Vikipediya ro'yxatidagi maqola
Bu ro'yxat xulosa chiqarish qoidalari, matematik formulalarga tegishli mantiqiy qonunlar.
Kirish
Xulosa chiqarish qoidalari sintaktik o'zgartirish argument yaratish uchun qandaydir asoslardan xulosa chiqarish uchun foydalaniladigan qoidalar. Qoidalar to'plamidan har qanday haqiqiy xulosa chiqarish uchun foydalanish mumkin, agar u to'liq bo'lsa, hech qachon bekor qilingan xulosani, agar u sog'lom bo'lsa. To'g'ri va to'liq qoidalar to'plami quyidagi qoidalarga har qanday qoidalarni kiritmasligi kerak, chunki ko'plab qoidalar ortiqcha va ularni boshqa qoidalar bilan tasdiqlash mumkin.
Chiqib ketish qoidalari vaqtinchalik taxmin asosida subderivatsiyadan xulosa chiqarish. Quyida yozuv
![{ displaystyle varphi vdash psi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5c3647f21c5149f25463fe5d81eb6fe3b784f41)
vaqtinchalik taxmindan bunday subderivatsiyani ko'rsatadi
ga
.
Klassik sentensial hisoblash qoidalari
Sententsial hisob-kitob sifatida ham tanilgan taklif hisobi.
Rad etish qoidalari
- Reductio ad absurdum (yoki Salbiy kirish)
![{ displaystyle varphi vdash psi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5c3647f21c5149f25463fe5d81eb6fe3b784f41)
![{ displaystyle { underline { varphi vdash lnot psi}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee4304a4bd25eed12585526d42d16ff0ae3f0b8f)
![{ displaystyle lnot varphi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a96766a23a9525c90f64bf05589c735b0d0e5c8d)
- Reductio ad absurdum (bilan bog'liq chiqarib tashlangan o'rta qonun )
![{ displaystyle lnot varphi vdash psi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4324190bdbcf764d068f048392c2dcf3890da621)
![{ displaystyle { underline { lnot varphi vdash lnot psi}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c38e2ee70c02f364d97ba7c9a38a004f5b890463)
![varphi](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e)
- Ex qarama-qarshilik quodlibet
![varphi](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e)
![{ displaystyle { underline { lnot varphi}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/419fb5ca2aa9ee4e04905ed81e95e6434c5bb88f)
![psi](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a)
- Ikkita inkorni yo'q qilish
![{ displaystyle { underline { lnot lnot varphi}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/640acc5c9874115931ea417a7c980e4f60c7ab0e)
![varphi](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e)
- Ikki marta inkor etish
![{ displaystyle { underline { varphi quad quad}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61d80884dfa13bc7b0ffc020c83eb837473f3750)
![{ displaystyle lnot lnot varphi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18e1ce562a300cae1503b8d6a39e810c1eb17b13)
Shartli shartlar uchun qoidalar
- Chegirma teoremasi (yoki Shartli kirish )
![{ displaystyle { underline { varphi vdash psi}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0325b53557cdce9b4f9f42adfac70056366d4acc)
![{ displaystyle varphi rightarrow psi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/724678bd1e82d05bae3d261b9160d514add0de3b)
- Modus ponenslari (yoki Shartli yo'q qilish)
![{ displaystyle varphi rightarrow psi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/724678bd1e82d05bae3d261b9160d514add0de3b)
![{ displaystyle { underline { varphi quad quad quad}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5366a62cef6641dd3e34aac25af3610b36ca6216)
![psi](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a)
- Modulli tollens
![{ displaystyle varphi rightarrow psi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/724678bd1e82d05bae3d261b9160d514add0de3b)
![{ displaystyle { underline { lnot psi quad quad quad}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/619e53ead7f0c65a16520a8e1c588dabdffe4007)
![{ displaystyle lnot varphi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a96766a23a9525c90f64bf05589c735b0d0e5c8d)
Bog'lanish uchun qoidalar
- Qo'shish (yoki Birlashtiruvchi kirish)
![varphi](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e)
