[]
̿ȤϡXYǤפȤʸޤϡXYǤϤʤפȤʸǤꡢְäƤ뤫ޤäƤʸΤȤǤ롣
Ĥޤꡢ̿ϿȵΤɤ餫ҤȤĤġϡP̿ΤȤϡPǤäƵǤϤʤ뤤PϿǤϤʤƵǤΤɤ餫ˤʤȤȤǤ롣ǡǤäƵǤϤʤ̿̿ȸƤӡǤϤʤǤ̿̿ȸƤ֤Ȥˤ롣
[]
̿ǤäƵǤϤʤ̿
̿ǤϤʤǤ̿
ؤоʪϡ̿̿ǤϤʤоʪ2Ĥʬ롣
ǡPؤоʪȤ롣ơΤ褦A,B֤ơ̿Ȳꤹ롣
̿AоʪPϿǤ롣
̿BоʪPϵǤ롣
ȡABοȤ߹碌ϼ4̤ˤʤ롣
A | B | ̣ |
1 | 1 | PϿĵǤ |
1 | 0 | PϿǤäơǤϤʤ |
0 | 1 | PϿǤϤʤơǤ |
0 | 0 | PϿǤʤеǤʤ |
ǡ㤨ñPϿǤȤȤϡ2ĤΥѥͤ롣
1PϿǤ롢ĵǤ
2PϿǤ롢ĵǤϤʤ
PϿǤפȤäƤ⡢PϵǤϤʤפȤϤʤȤˤʤ롣PоʪǤꡢ̿ǤϤʤȤա
äơŪˤPϿǤ뤳ȤꤷƤ⡢PϵǤ뤳ȤˤϤʤʤ
ȤP̿ȤȡΤȤPϿ̿ꡢ̿Ǥ뤫Τɤ餫Ǥ롣ȡPϿ̿ǤפꤹȡˤPϿ̿ǤϤʤפȤ櫓
[]
1ˢ̡PˡPˡʤ̤̣
2ˢ̡PˡP
PؤоʪȤȡP̿ǤפȡPϿ̿ǤפȤȤϰ㤦ȤΤϡPϿǤ롣P̿ǤפȤ̿˻Τ¸ߤ뤫Ǥ롣㤨С̷⤬Ǥ롣̷̷ʸˤȤϡʸǤ⤢ꡢְäƤʸǤ⤢ꡢ2ĤäƤ롣̷⤬̿ǤϤʤȤΤϼɡȤΤϤɤȤϡ̷ޤȤƺѤƤ˻פƤ˲ǽ뤫Ǥ롣
̿ˤϿ뤬դ˿ΤΤ̿ǤȤΩʤ
ǡ̿ǤϤʤоʪ̾ΤƤ
[]̿ǤϤʤоʪ̿ȸƤ֡
ơؤоʪ̿̿ʬव뤳Ȥˤʤ롣
ïǤʤʸ̿ȤϤʤͤǤ̤οͤǤʤʸ̿Τʤ̤͡פФȡŪȤïǤǤΤǤäĤޤꡢʸΰ̣ǤƤ̿Ȥߤʤ顢դΤ狼ʤ֤ʸΰ̣Ǥ櫓ʤ
Τ褦ˤʸ꤫̿̿Ǥʤμ¤ʬˡϤʤоʪ̿ȲꤷȤ̷⤬ФƤСоʪ̿Ȥ롣ˡƳˡ§Ǥ롣
ѤƵоݤˤƤΤϼ̿Ǥ롣ؤΤñPȽȤϡP̿ǤסP̿ǤʤסʡP̿ǤסˤȤʬƤ줾ˤĤʤʤФʤʤ
ʤߤˡؤǤϼ̿оݤȤȤäˤäƤ̿оݤˤǤ롣
PؤоʪȤ롣⤷ؤоʪ̿̿2ĤʬǤСʸX̿ˤʤ롣
XP̿Ǥ롣
P̿ǤXϿ̿ˤʤꡢP̿ǤXϵˤʤ롣ˤꡢ̿̿̿ȵ̿ѴǤ롣櫓
2Ĥ֤̿ʤСפȤȤӤĤ黻ǡABפȵҤ롣̿Aǡ̿BΤȤΤߡA͡פȤʤ롣
ͤɽηɽΤɽȤ
ɽϡ̿ο͡ʿȵˤΤ٤ƤȤ߹碌ˤĤơα黻̤οͤҤɽǤ롣
v(A) | v(A) |
T | F |
F | T |
ǤϤ̿οͤ뤿v( )ĤƤ뤬褯άƽƤ뤳Ȥ⤢롣 ޤTסTrueˡFסFalseˤȽ
v(A) | v(B) | v(AB) |
T | T | T |
T | F | F |
F | T | F |
F | F | F |
v(A) | v(B) | v(AB) |
T | T | T |
T | F | T |
F | T | T |
F | F | F |
v(A) | v(B) | v(AB) |
T | T | T |
T | F | F |
F | T | T |
F | F | T |
˼̿οɽĴ٤衣ʤ٤ʬƬǹͤ뤳ȡ
[̿]AB [̿]AB
嵭2Ĥ̿η̤Ӥơդʤ
[]
̿ο˴ؤ餺˿Ȥʤ褦̿̿ʥȡȥȸƤ֡
