
nirddeshaparamaaya calculus

nirddesha calculus yukthiyude oru shaakhayaanu. ithine propositional lojik, stettment lojik, senttanshyal calculus, senttanshyal lojik allengil chilappol seero-oordar lojik ennum vilikkunnu. nirddeshangalum (avaa shario thetto aakam) avaye adisthaanamaakkiyulla vaadangalude nirmmaanam ulppede nirddeshangal thammilulla bandhangalum ithu kaikaaryam cheiyunnu. logical connecteevukal upayogichu nirddeshangale bandhippichaanu samyuktha nirddeshangal roopappedunnathu. yukthisahamaya connecteevukal illaatha nirddeshangale aattomik nirddeshangal ennu vilikkunnu.
phastt-oordar lojikkil ninnu vyathyasthamayi, propositional lojik yukthirahithamaya vasthukkalo avayekkurichulla pravachanangalo allengil quandifierukalo kaikaaryam cheiyunnilla. ennirunnaalum, propositional lojikkinte ellah yanthrangalum phastt-oordar lojikkilum uyarnna oordar lojikkilum ulppeduthiyittundu. eee ardhathil, propositional lojik phastt-oordar lojikkinteyum uyarnna oordar lojikkinteyum aditharayaanu.
vishadeekaranam.
logical connecteevukal swaabhaavika bhashakalil kaanapedunnu. udaaharanathinu, engleeshil, chila udaaharanagal "aandu" (kanjakshan), allengil "(disjunction)", allah (nishedham), "iff" (ennaal metteeriyal contition soochippikkaan upayogikkumbol maathram) ennivayaanu.
propositional lojikkinte paridhikkullil valare lalithamaaya oru nigamanatthinte udaaharanamaanu inipparayunnathu
prameyavum nigamanavum nirddeshangalaanu. parisaram nissaaramaayi kanakkaakkappedunnu, modus ponense (oru anumana niyamam) prayogikkunnathiloode nigamanam pinthudarunnu.
nirddeshangalude ghadanaye sambandhichidatholam nirddeshangalude ghadanaye sambandhichidatholam, avaye logical connecteevukal vazhi vighadippikkaan kazhiyaatha poyintinappuratthu, eee anumaanam aaa "aattomik" prasthaavanakale stettment aksharangal upayogichu maattisthaapikkaam, avaa prasthaavanakale prathinidheekarikkunna veriyabilukalaayi vyaakhyaanikkappedunnu
ithu thaazhepparayunna reethiyil churukkatthil prasthaavikkam
mazha peiyunnu" ennum "meghaavrithamaanu" ennum vyaakhyaanikkumbol mukalil paranja pratheekathmaka padaprayogangal swaabhaavika bhaashayile yadhaartha padaprayogavumaayi krithyamaayi poruthappedunnuvennu kaanaan kazhiyum. athu mathramalla, eee "roopathinte" mattethenkilum anumaanavumaayi avaa poruthappedum, athu eee nigamanatthinte athe adisthaanathil saadhuthayullathaayirikkum.
nirddeshangale prathinidheekarikkunnathin oru aupachaarika bhaashayude soothravaakyangal vyaakhyaanikkaavunna oru aupachaarika samvidaanathiloode nirddeshaparamaaya yukthi padikkaam. svayamsidhathakaludeyum anumana niyamangaludeyum oru samvidaanam chila soothravaakyangal uruthirinjedukkaan anuvadikkunnu. eee soothravaakyangale sidhaanthangal ennu vilikkunnu, avaa yadhaartha nirddeshangalaayi vyaakhyaanikkaam. atharam soothravaakyangalude nirmmitha kramatthe "derivation" allengil "proof" ennum shreniyude avasana soothravaakyam sidhaanthavumaanennum ariyappedunnu. sidhaantham prathinidheekarikkunna nirddeshathinte thelivaayi derivation vyaakhyaanikkaam.
aupachaarika yukthiye prathinideekarikkaan oru aupachaarika samvidaanam upayogikkumbol, prasthavana aksharangal (saadhaaranayaayi formula _ 2, formula _ 3, formula _ 7 polulla valia roman aksharangal) maathrame nerittu prathinidheekarikkukayulloo. avaa vyaakhyaanikkumbol uyarnnuvarunna swaabhaavika bhasha nirddeshangal samvidaanathinte paridikk purathaanu, aupachaarika samvidaanavum athinte vyaakhyaanavum thammilulla bandhavum aupachaarika samvidaanathinu purathaanu.
classical sathyam-functional propositional lojikkil, soothravaakyangalkku saadhyamaaya rund sathya moolyangalil onnu, "sharee" yude sathya moolyam allengil "thettu" yude sathya moolyam ennivayundennu vyaakhyaanikkappedunnu. dwaithathwathinte thathwavum ozhivakkappetta madhyathinte niyamavum uyarthippidikkappedunna. sathya-pravartthana nirddesha logic athinulla sisttangal isomorphic aayum nirvachichirikkunnathu siroth-oordar lojikkaayi kanakkaakkappedunnu. ennirunnaalum, badal nirddeshaparamaaya yukthikalum saadhyamaan. kooduthal vivarangalkk thaazhe koduthirikkunna mattu yukthisahamaya kanakkukoottalukal kaanuka.
charithram.
muun thathwachinthakar nirddeshaparamaaya yukthiye (nirddeshaparamaaya calculusumaayi parasparam maattaavunna) soochippichirunnuvenkum, bisi moonnaam noottaandil chrysippus ithu oru aupachaarika yukthiyayi (stoik lojik) vikasippikkukayum addehathinte pingamiyaaya stoikkukal ithu vipuleekarikkukayum cheythu. nirddeshangalil shradha kendreekarichaayirunnu yukthi. eee purogathi nibandhanakalil shradha kendreekarichirunna parambaraagatha sylogistic lojikkil ninnu vyathyasthamaayirunnu. ennirunnaalum, mikka yadhaartha rachanakalum nashtappedukayum stoikkukal vikasippichedutha nirddeshaparamaaya yukthi pinneedu purathana kaalath manassilakkappedaathirikkum cheythu. thathfalamaayi, panthrandaam noottaandil peettar abellard eee samvidaanam punarnirmmichu.
pratheekathmakamaaya yukthi upayogichu nirddeshaparamaaya yukthi oduvil parishkarikkapettu. 17/18-aamm noottaandile ganithashaasthranjanaaya gottfried lebnis calculus ratiosinettarumaayulla addehathinte pravarthanatthinu pratheekathmaka yukthiyude sthaapakanaayi kanakkaakkappedunnu. addehathinte rachanakal itharathilulla aadyathethaanengilum, valia yukthisahamaya samoohathinu athu anjjaathamaayirunnu. thalphalamaayi, leibnis kaivaricha pala munnettangalum leebnisil ninnu poornnamaayum swathanthraraaya jorjju boole, augustus dii morgan thudangiya yukthivaadikalaanu punarnirmmichathu.
