; Modus Ponens 1
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- せとか わたぬき
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1 ; Modus Ponens 1
2 1 2 Modus Ponens Gentzen 13 Modus ponens P Q P Q Q 2 P Q P Q Modus ponens Y Y 1 Modus Ponens P 2
3 Y Y Y Y Y Y C D Y F (C D) F F C D 2 1 A (B C) 2 A B 3 A C 3 A (B C) A B A C A (B C) A B A C 3 3 A (B C) A A Y B C A (B C) B C A 3
4 A B A A Y B A B B A 2 B C B 3 A (B C) A B A B C B 5 5 C A (B C) B C A A B B C 5 C B C B B Y C B C C B C A A (B C) B C A A B A B C (A B) C 2 A B C 3 1 A (A B) 2 A B 2 1 A B 2 B C 3 A C 4 1 (A B) (C D) 2 (C D) 2 A B E 5 1 (A C) ( B (C D)) 2 (A C) B 3 (A C) C D 6 1 (A C) ((A C) (B D)) 2 A C 3 B D 4
5 P Q P Q Y Y I Y Y Y Y Y I (A B) C 2 1 (A B) C 2 A 2 (A B) D 3 B 3 A C 4 B C D 4 1 ((A B) C) D 5 1 A (B C) 2 E (A B) 2 A D 3 E C 3 A 4 E (B C) D D 7 1 A (B C) 2 A A (B C) 3 1 A B 2 A C 3 A B C 6 1 A (B C) 2 A 3 B A C 5
6 P Q P P Q Q Y Y Y E Y Y Y E1 Y Y E2 E1 1 Y E2 2 Y Y A B 2 A (C D) D 2 1 A (B C) 2 A B C 3 1 A (B C) (A B) C 4 1 A (A B) B 5 1 A (B C) 2 B D C D 6 1 (A B) (C D) 2 A C B D 7 1 (A (B E)) (C (B F )) 2 A C 8 1 (((A B) C) D) (A C) (B D) F 6
7 ( ) ( ) 2 [] 1 ( ) 1 [ ] [] 1 I 3 [] 1 [] 1 3 7
8 4 3 [] 1 [] 1 [] 1 [] 1 [] 1 [] 1 ( ) 1 5 [] 1 5 I [] 1 []1 I ( )
9 I I 2 I 9
10 [ ] [] 1 []1 I ( ) I, 1 [] 1 I [] 1 1 I, 1 1 I [] 1 Y Y Y [ ] Y I Y Y [ ] Y [] n Y Y I, n [ ] [ ] n 7 1 D (A (B C)) D (A B) 10
11 D (A (B C)) D (A B) [A] 2 [D] 1 D (A (B C)) B A B 2 D (A B) 1 [D] 1 D (A (B C)) A B D (A B) 1 D (A (B C)) [D] 1 A (B C) [A] 2 B C B E A B I, 2 D (A B) I, A B 2 B C A C 2 A (C D) A D 3 1 A (A B) A B 4 1 (A B) C A (B C) 5 1 A (B C) (A B) C 6 1 (A B) C 2 (A B) D A (B (C D)) 7 1 A (A B) B 8 1 A B ((A B) B) B 9 1 A (B C) (A B) (A C) 10 1 (A B) (A C) A (B C) 11
12 17 I,, I, E 1 2 (a) I (b) I 3 (a) E (b) 1 (A B) (A C) A (B C) 1 (A B) (A C) A (B C) 2 (a) I (A B) (A C) [A] 1 B C A (B C) I, 1 2 (A B) (A C) A B C (b) I 2 (A B) (A C) (A B) (A C) [A] 1 (A B) (A C) [A] 1 A B 2 (A B) (A C) A B C I B C C 2 A (B C) I, 1 12
13 3 (a) E (A B) (A C) A B E (A B) (A C) A C B (A B) (A C) E [A] 1 A B B C A (B C) I, 1 E (A B) (A C) A C C I E [A] A B A C A B 3 A B A C A C (b) (A B) (A C) A B B E [A] 1 B (A B) (A C) A C C (A B) (A C) E [A] 1 A B B B C (B A C) I, 1 E [A] 1 C I (A B) (A C) A C C E [A] B C B 2 B C C (A B) (A C) (A B) (A C) E A B [A] 1 E A C [A] 1 B C I B C A (B C) I, 1 18 I A (A (B C)) B C 2 1 A B (B C) (A C) 4 1 A (C D) 5 1 A (B C) (A B) D (A B) C 3 1 A B 2 A (B C) C 6 (A B) ((B C) (A C)) 7 1 A (B C) B (A C) 8 1 A (B C) (A B) (A C) 9 (B (A C)) (A (B C)) 13
14 19 10 n n 2 n Proof n n 2 I n n m n = 2m + 1 n 2 = (2m + 1) 2 = 4m 2 + 4m + 1 = 2(2m 2 + 2m) + 1 x x = 2y + 1 y x n 2 n 2 n n 2 n I [n 2 ] [ (n )] n m n = 2m + 1 n 2 = 2(2m 2 + 2m) + 1 (x = 2y + 1) (x ) (n 2 = 2(2m 2 + 2m) + 1) (n 2 ) n 2 n n 2 n I 110 P P 14
15 E E 11 (1) (2) (3) (4) 1 A B 1 A B 1 A (B C) 1 (C (A B)) D 2 A B 2 B 2 (A B) C 2 (C D) (A B) 3 A 3 A