դˡ˵Ȥʤ̷̿̿̿ȸƤ֡
̿I̿Oɽ롣
1AˡAϾ˿Ȥʤ뤫鹱̿Ǥ롣
2AʡAϾ˵Ȥʤ뤫鹱̿Ǥ롣
[]
1ˢIO
2ˢOI
[]IOIOɽɽȼΤ褦ˤʤ롣
I | O | I | O |
T | F | F | T |
[]
ABפФơBAפȤ֢A͢BפȤ֢B͢AפжȤ
[]
ABפȤжǤ֢B͢AפƱͤǤ롣
ǽƳ٤Ƴе§ˡ§rule of assumptionsǤ롣άAȸƤФ롣
ε§ϡڤǤդʳˤơڤβȤФ줿Ǥդ̿ƳФ뤳ȤΤ褦ˡԤκ¦ˡֹ̿עֹֹסʳդˢֲȤʤäƤ̿עAפȵҤ롣
1 (1) PQ A
̿PQפε§ȤƻȤȤ̣롣
ϤޤäʤۤŪˤϤ褤餫βȤʤоǤʤŪ¤ʤɤ¿DzѤ櫓Ȥʤ٤̿ϹƤ̤ǤʤФʤʤ㤨аŹǤʤ顢¿ΰŹؼԤ˹ǧΤƤǰʬϺסΥпϺפȤä̿Ǥ롣¾ˤǧΤޤǤˤϻäƤʤʲʤɤ⤢롣㤨С֥४饯벾סCRSפʤɤǤ롣ٹ礤ԤβΤۤĤޤꡢ᤹붯ǤȡԤΤۤȤȤˤʤ롣
ؼԤϲ椹櫓ǤϤʤ˴Ť٤Ƥη˴ŤƤ뤳ȤΤ櫓Ǥ롣
2ܤƳе§꼰MPPModus Ponendo PonensǤ롣MPPϡȤƾŪ̿ȤξŪ̿ȤͿƤʤСξŪ̿θȤƳȤȤΤǤ롣Ĥޤ黻ͤ˴ؤΤǤ롣MPPĥͳϼ̤Ǥ롣
A | B | AB | A(AB) | (A(AB))B |
T | T | T | T | T |
T | F | F | F | T |
F | T | T | F | T |
F | F | T | F | T |
η̡(A(AB))BפϥȡȥȤʤΤǡĤޤꡢAʡABפע͡BפɬΩ롣
꼰ϸȤ˻ȤʤɡؤΤƤϤäȤŪʤΤǤ롣ޤŪǼˡ꼰ʤа̣ʤ
MPPѤȤ
1
1 (1) PQ A 2 (2) P A 1,2 (3) Q 1,2 MPP
2
1 (1) Q(PQ) A 2 (2) Q A 1,2 (3) PQ 1,2 MPP
嵭ϤɤȤ⡢12ϲȤͿ줿ȤˡMPPѤ뤳ȤQƳȤ̣Ǥ롣
⤦ʣ롣
3
1 (1) PQ A 2 (2) QR A 3 (3) P A 1,3 (4) Q 1,3 MPP 1,2,3 (5) R 2,4 MPP
ǽ3ԤɬפʲƳƤǤ롣(1)ܤϾˡPQפͿƤ롣(3)ܤǤPͿȡMPPˤä(4)ܤǷQƳ롣
A | B | C | AB | BC | (AB)(BC) | AC | ( (AB)(BC) )(AC) |
T | T | T | T | T | T | T | T |
T | T | F | T | F | F | F | T |
T | F | T | F | T | F | T | T |
T | F | F | F | F | F | F | T |
F | T | T | T | T | T | T | T |
F | T | F | T | F | F | T | T |
F | F | T | F | T | F | T | T |
F | F | F | F | F | F | T | T |