muun sylogistic lojikkil ninnulla oru purogathiyaayi propositional lojikkine kanakkaakkunnathupole, gottlobe frejinte predicate logic munkaala propositional lojikkil ninnulla oru purogathiyaayi kanakkakkaam. sylogistic lojikkinteyum propositional lojikkinteyum savisheshathakal samyojippikkunnathaayi oru rachayithaav pravachana yukthiye vivarikkunnu."thalphalamaayi, pravachana yukthi yukthiyude charithratthil oru puthiya yugathinu thudakkamittu; ennirunnaalum, swaabhaavika nigamanam, sathyavrikshangal, sathyapattikakal ennivayulppede, nirddesha yukthiyile purogathi ippozhum freejinu sesham undaayi. gerhard jentsanum staanislaa jaskovsky chernnaanu swaabhaavika deduction kandupidichathu. evert villem bethaanu sathyavrikshangal kandupidichathu. ennirunnaalum, sathyappattikakalude kandupidutham anishchithathvamullathaanu.
frejinteyum bertrand rassalinteyum kruthikalil, sathya pattikakalude kandupiduthathe swaadheenikkunna aashayangalund. yadhaartha pattika ghadana (oru pattikayayi formet cheythirikkunnu), saadhaaranayaayi ludwig vitjanstain allengil emil poost (allengil randum, swathanthramaayi) aanu. frejinum russellinum purame, philoo, boole, chaals saandezhs peezhs, ernst schroeder enniva sathya pattikakalkk munbulla ashayangal ullavaraan. jaann lucassivics, alfred north whitehead, vilyam staanli jevons, joonn wenn, clarence irving levis ennivaraanu pattikayude gadanayude mattu bahumathikal. aathyanthikamayi, joonn shoskiyeppole chilar nigamanatthiletthi, "sathyappattikakalude 'kandupiduthakkaaran' enna padavi eathoru vyakthikkum nalkanamennu vyakthamalla.".
padavali.
pothuvaayi paranjaal, eee padaprayogangalude (svayamsidhavaakyangal) oru prathyeka upavibhagamaaya vaakyaghadana padaprayogangalude ("nannayi roopappeduthiya soothravaakyangal") oru koottam, koodaathe padaprayogangalude idathil yukthisahamaya thulyathayaayi vyaakhyaanikkaan uddeshichulla oru prathyeka bainari bandhathe nirvachikkunna aupachaarika niyamangalude oru koottam enniva ulkkollunna oru aupachaarika samvidaanamaanu calculus.
aupachaarika samvidaanam oru yukthisahamaya samvidaanamaayi uddeshikkumbol, padaprayogangal prasthavanakalaayi vyaakhyaanikkaan uddeshichullathaanu, koodaathe "anumana niyamangal" ennariyappedunna niyamangal saadhaaranayaayi sathyam samrakshikkaan uddeshichullathaanu. eee krameekaranatthil, svayamsidhavaakyangal ulppettekkaavunna niyamangal, yadhaartha prasthaavanakale prathinidheekarikkunna ("anumaanikkuka") soothravaakyangal uruthirinjedukkaan upayogikkaam-yadhaartha prasthaavanakale prathinidheekarikkunna thannirikkunna soothravaakyangalil ninnu.
svayamsidhathakalude koottam shoonyamo shoonyamallaatha parimithamaaya koottamo allengil ennaavunna ananthamaaya koottamo aakam (svayamsidhamaaya schema kaanuka). oru aupachaarika vyakaranam bhaashayude aavishkaarangaleyum nannayi roopappeduthiya soothravaakyangaleyum aavarthichu nirvachikkunnu. koodaathe, sathyatheyum moolyanirnnayangaleyum (allengil vyaakhyaanangal) nirvachikkunna oru ardhashaasthram nalkaam.
oru nirddesha kaalkkulasinte bhashayil ulppedunnu
vyaakarana niyamamanusarichu oopparettar chinnangal vazhi aattomik soothravaakyangalil ninnu nirmmikkaan kazhiyunna eathenkilum aattomik formulayo allengil eathenkilum formulayo aanu "nannayi roopappetta formula".
ganithashaasthranjar chilappol propositional sthiraangangal, propositional variableukal, skeematta enniva thammil verthirichariyunnu. nirddesha sthiraangangal chila prathyeka nirddeshangale prathinidheekarikkunnu, athesamayam nirddesha variableukal ellah aattomik nirddeshangaludeyum koottatthil vyathyaasappettirikkunnu. ennirunnaalum, ellah nirddeshangalilum skimatta vyathyaasappettirikkunnu. propositional sthiraangangale,,,,,,,,,,,,,,, ennivayaal prathinidheekarikkunnath saadhaaranamaanu, koodaathe schematic aksharangal palappozhum greak aksharangalaanu, mikkappozhum,,,,.
adisthaana ashayangal.
thaazhepparayunnava oru staanderd propositional calculus rooparekhappeduthunnu. pala vyathyastha formulationukalum nilavilundu, avayellaam erekkure thulyamaanu, pakshe avayude vishadaamshangalil vyathyaasamund
ganithashaasthrathile oru aksharam (udha.jii.). ellah nirddeshangalkkum rund sathya moolyangalil onnu krithyamaayi aavasyamaanu shario thetto. udaaharanathinu, purathu mazha peiyunnu enna nirddesham ayirikkatte. purathu mazha peiyukayaanengil ithu sariyaanu (), allaathapaksham thettayirikkum ().
eee vyathyastha oopparettarmaarkkulla sathya pattikakalum vishakalana tabloxinte reethiyum nokkunnathu valare sahaayakaramaan.
pravartthanatthin keezhil adachupoottal.