16 I I [] n I, n [ ] A [ ] n A B I B B B RAA I A B 2 B A 2 1 A A 3 1 (A B) A B 4 ( B (A B)) A 5 A A 6 1 A (B B) A 7 1 A B B A 8 1 A B ( A B) 9 1 A B (A B) 16
17 112,, I Y Y Y Y I E Y Y Y Y E1 Y Y E2 I Y Y [] n Y Y I, n Y Y Y Y I [] n I, n E E 17 E
18 14 (provability ) proof figure conclusion B 1,, B n A B 1,, B n A provable B 1,, B n A A A A (B C), A (B D) (A B) (C D) 2 (A B) (B C) A C 3 A (B C) (A B) (A C) 4 A ((A (B C)) C) A (A (B C)) C B 2 A B B A 3 (A B) C A ( C B) 7 (A B) (A B) 8 (A B) (A B) 9 (A B) (A B) 4 A (B C) C (A B) 5 A A 6 A A 10 A A 11 (A B) ( A B) 18
19 P P Q Q P Q Y Y Y I Y Y I1 Y Y Y Y I A (A B) 2 B (A B) 3 (A B) C A C 4 (A B) A B 5 ( A B) A B 6 (A B) A B 19
20 [] 1 [] (a) (b) 2 3 Y Y Z Z 20
21 E Z Ỵ Y Z Z Y [] n Z Z [Y ] n Z E, n 18 A B ( A B) 1,, A B ( A B) [ A B] 1 A B ( A B) [ A B] 1 [ A B] 1 ( A B) A B [ A B] 1 ( A B) 1 A B [ A] 2 [ B] 2 [ A B] 1 A B [ A] 2 ( A B) 1 A B [ B] 2 3 A B A E A B [ A] 2 E [ A B] 1 E [ B]2 B E E, 2 ( A B) I, 1 A B 2 21
22 19 1 A C, B C (A B) C 2 A B B A 3 A (A (A B)) 4 A (A B) A 5 A (A (A B)) 6 (A B) C (A C) (B C) 7 A B (A B) 8 (A B) C (A C) (B C) 9 (A A) A 10 A B (A B) 11 (A B) C (A C) (B C) 12 (A C) (B C) (A B) C 13 (A B) C B (C A) 14 (A C) (B C) (A B) C E E 20 1 A 2 A A B 4 A B A B 5 A (A B) B 3 A (A B) 22
23 118 A A A A A A A A E E RAA [ ] n RAA, n 23
24 Remark 21 RAA I [ ] n RAA, n [] n I, n [ ] n I, n 1 Remark 22 I [ A] 1 A I, 1 A E E A [ A] 1 E A RAA, A A 2 A (A B) A 3 A B B A 4 A B B A 5 A B A B 6 A A 7 (A B) A B 24
25 (A B) (B C) A C 2 (A B) (A C) A (B C) 3 (A (A B)) B Modus ponens 4 (A B) C A (B C) 5 A (B C) (A B) C 6 A A 7 A A 8 (B (A C)) (A (B C)) 9 (A A) 2 10 A B B A 11 B A A B 12 A B, A B 13 (A B) ( A B) 14 ( A B) (A B) 15 ( A B) (A B) 3 16 (A B) ( A B) 17 A A 18 A B A B 19 A B A B 20 (A B) C (A C) (B C) 21 (A C) (B C) (A B) C 22 (A B) C (A C) (B C) 23 (A C) (B C) (A B) C 25
26 120 I Y Y Y Y I E Y Y Y Y E1 Y Y E2 I Y Y [] n Y Y I, n Y Y Y Y I [] n I, n E E 26
27 I Y Y Y Y I1 Y Y I2 E Y Y Z Z Y [] n Z Z [Y ] n Z E, n E E E E RAA [ ] n RAA, n 27
U( xq(x)) Q(a) 1 P ( 1 ) R( 1 ) 1 Q( 1, 2 ) 2 1 ( x(p (x) ( y(q(x, y) ( z( R(z))))))) 2 ( z(( y( xq(x, y))) R(z))) 3 ( x(p (x) ( ( yq(a, y) ( zr(z))))
4 15 00 ; 321 5 16 45 321 http://abelardfletkeioacjp/person/takemura/class2html 1 1 11 1 1 1 vocabulary (propositional connectives):,,, (quantifires): (individual variables): x, y, z, (individual constatns):
Akito Tsuboi June 22, T ϕ T M M ϕ M M ϕ T ϕ 2 Definition 1 X, Y, Z,... 1
Akito Tsuboi June 22, 2006 1 T ϕ T M M ϕ M M ϕ T ϕ 2 Definition 1 X, Y, Z,... 1 1. X, Y, Z,... 2. A, B (A), (A) (B), (A) (B), (A) (B) Exercise 2 1. (X) (Y ) 2. ((X) (Y )) (Z) 3. (((X) (Y )) (Z)) Exercise
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Advent Calendar 2018 @Fukuso Sutaro,,, ( ) Davidson, 5, 1 (quantification) (open sentence) 1,,,,,, 1 1 (propositional logic) (truth value) (proposition) (sentence) 2 (2-valued logic) 2, true false (truth
