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- ゆりか にばし
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1 2017 (1) ( )
2 - -
3 1/63
4 ( ) / 1/63
5 ( ) 0 2/63
6 ( ) 3/63
7 " " ( etc.) ( ) 4/63
8 5/63
9 (Logical Reasoning) 6/63
10 7/63
11 8/63
12 9/63
13 2 1 10/63
14 X Y a X a Y 11/63
15 2 2 ( (?)) ( (?)) 12/63
16 A D O B C AB = DC AC = DB (1) ABC DCB (2) ABO DCO 13/63
17 (a) 3 (b) 2 (c) /63
18 1 1 ( ) AB = DC AC = DB (1) ( ) BC = BC (2) ((1),(2) (a) ) ABC DCB (3) ((3) ) AB = DC (4) ((3) ) OAB = ODC (5) ((3) ) ABC = DCB (6) ((3) ) ACB = DBC (7) ((6),(7) ) ABO = ABC DBC = DCB ACB = DCO (8) ((5),(6),(8) (c) ) ABO DCO 15/63
19 A B /63
20 ( ) 17/63
21 ( ) 18/63
22 (a,b,,;, ) k /63
23 (a,b,,;, ) k the least positive integer that is not denoted by an English expression containing fewer than 200 occurrences of symbols ( 200 ) 200 k+1 ( ) 19/63
24 ( OK) ( ) 20/63
25 - - ( ) 20/63
26 (Aristotle, B.C ) 17 21/63
27 4 syllogism All X is Y No X is Y Some X is Y Some X is not Y 22/63
28 (1) All X is Y Therefore, All non-y is non-x (2) All X is Y No Z is Y Therefore, No Z is X (3) All X is Y Some Z is Y Therefore, Some Z is non-x 23/63
29 Venn Diagrams (Venn, 1881) A 24/63
30 Venn Diagrams B 25/63
31 A B ( ) A B 26/63
32 A B ( ) A B 27/63
33 A A 28/63
34 B B 29/63
35 (1) (A B) = A B 30/63
36 (2) (A B) = A B 31/63
37 (A B) A B A B A B = A A B = B A B = 0 A B = 1 ( 0 ) 32/63 1
38 ( ) (2) All X is Y No Y is Z Therefore, No X is Z X Y Y Z = 0 Therefore, X Z = 0 33/63
39 Algebra al-jabr Algebra( ) al-jabr 1+x = 3 x 2x = 3 1 2x = 2 x = 1 34/63
40 Algebra al-jabr Algebra( ) al-jabr 1+x = 3 x 2x = 3 1 2x = 3 1 x = 1 34/63
41 X Y (1) Y Z = 0 (2) Therefore, X Z = 0 By (1) X Y = 0 (3) By (3) X Y Z = 0 (4) By (2) X Y Z = 0 (5) By (4),(5) (X Y Z) (X Y Z) = 0 (6) By (6) (X Z) (Y Y ) = 0 (7) By (7) (X Z) 1 = 0 (8) By (8) X Z = 0 35/63
42 (George Boole, ) ( ) 36/63
43 A. De Morgan (Formal Logic, 1874), George Boole (Calculus of Thought, 1854), C.S. Pierce, Ernst Schröder (The Algebra of Logic, 1890), W.S.Jevons (Pure Logic, 1864; Elementary Lessons in Logic, 1870). 1. X X = X 2. X X = X 3. X Y = Y X 4. X Y = Y X 5. X (Y Z) = (X Y) X 6. X (Y Z) = (X Y) X 37/63
44 7. X (X Y) = X 8. X (X Y) = X 9. X (Y Z) = (X Y) (X Z) 10. X (Y Z) = (X Y) (X Z) 11. X X = X X = (X ) = X 14. X 1 = X 1 = X 16. X 0 = X 17. X 0 = (X Y) = X Y 19. (X Y) = X Y 38/63
45 Peano Formulario mathematico Formulario mathematico ( ) al-jabr 39/63
46 (Giuseppe Peano, ) ( ) 40/63
47 ( ) 41/63
48 x (P x Q x) P a Q a ( ) 42/63
49 (Gottlob Frege, ) ( ) 43/63
50 Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position I was placed in by a letter of Mr. Bertrand Russell, just when the printing of this volume was nearing its completion. ( 2 ) 44/63
51 (Bertrand Russel, ) ( ) 45/63
52 ( ) X X A A X / X X A A A A A / A A A A / A A A A A A / A A A A / A ( ) 46/63
53 (theory of types) (Bertrand Russel, 1903) /63
54 (Alfred North Whitehead and Bertrand Russell, 1910, 1912, and 1913) 48/63
55 ( ) 49/63
56 ( ) (1) (P P) P (2) Q (P Q) (3) (P Q) (Q P) (4) (P (Q R)) (Q (P R)) (5) (Q R) ((P Q) (P R)) A B A B A Aθ θ: 50/63
57 1904: ( ) /63
58 (Hilbert and Ackermann, 1928)... (1) A (B A) (2) (A B) ((A (B C)) (A C)) (3) A (B (A B)) (4) A B A (4 ) A B B (5) A A B (5 ) B A B (6) (A C) ((B C) (A B C)) 52/63
59 (7) (A B) ((A B) A) (8) A A (9) A(t) xa(x) (10) xa(x) A(t) A B A B A(a) C C A(a) xa(x) C C xa(x) a C Hilbert ( ) ( ) 53/63
60 54/63
61 55/63
62 56/63
63 57/63
64 (Kurt, Gödel, 1930) 58/63
65 I (Kurt Gödel, 1931) ( ) 59/63
66 (Natural Deduction) (Gerhard Gentzen, 1935) Hilbert&Ackermann ( ) 60/63
67 x P Q x P a Q a [ x(p(x) Q(x)] P(a) Q(a) Q(a) [P(a)] ( ) 1 [ ] 61/63
68 (Gerhard Gentzen, ) ( ) 62/63
69 2 ( ) 63/63
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