sathya-pravartthana connectevukalkku keezhil propositional lojik adanjirikkunnu. athaayath, eathoru nirddeshathinum athu oru nirddeshamaanu. athupole, eathenkilum nirddeshangalkku, athupole thanne, vyavasthaapithavum, rund vyavasthakalullathumaaya vichedanam oru nirddeshamaanu. udaaharanathinu, ithu oru nirddeshamaanu, athinaal ithu mattoru nirddeshavumaayi samyojippikkaam. ithu prathinidheekarikkunnathin, eathu nirddeshamaanu eathaanu samyojippichirikkunne soochippikkaan nammal paraantheesukal upayogikkendathundu. udaaharanathinu, ithu nannayi roopappetta oru soothravaakyamalla, kaaranam nammal samyojippikkukayaano atho samyojippikkukayaano ennu namukkariyilla. athinaal nammal onnukil aadyathethine prathinidheekarikkaano randaamathethine prathinidheekarikkaano ezhuthanam. sathyaavasthakal vilayiruthunnathiloode, rund padaprayogangalkkum ore sathyaavasthakalundennu (ore kesukalil sathyamayirikkum), koodaathe ekapaksheeyamaaya samyojanangal vazhi roopappedunna eathoru nirddeshathinum, paraantheesukalude sthaanam pariganikkathe, ore sathyaavasthakal undaakumennum naam kaanunnu. ithinartham samyojanam associative aanennaan, ennirunnaalum, paraantheesukal orikkalum oru uddeshyavum niravettunnillennu oraal anumaanikkaruth. udaaharanathinu, vaakyathin ore sathyaavasthakalilla, athinaal avaa vyathyastha vaakyangalaanu, avaa paraantheesukalaal maathram verthirikkappedunnu. mukalil paraamarshichirikkunna sathya-pattika reethi upayogichu oraalkk ithu parisodhikkaan kazhiyum.
shradhikkuka eathu ekapaksheeyamaaya ennam propositional sthiraangangalkkum, avayude saadhyamaaya sathya-moolyangal pattikappedutthunna parimithamaaya ennam kesukal namukk roopappeduthaan kazhiyum. ithu srishtikkunnathinulla oru lalithamaaya maarggam sathyam-pattikakal aanu, athil oraal ezhuthunnu,,...,, nirddesha sthiraangangalude eathenkilum pattikaykku-athaayath, entrikalulla nirddesha sthiraangangalude eathenkilum pattika. eee pattikaykku thaazhe oraal varikal ezhuthunnu, onninu thaazhe varikalude aadya pakuthiyil sharee (allengil tii) ennum randaam pakuthi thettayi (allengil eph) ennum niraykkunnu. onninu thaazhe naalilonn varikalil tii, thudarnnu naalilonn eph, thudarnnu naalilonn tii, avasana paadatthil eph enniva niraykkunnu. oro varikaludeyum ettaam niraykkum sarikkum thetkkum idayil adutha nira marimari varunnu, thudarnnu pathinaarukal, angane, oro variyilum avasaanathe nirddesha sthiraangam t, f ennivaykkidayil vyathyaasappedunnu. ithu aaa nirddesha sthiraangangalkk saadhyamaaya kesukaludeyo sathya-moolya asainmentukaludeyo poornnamaaya pattika nalkum.
vaadangal.
propositional calculus oru "aargyument" nirddeshangalude pattikayayi nirvachikkunnu. saadhuvaaya oru vaadam nirddeshangalude oru pattikayaanu, athil avasaanatheth bakkiyullavayil ninnu pinthudarunnu-allengil soochippikkunnu. matellaa vaadangalum asaadhuvaayavayaanu. ettavum lalithamaaya saadhuvaaya vaadam modus ponense aanu, athinte oru udaaharanam inipparayunna nirddeshangalude pattikayaann
ithu moonnu nirddeshangalude pattikayaanu, oro variyum oru nirddeshamaanu, avasaanatheth bakkiyullavayil ninnu pinthudarunnu. aadyathe rund varikale parisaram ennum avasaanathe variye upasamharam ennum vilikkunnu. settu formula _ 9 leeii oro amgavum shariyaanenkil, eathu nirddeshavum eathenkilum settu nirddesha formula _ 9 ll ninnu pinthudarumennu njangal parayunnu. mukalilulla vaadathil, eathu samayathum sathyamaayirikkumbozhum, theerchayaayum sathyamaan. sathyam varumbol, kesukal 3,4 enniva (sathyam pattikayil ninnu) pariganikkaan kazhiyillennu shradhikkuka. eppozhaanu sathyam, namukk cases 2 pariganikkaan kazhiyilla. ithu cases 1 maathram avasheshikkunnu, athum sariyaanu. angane athu parisaram soochippikkunnu.
ithu kramatthil saamaanyavalkkarikkappedunu. athinaal, evideyengilum, evideyengilum enthengilum nirddeshangal undakam,
mattu vaada roopangal soukaryapradamaanu, pakshe aavasyamilla. oru samboornna svayamsidhaantham nalkiyaal (atharamoru koottatthinu chuvade kaanuka), mattu ellah vaada roopangalum nirddeshaparamaaya yukthiyil theliyikkaan modus ponense mathiyakum, athinaal avaa oru derivateevaayi kanakkakkaam. shradhikkuka, phastt-oordar lojik polulla mattu lojikkukalilekku propositional lojikkinte vipuleekaranathinte kaaryatthil ithu shariyalla. poornatha nedunnathinu phastt-oordar lojikkinu kuranjathu oru adhika anumana niyamamenkilum aavasyamaanu.
sthaapithamaaya sathyangalil ninnu oraalkk puthiya sathyangal nedaanaakum ennathaanu aupachaarika yukthiyile vaadathinte praadhaanyam. mukalilulla aadya udaaharanathil, rund parisarangalum kanakkiledukkumbol, athinte sathyam ithuvare ariyappedukayo prasthaavikkappedukayo cheythitilla. vaadam paranjathinushesham, athu anumaanikkappedunnu. eee reethiyil, mattoru koottam nirddeshangalil ninnu kanakkakkavunna ellah nirddeshangaludeyum oru koottamaayi njangal oru deduction systethe nirvachikkunnu. udaaharanathinu, nirddeshangalude formula _ 12 nalkiyaal, namukk oru deduction sisttam nirvachikkaan kazhiyum, ithu pinthudarunna ellah nirddeshangaludeyum koottamaanu. aavarthanam ellaayppozhum anumaanikkappedunnu, athinaal formula 13. avasana gadakathinteyum modus ponensinteyum aadya gadakathil ninnu, oru anantharafalamaanu, athinaal formula 14. kaaranam nammal vendathra poornnamaaya soothravaakyangal ulppeduthiyitla, ennirunnaalum, mattonnum anumanikkaan kazhiyilla. athinaal, propositional lojikkil padicha mikka deduction sistangalkkum formula _ 15 nirnnayikkaan kazhiyumenkilum, atharamoru nirddesham theliyikkaan kazhiyaathathra durbalamaanu ithu.
oru propositional kaalkkulasinte pothuvaaya vivaranam.