( š ) š 13,448 1,243,000 1,249,050 1,243,000 1,243,000 1,249,050 1,249, , , ,885
( š ) 7,000,000 191 191 6,697,131 5,845,828 653,450 197,853 4,787,707 577,127 4,000,000 146,580 146,580 64,000 100,000 500,000 120,000 60,000 60,000 60,000 60,000 60,000 200,000 150,000 60,000 60,000 100,000
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( œ ) œ 138,800 17 171,000 60,000 16,000 252,500 405,400 24,000 22 95,800 24 46,000 16,000 16,000 273,000 19,000 10,300 57,800 1,118,408,500 1,118,299,000 109,500 102,821,836 75,895,167 244,622 3,725,214
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( œ ) œ 2,000,000 20. 4. 1 25. 3.27 44,886,350 39,933,174 4,953,176 9,393,543 4,953,012 153,012 4,800,000 164 164 4,001,324 2,899,583 254,074 847,667 5,392,219 584,884 7,335 4,800,000 153,012 4,800,000
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Ÿ Ÿ ( œ ) 120,000 60,000 120,000 120,000 80,000 72,000 100,000 180,000 60,000 100,000 60,000 120,000 100,000 240,000 120,000 240,000 1,150,000 100,000 120,000 72,000 300,000 72,000 100,000 100,000 60,000
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23 59 13 23 24 0101 0001 0101 0002 0101 0001 0101 0002 0101 0007 0101 0009 0101 0012 0101 0026 0101 0031 0101 0033 0101 0056 0101 0059 0101 0075 0101 0076 0101 5001 0101 0002 0101 0003 0101 0008 0101 0010
š ( ) 27,315,969 () 12,730,389 j 990,990 j 13,721,379 () 54,000,000 j 6,336,275 j 5,000,000 j 10,181,806 j 9,679,530 30,000,000 j 3,091,200 j 10,000,0
š ( ) œ 3,796,206,527 9,479,829,721 8,349,567,362 5,988,144,548 96,044,592 549,455,601 1,715,922,621 9,086,209,371 1,981,199,867 928,853,525 5,808,149,465 1,054,132,789 4,752,717,669 1,299,007 368,003,049
( œ ) œ ,906,148,000 1,800,000,000 1,706,469,380 1,350,676, ,793,167 1,555,793,167 3,000,000, ,537, ,
¾ œ ( œ ) (Ÿ ) lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll J N I Q ( œ ) œ 14 15. 3.28 2,906,148,000 1,800,000,000 1,706,469,380 1,350,676,213 355,793,167 1,555,793,167 3,000,000,000
Ÿ ( ) Ÿ ,195,027 9,195,027 9,195, ,000 25, ,000 30,000 9,000,000 9,000, ,789, ,000 2,039,145 3,850,511 2,405,371
Ÿ ( ) Ÿ 540,000 980,000 300,000 700,000 1,200,000 1,100,000 1,300,000 980,000 400,000 220,000 280,000 400,000 300,000 220,000 1,300,000 460,000 260,000 400,000 400,000 340,000 600,000 1,500,000 740,000
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18 5 10 1 1 1.1 1.1.1 P Q P Q, P, Q P Q P Q P Q, P, Q 2 1 1.1.2 P.Q T F Z R 0 1 x, y x + y x y x y = y x x (y z) = (x y) z x + y = y + x x + (y + z) = (x + y) + z P.Q V = {T, F } V P.Q P.Q T F T F 1.1.3
š ( š ) ,400, , ,893, ,743, ,893, ,893, ,658,475 35,884,803 2,167,037 7,189,060 55,417,575 81,08
Ÿ š ( š ) 1,970,400 5,000,000 12. 3.26 180,553,493 9. 9.29 41,772,995 10. 9.28 50,075,163 13. 2. 2 1,000,000 10.12.27 j 19,373,160 13. 4. 1 j 1,200,000 38. 3.19 j 1,100,000 6. 9.22 14. 1. 8 0 0 14. 3.13