![{ displaystyle { underline { psi quad quad }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a36de6e71855251806814852eaa69a5c54f7896b)
![{ displaystyle varphi land psi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a406020b081624ed6df4349e583a404ad556ae31)
- Soddalashtirish (yoki Ulanishni bartaraf etish)
![{ displaystyle { underline { varphi land psi}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf7155f8aed0fdb0aac3f550b8228d2ccc1b7fd2)
![varphi](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e)
![{ displaystyle { underline { varphi land psi}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf7155f8aed0fdb0aac3f550b8228d2ccc1b7fd2)
![psi](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a)
Ajratish qoidalari
- Qo'shish (yoki Diskunktsiyani kiritish)
![{ displaystyle { underline { varphi quad quad }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdaf33fffde6d8d6fd908ba7ac01f278fc61b07d)
![{ displaystyle varphi lor psi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53732d566089f41274c3fc138c14cd87ba59febd)
![{ displaystyle { underline { psi quad quad }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a36de6e71855251806814852eaa69a5c54f7896b)
![{ displaystyle varphi lor psi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53732d566089f41274c3fc138c14cd87ba59febd)
- Ishni tahlil qilish (yoki Ishlar bo'yicha dalil yoki Ishlar bo'yicha tortishuv yoki Diskunktsiyani yo'q qilish)
![{ displaystyle varphi rightarrow chi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e2ddc0e6ab0cdd7ef401b4c1901b74ce5d61f44)
![{ displaystyle psi rightarrow chi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9d3b0f452cc028acabd9d8595e71245ed560509)
![{ displaystyle { underline { varphi lor psi}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/620f6edb47c5b6366765f8b81cddf31a06982c9b)
![chi](https://wikimedia.org/api/rest_v1/media/math/render/svg/656111758322ace96d80a9371771aa6d3de25437)
- Disjunktiv sillogizm
![{ displaystyle varphi lor psi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53732d566089f41274c3fc138c14cd87ba59febd)
![{ displaystyle { underline { lnot varphi quad quad}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6839c77dc5843f772ffcd444e49bcfaf639bc427)
![psi](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a)
![{ displaystyle varphi lor psi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53732d566089f41274c3fc138c14cd87ba59febd)
![{ displaystyle { underline { lnot psi quad quad}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b558450a9f279e27ad7aa370000672574c072a2a)
![varphi](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e)
- Konstruktiv dilemma
![{ displaystyle varphi rightarrow chi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e2ddc0e6ab0cdd7ef401b4c1901b74ce5d61f44)
![{ displaystyle psi rightarrow xi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2fda34c6ad13531fb16d3c8e61641a488034b1c)
![{ displaystyle { underline { varphi lor psi}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/620f6edb47c5b6366765f8b81cddf31a06982c9b)
![{ displaystyle chi lor xi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/314d002246285b90c6f44677d327f82c6ac969dd)
Ikki shartli qoidalar
- Ikki shartli kirish
![{ displaystyle varphi rightarrow psi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/724678bd1e82d05bae3d261b9160d514add0de3b)
![{ displaystyle { underline { psi rightarrow varphi}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee0df2d5845a0ce9f86745a6b198587c83a81489)
![{ displaystyle varphi leftrightarrow psi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e68c9c0e49df2cdfb5bbb241b0d3f6f10e20aa90)
- Ikki tomonlama shartli ravishda yo'q qilish
![{ displaystyle varphi leftrightarrow psi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e68c9c0e49df2cdfb5bbb241b0d3f6f10e20aa90)
![{ displaystyle { underline { varphi quad quad}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61d80884dfa13bc7b0ffc020c83eb837473f3750)