oru propositional calculus oru aupachaarika sisttam formula _ 16 aanu, avide
soothravaakyangalude" koottam ennum ariyappedunna soothravaakyam _ 17-nte "bhasha", "nannayi roopappetta soothravaakyangal", inipparayunna niyamangalal nirvachikkappedunnu
eee niyamangalude aavarthichulla prayogangal sankeernamaaya soothravaakyangalude nirmmaanatthe anuvadikkunnu. udaaharanathina
udaaharanam 1. lalithamaaya svayamsidhamaaya samvidaanam.
formula _ 27, ivide formula _ 18, formula _ 29, formula _ 30, formula _ 31 enniva inipparayunna reethiyil nirvachikkappedunnu
appol formula _ 41 formula _ 42 ennum formula _ 43 formula _ 44 ennum nirvachikkappedunnu.
eee samvidaanam mettaamathu settilaanu upayogikkunnathu.emm aupachaarika proof daattaabes.
udaaharanam 2. swaabhaavika kizhivu samvidaanam.
formula _ 53, ivide formula _ 18, formula _ 29, formula _ 30, formula _ 31 enniva inipparayunna reethiyil nirvachikkunnu
oru nirddesha kaalkkulasinte inipparayunna udaaharanathil, parivartthana niyamangal "swaabhaavika kizhivu samvidaanam" ennu vilikkappedunnathinte anumana niyamangalayi vyaakhyaanikkaan uddeshichullathaanu. ivide avatharippichirikkunna prathyeka samvidaanathinu prarambha pooyintukalonnumilla, athaayath logical aaplikkeshanukalkkaayulla athinte vyaakhyaanam athinte sidhaanthangal oru shoonyamaaya svayamsidha ganathil ninnaan uruthirinjathu.
nammude nirddesha calculusinu pathinonnu anumana niyamangalund. sariyaanennu karuthappedunna oru koottam soothravaakyangal nalki mattu yadhaartha soothravaakyangal uruthirinjedukkaan eee niyamangal namme anuvadikkunnu. mattu nannayi roopappetta soothravaakyangalil ninnu namukk nannayi roopappetta chila soothravaakyangal anumanikkaan kazhiyumennu aadyathe patthu lalithamaayi prasthaavikkunnu. ennirunnaalum, avasana niyamam saangalppika nyaayavaadam upayogikkunnu, niyamatthinte adisthaanathil mattoru soothravaakyam anumanikkaan kazhiyumo ennu nokkaan anumanikkappetta soothravaakyangalude oru koottatthinte bhaagamaanenn njangal thaalkkaalikamaayi anumaanikkunnu. aadyathe patthu niyamangal ithu cheyyaathathinaal avaye saadhaaranayaayi "oohaathmakamallaatha" niyamangalayum avasaanatheth "saangalppika" niyamamaayum visheshippikkunnu.
parivartthana niyamangal vivarikkumbol, namukk oru lohabhasha chihna formula _ 65 avatharippikkam. ithu adisthaanaparamaayi "athu anumaanikkuka" ennu parayunnathinulla soukaryapradamaaya oru hraswalekhanamaanu. formet formula _ 66 aanu, athil prakkras ennu vilikkunna oru koottam (orupakshe shoonyamaaya) soothravaakyangalundu, ithu nigamanam ennu vilikkunna oru soothravaakyamaanu. parivartthana niyama soothravaakyam _ 66 ennathinartham oro nirddeshavum oru sidhaanthamaanenkilo (allengil svayamsidhaanthangalude athe sathyamoolyam undengilo), athum oru sidhaanthamaanu. thaazhepparayunna niyama samyojana aamukham pariganikkumbol, onnil kooduthal soothravaakyangal ullappozhellaam, samyojanam upayogichu ellaayppozhum athine surakshithamaayi oru soothravaakyamaayi kuraykkaan kazhiyumennu namukk manassilakum. athinaal churukkatthil, aaa samayam muthal nammal oru settinu pakaram oru formulayayi prathinidheekarikkam. soukaryathinaayulla mattoru ozhivakkal eppozhaanu oru shoonyamaaya settu, eee saahacharyathil athu drishyamaakanamennilla.
propositional calculusile thelivukal.
oru nirddesha kaalkkulasinte pradhana upayogangalilonnu, logical aaplikkeshanukalkkaayi vyaakhyaanikkumbol, nirddesha soothravaakyangal thammilulla logical thulyathayude bandhangal nirnnayikkuka ennathaanu. eee bandhangal nirnnayikkunnath labhyamaya parivartthana niyamangaliloodeyaanu, avayude kramangale "derivations" allengil "proofs" ennu vilikkunnu.
thudarnnulla charchayil, oro variyilum orotta soothravaakyam adangiyirikkunna akkangalulla varikalude oru sreniyaayi oru thelivu avatharippikkunnu, thudarnnu aaa soothravaakyam avatharippikkunnathin oru "kaaranam" allengil "nyaayeekaranam" undu. vaadathinte oro aamukhavum, athaayath vaadathinte oru sidhaanthamaayi avatharippicha oru anumaanam, kramatthinte thudakkathil pattikappedutthukayum mattu nyaayeekaranangalkk pakaramaayi oru "aamukhamaayi" adayaalappeduthukayum cheiyunnu. avasana variyil nigamanam pattikappedutthiyirikku. oro variyum munpullavayil ninnu oru parivartthana niyamatthinte sariyaaya prayogatthiloode pinthudarukayaanenkil oru thelivu poortthiyaakum. (oru vipareeta sameepanathinaayi, proof-trease kaanuka).
swaabhaavika kizhivu sanbradaayathile oru thelivinte udaaharanam.
105 enna soothravaakyathe "anumaanikkuka, anumaanikkuka" ennu vyaakhyaanikkuka. onnum anumaanikkathe, athu soochippikkunnuvennu anumaanikkuka", allengil "ithu soochippikkunna oru swarashaasthramaanu", allengil "ellaayppozhum sariyaanu soochippikkunnathu" enningane soothravaakyam _ 106 vaayikkuka.
oru classical propositional calculus sisttathile oru thelivinte udaaharanam.
mukalil vivaricha jaann лukaseevicz swamedhayaa ulla sisttathil athe sidhaantha soothravaakyam _ 107 nammal ippol theliyikkunnu, ithu classical propositional kaalkkulasinayulla hilbert-stail deductive sisttathinte udaaharanamaanu.
eee sidhaanthangal ivayaanu
thelivu thaazhepparayunnavayaanu
niyamangalude dridathayum poornnathayum.
eee niyamangalude nirnaayaka savisheshathakal avaa "shabdavum" "poornnavumaanu" ennathaanu. anoupachaarikamaayi ithinartham niyamangal sariyaanennum mattu niyamangalonnum aavasyamillennum. eee avakaashavaadangal inipparayunna reethiyil kooduthal aupachaarikamaakkaam.
