F( z ), F( z ), ( ) F( v F F F F F ( z ), F( z ), ; ( ) F( v) F( z ), F( z ), v ) F( v v) F ( v F( z ), F( z ), F( z ), U( z u( z U( z)

368 (1) U( z U( z k P ( k, Sb ( p, p) ) S ( u, v S b( a, b), B( u, v P (y, a ) B( z, S( z, z ) U( z v), z, B( v, S( z ) B( z, S( z z ) ) U( z ( 2 U( z ( 1), U( z U( z U( z Sb( p, p) P (0, Sb( p, p ) ), P (1, Sb( p, p) ), P (2, Sb( p, p) ), B( z, S( z, z )) B ( z S( z z ) ) B( z, S( z, z v) B( v, S( z ) u( z (, z ) ) Neg ( a = A a A Neg( a =0 a Neg( a) = 2 3 N( u)

369 a) [ b) P (b, a c) P (c, N eg( a) ) ] a b, b a A c c A u) [ v) B( v u w) B ( w, N( u) ) ] Cons i s Cons i s { } { U( z a) [ b) P ( b, a) c) P (c, Neg ( a) ) ] ( x ) P x (p, p) ) Cons i s U( z ) Cons i s U( z z n) n z 3. 4

370 5 J. B. Ros s er (J. Par i s L. Har r i ngt on 20 30

371 U( z U ( z U ( z )

372

373 1936

374

375

376 ,

377 D. R. Ho f s t adt er 1979 Es cher ( Ba ck GEB 1984

378 BG =, ( J. van Hei j enoor t

379 A D E X, Y F1 (x, ) ={x, } F (x, ) = {z r s z = r, s z x r s} F ( x, x x r s r s F (x, ) =x- ; F x = { r s r, s x s } F ( x, ) ={ s s x r ) ( r s ) } ; F (x, ={ r s r, s x s, r } F (x, ) ={ r, s, t r, s, t x s, t, r y} ; F (x, y ={ r, s, t. r, s, t x r, t, s y} L F. G. pp The Co ns i s t ef n y of t he Cont i nu umhypot he s i s, Pr i ncet on, 1940