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( ) 6,364 6,364 8,884,908 6,602,454 218,680 461,163 1,602,611 2,726,746 685,048 2,022,867 642,140 1,380,727 18,831 290,000 240,000 50 20. 3.31 11,975,755 1,215,755 10,760,000 11,258,918 (68) 160,000 500,000
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š ( ) 300,000 180,000 100,000 120,000 60,000 120,000 240,000 120,000 170,000 240,000 100,000 99,000 120,000 72,000 100,000 450,000 72,000 60,000 100,000 100,000 60,000 60,000 100,000 200,000 60,000 124,000
Ÿ ( ) ,166,466 18,586,390 85,580,076 88,457,360 (31) 1,750,000 83,830,000 5,000,000 78,830, ,388,808 24,568, ,480 6,507,1
( ) 60,000 120,000 1,800,000 120,000 100,000 60,000 60,000 120,000 10,000,000 120,000 120,000 120,000 120,000 1,500,000 171,209,703 5,000,000 1,000,000 200,000 10,000,000 5,000,000 4,000,000 5,000,000
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24 591324 25 0101 0002 0101 0005 0101 0009 0101 0012 0101 0013 0101 0015 0101 0029 0101 0031 0101 0036 0101 0040 0101 0041 0101 0053 0101 0055 0101 0061 0101 0062 0101 0004 0101 0006 0101 0008 0101 0012
Microsoft PowerPoint - logic ppt [互換モード]
![Microsoft PowerPoint - logic ppt [互換モード]](/thumbs/94/118469557.jpg "Microsoft PowerPoint - logic ppt [互換モード]")
寄せられた質問 : 演習問題について この講義の範囲に含まれる適切な演習問題が載っている参考書がありますか? できれば解答や解説が付いているものがあると良いのですが 第 回の授業の中で 演習問題に取り組む方法を説明しますこの授業は 回だけ行うもので 書籍の1 冊分に比べると少ない分量しかカバーしていません 回の講義の概観 : 1 完全性と不完全性 命題論理 命題論理 ( 真理値 ) ( 公理と推論規則
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š ( š ) 60,000 240,000 120,000 60,000 120,000 360,000 72,000 1,128,000 56,380,000 14. 2.20 35,492,337 17,401,486 18,090,851 32,141,906 11,070,000 3,570,000 7,500,000 7,020,000 7,020,000 851 851 9,778,644
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( š ) 557,319,095 2,606,960 31,296,746,858 7,615,089,278 2,093,641,212 6,544,698,759 936,080 3,164,967,811 20. 3.28 178,639,037 48,288,439 170,045,571 123,059,601 46,985,970 55,580,709 56,883,178 19. 4.20
( ) œ œ 50, , , , ,000 f 240, ,000 f21 30,000 1,000, ,725,367 18,680,993 9,044,374 11,219,342 9,000,000 9,
( ) œ 58,287,360 13. 4.25 15. 3.14 513,273,280 131,411,617 381,861,663 329,679,509 7,100,000 1,500,000 5,600,000 374,761,663 374,742,000 19,663 18,006,875 12,382,661 114,390 804,787 4,705,037 311,672,634
( ) ,930,658,000 3,000,000,000 3,380,845,133 2,160,275,757 1,220,569, ,569,376 2,550,000, ,308, ,308,
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Ÿ ( ) Ÿ 7,488,161,218 7,396,414,506 91,708,605 38,107 4,376,047 2,037,557,517 1,000,000 i 200,000,000 1,697,600, ,316.63fl 306,200,000 14
Ÿ ( ) (Ÿ ) Ÿ J lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll ¾ 17 18. 3.30 24,222,550,856 8,088,715,093 16,133,835,763 14,673,176,237 (400,000) 1,265,253,000 201,000,000 1,000,000 200,000,000
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Ÿ ( ) œ 1,000,000 120,000 1,000,000 1,000,000 120,000 108,000 60,000 120,000 120,000 60,000 240,000 120,000 390,000 1,000,000 56,380,000 15. 2.13 36,350,605 3,350,431 33,000,174 20,847,460 6,910,000 2,910,000