äơ( (AB)(BC) )(AC)פϺǽŪ˥ȡȥ¨ĤǤΩäƤ롣ϡ֤֡⤷ABǤꡢġBCǤפʤСACǤפȤȤ̣롣
[]
μ³ˡȸƤ֤Ȥˤ롣
̷ȤϡȵƱ˻ĤȤǤä
ȤǡP̿ǤȤPϿΤɤ餫Ǥ롣ΤȤPϿǤäƵǤϤʤפ뤤ϡPϵǤäƿǤϤʤפΤɤ餫Ǥ롣
äơP̷⤷Ƥʤ
̿ǡǤ̷̿⤷ƤʤȤħͭƤȤ롣
äơ̿θΤҤȤĤȤƻȤ
ѸȤơ̷⤷̿פȤդȤ뤳Ȥ롣̷̿餢ꤨʤȤʸνˡ̿פȤդդƤ뤫顢̷⤷̿פȤ̿ʤΰˤǤϤʤȤȿ뤫⤷ʤ
ϴְ㤤Ǥ롣ȤΤ⡢̿ϤȤ̷⤷ƤʤΤ顢ˡ̷⤷פȤ֤뤳ȼΤꤨʤȤȤǤ롣ĤޤꡢΡ̷⤷̿פȤѸ2ĤڤΥʤҤȤĤñˤʤäƤΤǤ롣㤨С֥ߥͥϳˤǭǤϤʤפȤȤ狼Ȼפ
̷⤬褿顢˴ϢΤȤˡ롣¤Ϥ⤤ĤǤΩϤʤΤ
[]
ˡȤϡ̿AǤȲꤷ³Ȥ̷⤷ФƤ褦ʾ硢ȲꤷΤ̿AϵǤȷդˡǤ롣
ؤ˴ƤͤʤˡζϤϼ´ƤȻפ뤬ΤʤͤǤäƤˡޤȤȤϼºݤ˿ɽƤߤȤ狼Ȼפ
A | B | A | B | AB | A͢B | (AB)(A͢B) |
T | T | F | F | T | F | F |
T | F | F | T | F | T | F |
F | T | T | F | T | T | T |
F | F | T | T | T | T | T |
äơAȡ(AB)(A͢B)פƱͤˤʤäƤ롣
ʾΤȤޤȤȡϤǤ롣LPȸƤ֤Ȥˤ롣
[]̿θLP
[§1]ʱ赬§DRdeduction ruleˡ
AꤷBƳФAȤʤABƳФƤ褤
[§2]꼰MPʥƥ졧modus poneusˡ
A,ABBƳФƤ褤
[§3]ˡ
ADʢPˤAƳФƤ褤
[§4]
̢AAƳФƤ褤~
[§5]ʷ
A,BABƳФƤ褤
[§6]ʬΥ
ʣABAƳФƤ褤
ʣABBƳФƤ褤
[§7]ʢ
2AABƳФƤ褤
2BABƳФƤ褤
[§8]ʢ˼
AB,AC,BCCƳФƤ褤
ABϡABˢʡBAˤάǤ롣
ǡA,B,C,DǤդɽ
嵭θޤȤȼΤ褦ˤʤ롣
[§1]Ƴ§ | [§2]ν§ |
[§3]̤Ƴ§ | [§4]̤ν§ |
[§5]ʤƳ§ | [§6]ʤν§ |
[§7]ˤƳ§ | [§8]ˤν§ |
[§3']
AꤷơDʢDƳФAȤʤˢAƳФƤ褤
[§3' ']
AꤷơDʢDƳФAȤ̵ˡAƳФƤ褤
[§4']
A̢AƳФƤ褤
[§3][3'][3' ']碌ˡȸƤ֡ޤ[§4][4']碌ȸƤ֡
[]§ؤɤߴ
A1ʡĢAnˢBηȡA1,,AnBƳФƤ褤פȤ§ȤϡŪƱΤȸʤƤ褤
椨A1ʡĢAnˢBη줿ʤСľ§A1,,AnBƳФƤ褤פ줿ΤȤߤʤȤˤơ§ޤƱƤ̾ǸƤӡʸξѤ뤳Ȥˤ롣
[]
ABˢBA
[]
ABȤʬƻˤ롣
ŪϤBAǤ롣
ǡAƻ̤äBA夭Bƻ̤äƤBAˤĤȤ롣
[]
ʣAB[1]ꭡ
ʣA[2]ꭢ
ʣBA[2]
ʣABAˡʣˡʣˤDR
ʣB[5]ꭣ
ʣBA[5]
ʣBBAˡʣˡʣˤDR
ʣBA[1]ʣˡʣˡʣˤȢ˼
ʣˡABˢBAˡʣˡʣˤDR