nirddeshaparamaaya yukthiyude dridathaykkum poornnathaykkum ulla thelivukal nirddeshaparamaaya yukthiyile thelivukalallenn shradhikkuka; iva nirddeshaparamaaya yukthiyude gunangal theliyikkaan mettathariyaayi upayogikkunna zfc-yile sidhaanthangalaanu.
propositional variableukal sharee allengil thettu ennu mapp cheiyunna oru functionaayi njangal oru "trooth assainment" nirvachikkunnu. anoupachaarikamaayi atharam oru sathyanirnayathe chila prasthavanakal sathyavum mattullava sathyamallathathumaaya oru saadhyamaaya avasthayude (allengil saadhyamaaya lokam) vivaranamaayi manassilakkaam. soothravaakyangalude ardhashaasthram eathu "avastha" yaanu shariyenn kanakkakkunnathennu nirvachichukondu aupachaarikamaakkaam, ithaanu inipparayunna nirvachanam cheiyunnathu.
atharam oru sathyanirnayam thaazhepparayunna niyamangal upayogichu nannayi roopappeduthiya oru soothravaakyam thripthippeduthumbol njangal nirvachikkunnu
eee nirvachanathiloode oru nischitha koottam soothravaakyangal soochippikkunna oru soothravaakyathinte ardhamenthaanennu namukk ippol aupachaarikamaakkaan kazhiyum. soothravaakyangalude koottam kanakkiledukkumbol saadhyamaaya ellah lokangalilum ithu undengil anoupachaarikamaayi ithu sariyaanu. ithu inipparayunna aupachaarika nirvachanathilekku nayikkunnu ellah soothravaakyangaleyum thripthippeduthunna ellah sathya assainmentukalum thripthippeduthukayaanel, nannayi roopappetta oru koottam soothravaakyangal "ardhaparamaayi ulkkollunnu" (allengil "soochippikkunnu") ennu njangal parayunnu.
parimithamaaya ghattangalil mukalil avatharippicha anumana niyamangal upayogichu athu nedaan kazhiyumenkil maathram vaakyaghadanayil ulppedunna "vaakyaghadana ulcherkkal" ennu avasaanamaayi nammal nirvachikkunnu. anumana niyamangalude koottam shariyum poornnavumaayirikkuka ennathinte ardham krithyamaayi roopappeduthaan ithu namme anuvadikkunnu
soundnesa nannayi roopappetta soothravaakyangalude koottam "vaakyaghadanayil" nannayi roopappetta soothravaakyam ulkkollunnuvenkil "ardhathil" ulppedunnu.
poornnatha nannayi roopappetta soothravaakyangalude koottam "ardhaparamaayi" nannayi roopappetta soothravaakyam ulkkollunnuvenkil "vaakyaghadanayil" ulppedunnu.
melpparanja niyamangalude kaaryatthil ithu yadhaarthathil sariyaanu.
soundness proofinte rekhaachithram.
notteshanal kanvenshanukal vaakyangalude koottatthil oru variable ayirikkatte. vaakyangalkk mukaliloode paridhi nischayikkuka. kaaranam "vaakyaghadanayil" naam ezhuthunnu "ennathu theliyikkunnu. ardhathil" "njangal ezhuthunnu" soochippikkunnu ".
njangal kaanikkan aagrahikkunnu (theliyikkukayaanenkil, soochippikkunnu).
theliyikkunnu" ennathinu oru inductive nirvachanamundennum athu theliyikkukayaanenkil "foominte avakaashavaadangal prakadippikkunnathinula udanadi vibhavangal nalkunnuvennum njangal shradhikkunnu...". athinaal nammude thelivu induction vazhiyaan munnott pogunnathu.
swaabhaavika kizhivu samvidaanangalkku adisthaana ghattam II ozhivaakkaamenn shradhikkuka, kaaranam avaykku svayamsidhathakalilla. upayogikkumbol, ghattam II ll oro soothrangalum oru (ardhaathmaka) yukthisahamaya sathyamaanenn kaanikkunnath ulppedunnu.
theliyikkaavunna ettavum lalithamaaya vaakyangalum eathenkilum vaakyangalkk soochippikkunnuvennu adisthaana gattangal theliyikkunnu. (thelivu lalithamaanu, kaaranam oru settu athinte eathenkilum amgangale soochippikkunnu enna ardha vasthuthayum nissaaramaan.) inductive steppu theliyikkaavunna ellah vaakyangalum vyavasthaapithamaayi ulkkollum-oru anumana niyamam upayogichu yukthisahamaya nigamanathiletthaavunna oro kesum pariganikkunnathiloode-oru puthiya vaachakam theliyikkaavunnathaanenkil, athu yukthisahamaayi soochippikkunnuvennum kaanikkunnu. (udaaharanathinu, "namukk uruthirinjedukkaam" allengil "ennathil ninnu" namukk uruthirinjedukkaam "ennu parayunna oru niyamam namukkundakaam. 3.a theliyikkaavunnathaanenkil athu soochippikkunnathaanenna njangal anumaanikkunnu. theliyikkaavunnathaanenkil "allengil" theliyikkaavunnathaanennum namukkariyam. appol "allengil" umm soochippikkunnundenn naam kaanikkanam. ardha nirvachanathodum nammal ippol nadatthiya anumaanathodum appeel cheythaanu nammal angane cheiyunnathu. athu theliyikkaavunnathaanenna njangal anumaanikkunnu. athinaal ithu soochippikkukayum cheiyunnu. athinaal ellah kaaryangalum sariyaakkunna eathoru ardha moolyanirnnayavum sariyaakkunnu. ennaal sharee akkunna eathoru moolyanirnnayavum "allengil" allengil "ennathinu nirvachikkapetta ardhashaasthram anusarichu" allengil "sariyaakkunnu. athinaal ellah kaaryangalum sariyaakkunna eathoru moolyanirnnayavum "allengil" sariyaakkunnu. athinaal "allengil" soochippikkunnu.) saadhaaranayaayi, inductive steppil ellah anumana niyamangaludeyum dairkhyameriyathum ennaal lalithavumaaya cases-bye-cases vishakalanam adangiyirikkum, ithu oro ardhaparamaaya soochanayum "samrakshikkunnu" ennu kaanikkunnu.
provabilityude nirvachanamanusarichu, oru svayamsidhathayil amgamaayirikkunnathiloodeyo oru niyamam pinthudarunnathiloodeyo allathe mattoru vaakyavum theliyikkanaavilla; athinaal avayellaam ardhaparamaayi soochippikkukayaanenkil, deduction calculus mikachathaanu.
poornnathayude thelivu.