![psi](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a)
![{ displaystyle varphi leftrightarrow psi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e68c9c0e49df2cdfb5bbb241b0d3f6f10e20aa90)
![{ displaystyle { underline { psi quad quad}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/152e8920f390186ddf8105319ac64a2007909a59)
![varphi](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e)
![{ displaystyle varphi leftrightarrow psi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e68c9c0e49df2cdfb5bbb241b0d3f6f10e20aa90)
![{ displaystyle { underline { lnot varphi quad quad}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6839c77dc5843f772ffcd444e49bcfaf639bc427)
![{ displaystyle lnot psi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e95306a224ed882b91ba0a7f0e7f297f49b9023d)
![{ displaystyle varphi leftrightarrow psi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e68c9c0e49df2cdfb5bbb241b0d3f6f10e20aa90)
![{ displaystyle { underline { lnot psi quad quad}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b558450a9f279e27ad7aa370000672574c072a2a)
![{ displaystyle lnot varphi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a96766a23a9525c90f64bf05589c735b0d0e5c8d)
![{ displaystyle varphi leftrightarrow psi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e68c9c0e49df2cdfb5bbb241b0d3f6f10e20aa90)
![{ displaystyle { underline { psi lor varphi}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8a5c27d415b3e0f5ef2adb9c4fe96a94d6753e6)
![{ displaystyle psi land varphi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30d203e4c3edb5302fa396b92d8cbc14845e32a3)
![{ displaystyle varphi leftrightarrow psi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e68c9c0e49df2cdfb5bbb241b0d3f6f10e20aa90)
![{ displaystyle { underline { lnot psi lor lnot varphi}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c09fa9ca3a72b82c53776f4db159d4f96452685b)
![{ displaystyle lnot psi land lnot varphi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c65e7d32b78edcc7233e35b805109283c63b2e4c)
Quyidagi qoidalarda,
xuddi shunga o'xshash
muddat bundan mustasno
qayerda bo'lmasin
erkin o'zgaruvchiga ega
.
- Umumjahon umumlashtirish (yoki Umumjahon kirish )
![{ displaystyle { underline { varphi {( beta / alpha)}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e84d26bb39cf5246300f075d6505880e44ebb5ca)
![{ displaystyle forall alpha , varphi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a56edc03bd9670c38ca5c1674f766070f348cfff)
1-cheklash:
sodir bo'lmagan o'zgaruvchidir
.
Cheklov 2:
hech qanday gipotezada yoki bekor qilinmagan taxminlarda qayd etilmagan.
- Umumjahon misol (yoki Umumjahon yo'q qilish )
![{ displaystyle forall alpha , varphi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a56edc03bd9670c38ca5c1674f766070f348cfff)
![{ displaystyle { overline { varphi {( beta / alpha)}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4a81da77b0306e5736a9fac6d4ec10990de51d2)
Cheklov: bepul yuzaga kelishi mumkin emas
yilda
ichida sodir bo'lgan o'zgaruvchini miqdoriy aniqlovchi miqdor doirasiga kiradi
.
- Mavjud umumlashtirish (yoki Mavjud kirish )
![{ displaystyle { underline { varphi ( beta / alpha)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea2f57bf87db5b160aeee01683b7eb0596d91a92)
![{ displaystyle mavjud alfa , varphi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13b542f075bec773eda1746c66c23054d618e986)
Cheklov: bepul yuzaga kelishi mumkin emas
yilda
ichida sodir bo'lgan o'zgaruvchini miqdoriy aniqlovchi miqdor doirasiga kiradi
.
- Mavjud instantatsiya (yoki Mavjud bartaraf etish )
![{ displaystyle mavjud alfa , varphi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13b542f075bec773eda1746c66c23054d618e986)
![{ displaystyle { underline { varphi ( beta / alpha) vdash psi}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b16e383bda7e2569c7405344a486d3a92861e6c)
![psi](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a)
1-cheklash:
sodir bo'lmagan o'zgaruvchidir
.
Cheklov 2: hech qanday erkin yoki majburiy hodisa mavjud emas
yilda
.