03 04 WATER PROOF SHOCK PROOF DUST PROOF SCRATCH PROOF STAIN PROOF 05 06
MEDIAS U N-02E MEDIAS U N-02E 1 2 3 01 02 03 04 WATER PROOF SHOCK PROOF DUST PROOF SCRATCH PROOF STAIN PROOF 05 06 100Mbps 14Mbps 37.5Mbps 5.7Mbps 07 08 09 10 11 12 Wi-Fi 13 14 15 16 17 18 INTERFACE 19
š ( š ) 2,973,655 3,774,545 4,719,254 1,594,319 3,011,432 1,517,982 1,493, ,503 2,591, , , , , ,000 f21 500,000 24
š ( š ) 812,488 8,633,171 390,374,410 324,279,452 9,953,269 17,329,976 2,944,796 2,944,796 6,866,917 341,279,452 12,000,000 12,000,000 2,000,000 2,000,000 1,000,000 1,000,000 500,000 600,000 I 1,000,000
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š ( ) 200,000 100,000 180,000 60,000 100,000 60,000 120,000 100,000 240,000 120,000 120,000 240,000 100,000 120,000 72,000 300,000 72,000 100,000 100,000 60,000 120,000 60,000 100,000 100,000 60,000 200,000
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š ( ) œ 1 1,128,321 24,000 3,347 359,874 741,100 5,519,882 1,863,153 10,000 10,000 10,000 36,729 3,600,000 107,000 33,749,250 24. 5.29 1,518,131 18,131 1,500,000 1,511,729 1,500,000 1,500,000 769,268 100,000
š ( š ) 7,930,123,759 7,783,750, ,887, ,887 3,800,369 2,504,646,039 i 200,000,000 1,697,600, ,316.63fl 306,200,
š ( š ) (Ÿ ) J lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll ¾ 907,440,279 16 17. 3.30 23,805,381,307 7,603,591,483 16,201,789,824 15,716,666,214 (400,000) 1,205,390,461 200,000,000 200,000,000
Coq/SSReflect/MathComp による定理証明 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
Coq/SSReflect/MathComp による定理証明 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/006241 このサンプルページの内容は, 初版 1 刷発行時のものです. i 1611 400 Coq/SS- Reflect/MathComp 1 1 Coq/SSReflect/MathComp
官報（号外第196号）
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Ÿ ( Ÿ ) Ÿ šœš 100,000 10,000,000 10,000,000 3,250,000 1,000,000 24,350,000 5,000,000 2,500,000 1,200,000 1,000,000 2,960,000 7,000,000 1,500,000 2,200
šœ Ÿ ( Ÿ ) Ÿ 3,658,819,708 612,940,933 1,441,054,976 1,536,693,282 369,033,491 1,167,659,791 68,105,057 25,460 7,803,540,263 1,713,934,550 541,531,413 702,848,302 11,827 1,552,629,488 23,421,737,374 2,572,144,704
š ( š ) (6) 11,310, (3) 34,146, (2) 3,284, (1) 1,583, (1) 6,924, (1) 1,549, (3) 15,2
š ( š ) ( ) J lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll ¾ 13 14. 3.29 23,586,164,307 6,369,173,468 17,216,990,839 17,557,554,780 (352,062) 1,095,615,450 11,297,761,775 8,547,169,269
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š ( œ ) 4,000,000 52. 9.30 j 19,373,160 13. 4. 1 j 1,400,000 15. 9.24 i 2,000,000 20. 4. 1 22. 5.31 18,914,932 6,667,668 12,247,264 13,835,519 565,000 565,000 11,677,790 11,449,790 228,000 4,474 4,474
Ÿ Ÿ ( ) Ÿ , , , , , , ,000 39,120 31,050 30,000 1,050 52,649, ,932,131 16,182,115 94,75
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官報（号外第197号）
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