mukalil paranja athe notteshanal kanvenshanukalaanu njangal sweekarikkunnathu.
nammal kaanikkan aagrahikkunnu soochippikkunnuvenkil, theliyikkuka. nammal kontraaposishaniloode munnott pookunnu pakaram theliyikkunnillenkil athu ardhamaakkunnillennu nammal kaanikkunnu. sathyam aanenkilum nilanilkkaatha oru maathruka undennu nammal kaanikkukayaanengil, athu theerchayaayum ardhamaakkunnilla. theliyikkaatha nammude anumaanathil ninnu atharamoru maathruka nirmmikkuka ennathaanu ashayam.
athinaal oru anumana niyamamaayi modus ponense ullathum inipparayunna sidhaanthangal (avayude pakarakkar ulppede) theliyikkunnathumaaya oro samvidaanavum poortthiyaayi
aadya anchu vyavasthakal mukalilulla moonnaam ghattatthile anchu vyavasthakalude samthrupthikkum avasana moonnu vyavasthakal deduction sidhaantham theliyikkunnathinum upayogikkunnu.
udaaharanam.
udaaharanathinu, mattethoru tattolajiyeyum pole, neratthe vivaricha classical propositional calculus sisttathinte moonnu sidhaanthangal melpparanjavaye thripthippeduthunna eathu samvidaanathilum theliyikkaan kazhiyumennu kaanikkam, athaayath oru anumana niyamamaayi modus ponense undu, koodaathe melpparanja ettu sidhaanthangal (avayude pakarakkar ulppede) theliyikkunnu. ettu sidhaanthangalil, avasaanathe rund sidhaanthangal moonnu sidhaanthangalil randaanu; moonnaamathe soothravaakyam, formula _ 124, nammal ippol kaanikkunnathupole theliyikkaan kazhiyum.
thelivinaayi namukk saangalppika silojism sidhaantham (eee svayamsidha samvidaanathinu prasakthamaaya roopathil) upayogikkaam, kaaranam ithu ithinakam mukalil paranja ettu sidhaanthangalude koottatthilulla rund svayamsidhathakale maathrame aashrayikkunnulloo.
appol thelivu iprakaaramaanu
classical propositional calculus sisttathinte poornnatha parishodhikkunnu.
neratthe vivaricha classical propositional calculus systethinu mukalil soochippicha aavashyamaaya ettu sidhaanthangal theliyikkaan kazhiyumennu nammal ippol parishodhikkunnu. ivide theliyikkapetta niravadhi lemmakal njangal upayogikkunnu
niravadhi thelivu ghattangalkkayi saangalppika silojism mettaathiyoreminte reethiyum nammal upayogikkunnu.
poornnathayude thelivinulla mattoru rooparekha.
oru formula oru totology aanenkil, oro moolyanirnnayavum formulakku sariyaaya moolyam nalkunnuvennu kaanikkunna oru sathya pattikayund. atharamoru moolyanirnnayam pariganikkuka. upasoothrangalude dairkhyathekkurichulla ganithashaasthraparamaaya induction vazhi, upasoothrathile oro nirddesha veriyabilinteyum sathyathil ninno asathyathil ninno (moolyanirnayathinu anuyojyamaaya) upasoothrathinte sathyamo asathyamo pinthudarunnuvennu kaanikkunnu. (shariyaano soochippikkunnathu) soochippikkunnu (thettaano soochippikkunnathu)" upayogichu sathyappattikayude varikal oru samayam rund aayi samyojippikkuka. propositional veriyabilukalekkurichulla ellah aashrithathvangalum illaathaakkunnathuvare ithu aavarthikkuka. thannirikkunna swaraghadana naam theliyichathaanu phalam. oro swaravum theliyikkavunnathinaal, yukthi poornnamaanu.
oru sathya-pravartthana nirddesha kaalkkulasinte vyaakhyaanam.
oru sathyam-pravarthanaparamaaya nirddesha calculus formula _ 182-nte vyaakhyaanam, sathyam (tii), thettu (eph) ennivayude moolyangal nalkunna onno allengil mattonninte (pakshe randum allah) formula _ 182-nte oro nirddesha chinnathinum, avayude saadhaarana sathyam-pravarthanaparamaaya ardhangalude soothravaakya _ 182-nte connective chinnangalkkulla oru assainmentum aanu. oru sathya-pravartthana nirddesha kaalkkulasinte vyaakhyaanavum sathyappattikakalude adisthaanathil prakadippikkaam.
formula _ 185 vyathyastha nirddesha chinnangalkk formula _ 186 vyathyastha saadhyamaaya vyaakhyaanangalundu. eathenkilum prathyeka chihna soothravaakyam _ 187 nuu, udaaharanathinu, formula _ 188 saadhyamaaya vyaakhyaanangalundu
jody soothravaakyam _ 187, soothravaakyam _ 192 nuu saadhyamaaya vyaakhyaanangal _ 193 undu
formula _ 182 nuu formula _ 199 ullathinaal, athaayath, niravadhi nirddesha chinnangal ullathinaal, formula _ 200 undu, athinaal formula _ 182-nuu ennamatta vyathyasthamaaya vyaakhyaanangalundu.
sathyam-pravarthanaparamaaya nirddesha yukthiyude oru vaakyathinte vyaakhyaanam.
formula _ 182-nteyum formula _ 203-nteyum soothravaakyangal formula _ 182-nte vyaakhyaanamaanenkil inipparayunna nirvachanangal baadhakamaanu
eee nirvachanangalude chila anantharafalangal
badal calculus.
logical oopparettarmaarude vaakyaghadanaye mikathum svayamsidhangal vazhi nirvachikkukayum oru anumana niyamam maathram upayogikkukayum cheiyunna propositional kaalkkulasinte mattoru pathipp nirvachikkaan kazhiyum.
svayamsidhangal.
nannayi roopappeduthiya soothravaakyangalkku vendi nilakolluka. (nannayi roopappeduthiya soothravaakyangalil greak aksharangalonnum undaakilla, marichu valia roma aksharangal, connective oopparettarmaar, paraantheesukal enniva maathrame undaakoo.) appol soothravaakyangal inipparayunnavayaanu
anumana niyamam.
anumana niyamam modus ponense aannu
mettaa-inference niyamam.
tenstailinte idathuvashathu sidhaanthangalum tenstailinte valathuvashath nigamanavumulla oru kramam oru prakadanatthe prathinideekarikkatte. appol deduction sidhaantham inipparayunna reethiyil prasthaavikkam
eee deduction sidhaantham (didi) svayam propositional calculus upayogichu roopappeduthiyittilla ithu propositional kaalkkulasinte sidhaanthamalla, marichu propositional kaalkkulasinekkurichula oru sidhaanthamaanu. eee ardhathil, ithu oru mettaa sidhaanthamaanu, ithu propositional kaalkkulasinte soundness allengil poornnathayekkurichulla sidhaanthangalumaayi thaarathamyappeduthaavunnatha.