Cheklov 3:
hech qanday gipotezada yoki bekor qilinmagan taxminlarda qayd etilmagan.
Quyida universal umumlashtirish va ekzistensial yo'q qilishning alohida holatlari keltirilgan; kabi substruktiv mantiqlarda uchraydi chiziqli mantiq.
- Zaiflash qoidasi (yoki jabrlanuvchining monotonligi ) (aka klonlashsiz teorema )
![{ displaystyle alpha vdash beta}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42d75c7787c9df6850070baf7cb2432878995c74)
![{ displaystyle { overline { alpha, alpha vdash beta}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58b37a0cc078bb188cec95275e329bc497f4d8d6)
- Kasılma qoidasi (yoki jabrlanuvchining beparvoligi ) (aka yo'q qilinmaydigan teorema )
![{ displaystyle { underline { alfa, alfa, gamma vdash beta}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3b4eee32e0c35172a3615c6ea92bb2a24a51786)
![{ displaystyle alfa, gamma vdash beta}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e0a8415b75790c596160ed4f997d80ef9280567)
Jadval: xulosa qilish qoidalari
Yuqoridagi qoidalar quyidagi jadvalda umumlashtirilishi mumkin.[1] "Tavtologiya "ustunida berilgan qoidaning notasini qanday izohlash mumkinligi ko'rsatilgan.
Xulosa chiqarish qoidalari | Tavtologiya | Ism |
---|
![{ start {aligned} p p rightarrow q shuning uchun { overline {q quad quad quad}} end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95f187a2ebcc213400c7477e77e5638a802223fd) | ![{ displaystyle (p wedge (p rightarrow q)) rightarrow q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72db2e53d3ba51a816fa2ec217745350914af931) | Modus ponenslari |
![{ begin {aligned} neg q p rightarrow q shuning uchun { overline { neg p quad quad quad}} end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b586b432145f2ff65ce832693f2ff7d0ebc0bdc9) | ![{ displaystyle ( neg q wedge (p rightarrow q)) rightarrow neg p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07dd41490dad812dd9e2a11eb67df696ae9f8680) | Modulli tollens |
![{ begin {aligned} (p vee q) vee r shuning uchun { overline {p vee (q vee r)}} end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0141e10a72504ec1a042c8a1e2f2bc47bb24b87f) | ![((p vee q) vee r) rightarrow (p vee (q vee r))](https://wikimedia.org/api/rest_v1/media/math/render/svg/482d8a774fbeaf3dc28d5ec4094d9b58a5b66fbe) | Assotsiativ |
![{ begin {aligned} p wedge q shuning uchun { overline {q wedge p}} end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fca8ed6d1af319ec767d07e4152d7152bf4d442) | ![(p wedge q) rightarrow (q wedge p)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b6f83ff4ae5c64a3ff166a4f8c07c59c5567e5) | Kommutativ |
![{ start {aligned} p rightarrow q q rightarrow p shuning uchun { overline {p leftrightarrow q}} end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b2a12805bdbd7f44b24b6da6232778df025988a) | ![((p o'ng tirnoq q) xanjar (q o'ng tomondagi p)) o'ng chiziq ( p chap tomondagi q q)](https://wikimedia.org/api/rest_v1/media/math/render/svg/fba296f7bb432a1c7772c2c8491bccb71851dec5) | Ikki shartli takliflar qonuni |
![{ start {aligned} (p wedge q) rightarrow r shuning uchun { overline {p rightarrow (q rightarrow r)}} end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9453a89c178c20a958fe368c934d696e709cda5) | ![((p wedge q) rightarrow r) rightarrow (p rightarrow (q rightarrow r))](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc62e1cf55fcee505f4e92c31af12466a140dc18) | Eksport |
![{ start {aligned} p rightarrow q shuning uchun { overline { neg q rightarrow neg p}} end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8b9adc014b827cb1c58b5a4b9589ebd31dabb59) | ![(p rightarrow q) rightarrow ( neg q rightarrow neg p)](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b37d9f299a4298239f4eaf69aec38361b24e68e) | Transpozitsiya yoki kontrapozitsiya qonuni |
![{ start {aligned} p rightarrow q q rightarrow r shuning uchun { overline {p rightarrow r}} end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36a42de968b4729ba71118f33d2188f010edfd02) | ![((p rightarrow q) wedge (q rightarrow r)) rightarrow (p rightarrow r)](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb986d04778f181ae5d3f62b6463609a644ad003) | Gipotetik sillogizm |
![{ begin {aligned} p rightarrow q shuning uchun { overline { neg p vee q}} end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77cf73ab67588e93b2f85f203ce473ddb2d6e451) | ![(p rightarrow q) rightarrow ( neg p vee q)](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f60c2e786486b431a16c653e6ba228a71dd326f) | Moddiy ma'no |
![{ start {aligned} (p vee q) wedge r shuning uchun { overline {(p wedge r) vee (q wedge r)}} end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/181885d3de61a0bf31512de43e37a9cdc86786ba) | ![((p vee q) wedge r) rightarrow ((p wedge r) vee (q wedge r))](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2cff986517b87dd8614570474b5fc954692422c) | Tarqatish |
![{ begin {aligned} p rightarrow q shuning uchun { overline {p rightarrow (p wedge q)}} end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78bf2ccda66a618bc9c298354e4e8eaf7726d947) | ![(p rightarrow q) rightarrow (p rightarrow (p wedge q))](https://wikimedia.org/api/rest_v1/media/math/render/svg/ceb7842ea2c4ec46463d37bd0c05f788f551ccd4) | Absorbsiya |
![{ begin {aligned} p vee q neg p shuning uchun { overline {q quad quad quad}} end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a1abc4d3fc1ebe3150c05c6ecf3cfd6d587fc3c) | ![((p vee q) wedge neg p) rightarrow q](https://wikimedia.org/api/rest_v1/media/math/render/svg/b42226c64a9e6285bea4b09052a8286582e21a1d) | Disjunktiv sillogizm |
![{ begin {aligned} p shuning uchun { overline {p vee q}} end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1353f8d173a4b7467f9fb8a447ab96afd527eef) | ![p rightarrow (p vee q)](https://wikimedia.org/api/rest_v1/media/math/render/svg/68e4f32d8db48f50132651abbd6a3a76e94404c7) | Qo'shish |
![{ start {aligned} p wedge q shuning uchun { overline {p quad quad quad}} end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85ca65c99279a85351ec097b058a7b8784b0d068) | ![(p wedge q) rightarrow p](https://wikimedia.org/api/rest_v1/media/math/render/svg/f35dc5793741bc68c747903802f42a1471c866c3) | Soddalashtirish |
![{ begin {aligned} p q shuning uchun { overline {p wedge q}} end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f491b93b4782661f0a96bb9f7a9476dd4dbe041) | ![((p) xanjar (q)) o'ng chiziq (p xanjar q)](https://wikimedia.org/api/rest_v1/media/math/render/svg/6462d2bb765362c23a0b7dc5feac8d94d5488c51) | Birlashma |
![{ begin {aligned} p shuning uchun { overline { neg neg p}} end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cd73c79ef971f1378fa59ee971a3d9bfc1a097f) | ![p rightarrow ( neg neg p)](https://wikimedia.org/api/rest_v1/media/math/render/svg/51b7354013e157dc44005592cbfacfcdaf3c0f6c) | Ikkala inkor |