maruvashathu, vaakyaghadanayude thelivu prakriya lalithamaakkunnathin didi valare upayogapradamaanu, ithu modus ponenukalodoppam mattoru anumana niyamamaayi kanakkakkaanum upayogikkaanum kazhiyum. eee ardhathil, eee lekhanathil avatharippicha propositional kaalkkulasinte aadya pathippinte bhagamaaya naachural contitional proof inference niyamavumaayi didi poruthappedunnu.
dii. tii. yude vipareethavum saadhuthayullathaanu
vaasthavathil, dii. tii. yude vipareethamaayi saadhuta dii. tii. yumayi thaarathamyappeduthumbol eathaandu nissaaramaan
dii. tii. yude vipareethathinu sakthamaaya prathyaaghaathangalundu oru svayamsidhathaye oru anumana niyamamakki maattaan ithu upayogikkaam. udaaharanathinu, svayamsidhavum 1-umm namukk undu,
deduction sidhaanthathinte vipareethathiloode ithu parivartthanam cheyyaan kazhiyum
ithu nammodu parayunnath anumana niyamam
sveekaaryamaanu. eee anumana niyamam samyojana nirmmarjjanamaanu, ithu propositional kaalkkulasinte aadya pathippil (eee lekhanathil) upayogikkunna patthu anumana niyamangalil onnaanu.
oru thelivu.
svayamsidhathakal maathram ulppedunna (vaakyaghadana) prakadanatthinte oru udaaharanamaanu inipparayunnava
theliyikkuka formula _ 238 (implicationte reflexivity).
thelivu
samavaakya yukthikalumaayulla thulyatha.
hilbert shailiyilulla deduction sisttathinte oru udaaharanamaanu munbathe badal calculus. nirddesha samvidaanangalude kaaryatthil, yukthisahamaya connecteevukal upayogichu nirmmicha padangalaanu svayamsidhangal, eaka anumana niyamam modus ponense aanu. haiskool beejaganithathil anoupachaarikamaayi upayogikkunna ekweshan lojik hilbert sisttangalil ninnu vyathyasthamaaya oru calculus aanu. athinte sidhaanthangal samavaakyangalaan, athinte anumana niyamangal samathvathinte gunangal prakadippikkunnu, athaayath ithu pakarakkaarane sweekarikkunna padangalude oru samanvayamaanu.
mukalil vivarichathupole classical propositional calculus booliyan algibraikku thulyamaanu, athesamayam avabodhajanya propositional calculus highting algibraikku thulyamaanu. athath sisttangalude sidhaanthangalude oro dishayilum vivarthanathiloode thulyatha kaanikkunnu. classical allengil intutionistic propositional kaalkkulasinte prameya soothravaakyam _ 247 yathaakramam booliyan allengil highting beejaganithatthinte samavaakya soothravaakyam _ 248 aayi vivarthanam cheyyappedunnu. booliyan allengil highting beejaganithatthinte sidhaantha soothravaakyam _ 249 yathaakramam classical allengil avabodhajanya kaalkkulasinte sidhaantha soothravaakyam _ 250 aayi vivarthanam cheyyappedunnu, ithinu soothravaakyam _ 251 oru staanderd churukkamaanu. booliyan beejaganita soothravaakyathinte kaaryatthil _ 249 enna formula _ 253 ennum vivarthanam cheyyaavunnathaanu, ennaal eee vivarthanam avabodhajanyamaayi thettaanu.
booliyan, highting beejaganithathil, asamathva soothravaakyam samathwathinu pakaram _ 254 upayogikkaam. samathwa soothravaakyam _ 249 asamathvangalude oru jody soothravaakyamaaya _ 254 umm soothravaakyamaaya _ 257 umm aayi prakadippikkaavunnathaan. neremarichu asamathva soothravaakyam _ 254 samathwa soothravaakyam _ 259 allengil soothravaakyam _ 260 aayi prakadippikkaavunnathaan. hilbert-stail sisttangalude asamathvathinte praadhaanyam athu randaamathethinte deduction allengil entteyilment simbal formula _ 65-maayi poruthappedunnu ennathaanu.
beejaganita chattakkoodinte asamathva pathippil ithu vivarthanam cheyyappedunnu
neremarichu beejaganita asamathva soothravaakyam _ 254 enna padatthe ulcherkkal ennu vivarthanam cheiyunnu.
266 enna implication formula asamathvam allengil 254 enna entilement formula thammilulla vyathyasam aadyathethu yukthiyude aantharikavum randaamathethu baahyavumaan ennathaanu. rund padangal thammilulla aantharika ardham ore tharatthilulla mattoru padamaanu. rund padangal thammilulla bahyamaaya ardhamaayi ulcherkkal yukthiyude bhaashaykk purathulla oru mettaatrathu prakadippikkunnu, ithu lohabhaashayude bhaagamaayi kanakkaakkappedunnu. padanatthilulla yukthi avabodhajanyamaanenkilum, ulcherkkal saadhaaranayaayi classicalaayi rund moolyamullathaayi manassilaakkunnu idathu vasham valathuvasham ulkkollunnu, allengil thulyamalla, allengil allah.
mukalil vivarichathupole swaabhaavika deduction sistangalkkum thudarchayaaya calculusinum beejaganita lojikkukalilekkum athil ninnum samanamaayathum ennaal kooduthal sankeernavumaaya vivarthanangal saadhyamaan. randaamathethinte ulkkaazhchakale rund moolyamullathaayi vyaakhyaanikkaam, ennaal kooduthal ulkkazhchayulla vyaakhyaanam oru koottam aanu, athinte gadakangal oru vibhaagathinte roopeekaranangalaayi krameekarichirikkunna amoortha thelivukalaayi manassilakkaan kazhiyum. eee vyaakhyaanathil thudarchayaaya kaalkkulasinte katt niyamam vibhaagathile gadanayumaayi poruthappedunnu. booliyan, highting aaljibrakal eee chithratthil oro homsettinum paramavadhi oru morphism ulla prathyeka vibhagangalaayi praveshikkunnu, athaayath.iii.oro ulppeduthalinum oru thelivu, thelivukalude nilanilpp maathramaan pradhaanamenna aashayathin anusrithamaayia eathu thelivum cheyyum, avaye verthirichariyunnathil ardhamilla.
graaphikkal kalkuli.