![{ start {aligned} p vee p shuning uchun { overline {p quad quad quad}} end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bea4b40aff659f082f8738957d8bc7e7321b81d) | ![(p vee p) rightarrow p](https://wikimedia.org/api/rest_v1/media/math/render/svg/73dc29bc5fef2586a34c360803dd4d3fbf9f5356) | Disjunktiv soddalashtirish |
![{ begin {aligned} p vee q neg p vee r shuning uchun { overline {q vee r}} end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce0c36f39726afc825d9407a61eb0bd9ff16662) | ![((p vee q) wedge ( neg p vee r)) rightarrow (q vee r)](https://wikimedia.org/api/rest_v1/media/math/render/svg/db81f5a7ec7c2cbd79b95fc999fdced4a70010da) | Qaror |
![{ displaystyle { begin {aligned} p rightarrow q r rightarrow q p vee r shuning uchun { overline {q quad quad quad}} end {aligned}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f64b0dd548c42117d9fefc15d97fdbc43efcf2bf) | ![{ displaystyle ((p rightarrow q) wedge (r rightarrow q) wedge (p vee r)) rightarrow q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b7904a7d82cd4b7aae168d79708d18e2a9faf3d) | Ajratishni bartaraf etish |
Barcha qoidalar asosiy mantiqiy operatorlardan foydalanadi. "Mantiqiy operatorlar" ning to'liq jadvali a tomonidan ko'rsatilgan haqiqat jadvali, 2 ning barcha mumkin bo'lgan (16) haqiqat funktsiyalarining ta'riflarini berish mantiqiy o'zgaruvchilar (p, q):
p | q | | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|
T | T | | F | F | F | F | F | F | F | F | | T | T | T | T | T | T | T | T |
---|
T | F | | F | F | F | F | T | T | T | T | | F | F | F | F | T | T | T | T |
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F | T | | F | F | T | T | F | F | T | T | | F | F | T | T | F | F | T | T |
---|
F | F | | F | T | F | T | F | T | F | T | | F | T | F | T | F | T | F | T |
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bu erda T = true va F = false, va ustunlar mantiqiy operatorlar: 0, yolg'on, Qarama-qarshilik; 1, NOR, Mantiqiy NOR (Peirce o'qi); 2, Noqulayliklarni o'zgartiring; 3, ¬p, Salbiy; 4, Moddiy soddalashtirmaslik; 5, ¬q, Salbiy; 6, XOR, Eksklyuziv disjunktsiya; 7, NAND, Mantiqiy NAND (Sheffer zarbasi); 8, Va Mantiqiy birikma; 9, XNOR, Agar shunday bo'lsa, Mantiqiy ikki shartli; 10, q, Proyeksiya funktsiyasi; 11, agar / keyin, Mantiqiy xulosa; 12, p, Proektsiya funktsiyasi; 13, keyin / agar, Buning teskari ma'nosi; 14, Yoki, Mantiqiy disjunktsiya; 15, rost, Tavtologiya.
Har bir mantiqiy operator o'zgaruvchilar va amallar to'g'risida tasdiqlashda, xulosaning asosiy qoidasini ko'rsatishda ishlatilishi mumkin. Misollar:
- Ustun-14 operatori (OR), ko'rsatadi Qo'shish qoidasi: qachon p= T (gipoteza jadvalning dastlabki ikkita satrini tanlaydi), biz buni (14-ustunda) ko'rayapmiz p∨q= T.
- Yana shuni ko'rishimiz mumkinki, xuddi shu asos bilan yana bir xulosalar haqiqiydir: 12, 14 va 15 ustunlar T.
- 8-ustunli operator (AND), ko'rsatadi Soddalashtirish qoidasi: qachon p∧q= T (jadvalning birinchi satri), biz buni ko'ramiz p= T.
- Ushbu shart bilan biz ham shunday xulosaga keldik q= T, p∨q= T va boshqalar 9-15 ustunlar ko'rsatilgandek.
- 11-ustunli operator (IF / THEN), ko'rsatadi Modus ponens qoidalari: qachon p→q= T va p= T haqiqat jadvalining faqat bitta satri (birinchi) bu ikki shartni qondiradi. Ushbu yo'nalishda, q bu ham to'g'ri. Shuning uchun har doim p → q rost va p rost bo'lganda, q ham rost bo'lishi kerak.