parimithamaaya vasthukkalil ninnu parimithamaaya maargangaliloode nirmmikkunnidatholam kaalam, parimithamaaya adisthaanathil parimithamaaya oru koottam sequenceukalil ninnu oru aupachaarika bhaashayude nirvachanam saamaanyavalkkarikkaan kazhiyum, mattu niravadhi ganitha gadanakal ulppeduthunnu. enthinadhikam, aupachaarika gadanakalulla eee kudumbangalil palathum yukthiyil upayogikkaan prathyekichum anuyojyamaanu.
udaaharanathinu, aupachaarika bhaashakalude saadrishyangal valare aduthulla graafukalude niravadhi kudumbangalundu, avayil calculus enna ashayam valare eluppathilum swaabhaavikamaayum vyaapippichirikkunnu. vaachaka gadanakalude anubandha kudumbangalude vaakyaghadana vishakalanathil "paazhs graafukal" aayi niravadhi graafukal uyarnnuvarunnu. aupachaarika bhaashakalile praayogika kanakkukoottalinte aavasyakathakal palappozhum aavashyappedunnathu vaachaka stringukal paazhs graafukalude pooyintar ghadana rendareshanukalaayi parivartthanam cheyyanamennu, stringukal nannayi roopappetta soothravaakyangalaano allayo ennu parishodhikkunnathinulla oru kaaryamaayi. ithu cheythukazhinjaal, stringukalil kaalkkulasinte graaphikkal analog vikasippikkunnathil ninnu niravadhi gunangal labikkum. stringukalil ninnu paazhs graafukalilekkulla mappingine paazhsing ennum paazhs graafukalil ninnu stringukalilekkulla vipareeta mapping graph "traversing" ennu vilikkunna oru pravartthanatthiloode nedunnu.
mattu logical kanakkukoottalukal.
nilavile upayogathile ettavum lalithamaaya logical kaalkkulasinekkurichaanu propositional calculus. ithu pala tharatthil vipuleekarikkaam. (aadhunika yukthiyil pradhaanamaayum maattisthaapikkappedunna aristotaliyan "sylogistic" calculus, "chila" vazhikalil lalithamaanu-ennaal mattu vazhikalil kooduthal sankeernamaanu-propositional kaalkkulasinekkaal.) kooduthal sankeernamaaya logical calculus vikasippikkunnathinulla ettavum pettennulla maargam upayogikkunna vaakyangalude kooduthal sookshumamaaya vishadaamshangalodu sensitiivu aaya niyamangal avatharippikkuka ennathaanu.
phastt oordar lojik (eiii.keey.eiii. phastt-oordar predicade lojik) propositional lojikkinte "aattomik vaakyangal" padangal, variableukal, pravachanangal, quandifierukal enningane vibhajikkumbol phalangal nalkunnu, ivayellaam chila puthiyava avatharippikkunnathinoppa propositional lojikkinte niyamangal paalikkunnu. (udaaharanathinu, "ellah naaykkalum sasthanikalaan" ennathil ninnu namukk anumanikkaam "rover oru naayayaanengil rover oru sasthaniyaan".) phastt-oordar lojikkinte upakaranangal upayogichu vyakthamaaya sidhaanthangalo anumana niyamangalo upayogichu niravadhi sidhaanthangal roopappeduthaan kazhiyum, avaye thanne logical kaalkkuliyaayi kanakkakkaam. ivayil ettavum kooduthal ariyappedunnathu ganithashaasthramaanu; mattullavayil settu sidhaanthavum kevalashasthravum ulppedunnu. randaam-krama yuktiyum mattu uyarnna-krama yukthikalum onnaam-krama yukthiyude aupachaarika vipuleekaranangalaanu. athinaal, eee yukthikalumayi thaarathamyappeduthumbol propositional lojikkine "siroth-oordar lojik" ennu paraamarshikkunnath ardhavathaaya kaaryamaan.
propositional calculusil pidichedukkaan kazhiyaatha vividha nigamanangalum model lojik vaagdaanam cheiyunnu. udaaharanathinu, "anivaaryamaayi" ninnu namukk athu anumanikkaam. athu saadhyamaan" ennu nammal anumaanichekkaam. model lojikkukalum beejaganita lojikkukalum thammilulla vivarthanam classical, avabodhajanya lojikkukalumaayi bandhappettathaanenkilum booliyan allengil highting aaljibrakalil oru yoonari oopparettar avatharippikkunnathiloode, booliyan pravarthanangalil ninnu vyathyasthamayi, saadhyatha reethi vyaakhyaanikkunnu, koodaathe beejaganithathe highting cheiyunna kaaryatthil aavasyakatha vyaakhyaanikkunna randaamathe oopparettar (booliyan beejaganithathinu ithu anaavasyamaanu, kaaranam aavasyakatha dii morgan duval oof saadhyathayaanu). aadya oopparettar 0 umm disjungshion samrakshikkumbol randaamathethu 1 umm samyojanavum samrakshikkunnu.
vaakyangalkk "sharee", "thettu" enniva ozhikeyulla moolyangal anuvadikkunnavayaanu pala moolyamulla yukthikal. (udaaharanathinu, "randum", "randum" enniva staanderd "adhika moolyangalaanu"; "thudarchayaaya yukthi" oro vakyathinum "sharee", "thettu" ennivaykkidayil ananthamaaya "sathyathinte alavukal" undakaan anuvadikkunnu.) eee yukthikalkk palappozhum propositional calculusil ninnu thikachum vyathyasthamaaya calculational upakaranangal aavasyamaanu. moolyangal oru booliyan beejaganitam roopappeduthumbol (randiladhikam allengil ananthamaaya niravadhi moolyangal polum ulla), pala moolyamulla lojik classical lojikkaayi kurayunnu; athinaal moolyangal booliyan allatha oru beejaganitam roopappeduthumbol pala moolyamulla lojikkukal swatantra thaalpparyamullavayaanu.
pariharangal.
propositional calculus predicate calculus thammilulla oru sradheyamaaya vyathyasam oru propositional formulayude thripthikaratha nirnnaayakamaan ennathaanu. nirddeshaparamaaya yukthiyude soothravaakyangalude thripthikaratha theerumaanikkunnathu oru enn. pii-poornnamaaya prasnamaanu. ennirunnaalum, praayogika reethikal nilavilundu (udha.jii., dpll algoritam, 1962; chaaf algoritam, 2001) pala upayogapradamaaya kesukalkkum valare vegathilaanu. ganitha padaprayogangal adangiya nirddeshangalumaayi pravartthikkaan sameepakaala pravartthanangal saatt solver algorithangal vipuleekarichu; ivayaanu smt solverukal.