Mashinalar va yaxshi o'qitilgan odamlar bundan foydalanadilar stolga yaqinlashishga qarang asosiy xulosalar qilish va boshqa xulosalarni (xuddi shu binolar uchun) olish mumkinligini tekshirish.
1-misol
Quyidagi taxminlarni ko'rib chiqing: "Agar bugun yomg'ir yog'sa, biz bugun kanoeda chiqmaymiz. Agar bugun kanoeda sayohat qilmasak, unda ertaga kanoeda sayohat qilamiz. Shuning uchun (" shuning uchun "matematik belgisi bu
), agar bugun yomg'ir yog'sa, biz ertaga kanoeda sayohat qilamiz ".Yuqoridagi jadvalda xulosa qilish qoidalaridan foydalanish uchun biz ruxsat beramiz
"Bugun yomg'ir yog'sa" taklifi bo'ling,
bo'l "Biz bugun kanoeda chiqmaymiz" va ruxsat bering
be "Biz ertaga kanoeda sayohat qilamiz". Keyin ushbu argument quyidagi shaklga ega:
![{ start {aligned} p rightarrow q q rightarrow r shuning uchun { overline {p rightarrow r}} end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36a42de968b4729ba71118f33d2188f010edfd02)
2-misol
Keyinchalik murakkab taxminlarni ko'rib chiqing: "Bugun quyoshli emas va kechagiga qaraganda sovuqroq". "Biz faqat quyoshli bo'lsa suzishga boramiz", "Agar biz suzmasak, u holda bizda barbekyu bo'ladi", "Agar bizda barbekyu bo'lsa, u holda quyosh botguncha uyda bo'lamiz" xulosaga olib boring Biz quyosh botguncha uyga boramiz. "Xulosa qilish qoidalari bilan isbot: Keling
"Bugun quyoshli" taklifi bo'ling,
"Kechagidan sovuqroq" taklifi,
"Biz suzishga boramiz" taklifi,
taklif "Bizda barbekyu bo'ladi" va
taklif "Biz quyosh botguncha uyda bo'lamiz". Keyin gipotezalar paydo bo'ladi
va
. Sezgimizdan foydalanib, xulosa bo'lishi mumkin deb taxmin qilamiz
. Xulosa qilish qoidalari jadvalidan foydalanib, taxminlarni osongina isbotlashimiz mumkin:
Qadam | Sabab |
---|
1.![neg p wedge q](https://wikimedia.org/api/rest_v1/media/math/render/svg/626a03793fa20360c8a0465d20f0dd0ada57826e) | Gipoteza |
2. ![neg p](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2b198c79234d926cbee42c0f271d903ea55dc21) | 1-qadam yordamida soddalashtirish |
3. ![r rightarrow p](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e7e07c74d977b8a462efe28b9f6407e8e767ca3) | Gipoteza |
4. ![neg r](https://wikimedia.org/api/rest_v1/media/math/render/svg/51911343f3783285ef263346bc01d599b76766ad) | 2 va 3-bosqichlardan foydalangan holda modalar tollensi |
5. ![neg r rightarrow s](https://wikimedia.org/api/rest_v1/media/math/render/svg/eba0db689f6c4e4d1b889fa6038b35ceff0edfba) | Gipoteza |
6. ![s](https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632) | 4 va 5-qadamlardan foydalangan holda ponenslar |
7. ![s rightarrow t](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef033ab0c51ddd01279dd3e385f9da87aaa67892) | Gipoteza |
8. ![t](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560) | 6 va 7-qadamlardan foydalangan holda ponenslar |
Adabiyotlar
- ^ Kennet H. Rozen: Diskret matematika va uning qo'llanilishi, Beshinchi nashr, p. 58.
Shuningdek qarang
Falsafa portali
Mantiqiy tizimlar ro'yxati
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