
randaam kramam yukthi

yukthiyilum ganithathilum, randaam krama lojik phastt-oordar lojikkinte vipuleekaranamaanu, athu thanne propositional lojikkinte vipuleekaranamaanu. second oordar lojik uyarnna oordar logic type sidhaanthavum vazhi vipuleekarikkappedunnu.
phastt-oordar lojik vyakthikalil (vyavahaarathinte domaininte gadakangal) vyathyaasappedunna variableukal maathram alakkunnu; second-oordar lojik, koodaathe, bandhangale alakkunnu. udaaharanathinu, randaam krama vaakya soothravaakyam _ 1 parayunnath oro soothravaakyathinum "p", oro vyakthigatha "x" ennivaykku, onnukil "px" shariyaano allayo ("px") sariyaanennaanu (ithu ozhivakkiya madhyathinte niyamamaan). settukal, functionukal, mattu variableukal ennivayude alavum randaam krama lojikkil ulppedunnu (chuvadeyulla vibhaagam kaanuka). phastt-oordar, second-oordar lojik enniva oru prabhaashana mekhalayude ashayam upayogikkunnu (palappozhum "domaine" allengil "prapancham" ennu vilikkunnu). vyakthigatha gadakangal alakkaan kazhiyunna oru koottamaanu domaine.
udaaharanagal.
phastt-oordar lojikkinu vyakthikale alakkaan kazhiyum, pakshe gunangalekkaal allah. athaayath, namukk cube ("bii") polulla oru aattomik vaachakam eduthu peru oru variable upayogichu maattisthaapikkukayum oru quantifier attaachucheyyukayum cheythukondu oru alavu vaachakam nedaam
ennirunnaalum, pravachanatthinte kaaryatthilum namukk angane cheyyaan kazhiyilla. athaayath, thaazhepparayunna prayogam
ithu phastt oordar lojikkinte oru vaakyamalla, marichu ithu second oordar lojikkinte niyamaanusrithamaaya vaakyamaan. ivide, "p" oru predicate variablaanu, ardhathil ithu oru koottam vyakthikalaanu.
thalphalamaayi, randaam kramatthilulla yukthiykk onnaam kramatthilulla yukthiyekkaal kooduthal aavishkaarasheshiyundu. udaaharanathinu, ellah cubukaludeyum tetrahedronukaludeyum koottam thirichariyaan phastt-oordar lojikkil oru margavumilla. ennaal eee koottatthinte nilanilpp randaam nira yukthiyil ingane urappikkaam
appol namukk eee koottatthinte savisheshathakal urappikkaam. udaaharanathinu, ellah cubukaludeyum tetrahedronukaludeyum koottatthil dodecahedronukalonnum adangiyittillennu inipparayunnava parayunnu
randaam kramatthil alakkunnathu prathyekichum upayogapradamaanu, kaaranam ithu athicheraanulla gunangal prakadippikkaanulla kazhivu nalkunnu. udaaharanathinu, paarant ("x", "y") soochippikkunnathu "x" ennathu "y" yude paarant aanenkil, phastt-oordar lojikkinu "x" ennathu "y" yude poorvvikanaanenna property prakadippikkaan kazhiyilla. randaam kramatthil, "y" adangiyathum, rakshaakarthru bandhathinu keezhil adanjathumaaya oro koottam aalukalilum "x" adangiyittundennu paranjukondu namukk ithu prakadippikkaan kazhiyum
second oordar lojikkil pravachanagalkkulla variableukal undengilum pravachanangalude gunangalkk namukk variableukal illa ennathu sradheyamaanu. udaaharanathinu, "p" cube, tettu, dodec ennee pravachanangalkk oru property aakruthi ("p") undennu namukk parayaan kazhiyilla. ithinu moonnaam kramam yukthiyundaakum.
vaakyaghadanayum shakalangalum.
second oordar lojikkinte vaakyaghadana eathokke padaprayogangalaanu nannayi roopappetta soothravaakyangalennu parayunnu. phastt-oordar lojikkinte vaakyaghadanaykku purame, second-oordar lojikkil niravadhi puthiya "tharangal" (chilappol "tharangal" ennu vilikkunnu) variableukal ulppedunnu. ivayaanu
soothravaakyangal nirmmikkunnathin ippol nirvachichirikkunna oro variabilukalum saarvathrikamaayum koodaathe/allengil asthithvaparamaayi alakkappedaam. athinaal pala tharatthilulla kwaandifayarukalundu, oro tharatthilulla veriyabilukalkkum rund. phastt oordar lojikkilennapole, second oordar lojikkile oru "vaakyam" ennathu swatantra veriyabilukalillaatha (eathenkilum tharatthilulla) nannayi roopappetta oru soothravaakyamaanu.
mukalil nalkiyirikkunna nirvachanathil function veriyabilukalude aamukham upekshikkaan kazhiyum (chila rachayithakkal ithu cheiyunnu) kaaranam oru "n"-ary function variable aratti "n" + 1 nte relation variable relationsinte "n" + 1 aargyumentile "phalam" ennathinte savisheshathaykk anuyojyamaaya oru soothravaakyavum prathinidheekarikkam. (shapiro 2000, peju. 63)
monadic second oordar lojik (emm. ezz. oo) ennathu second oordar lojikkinte oru niyanthranamaan, athil uniry bandhangalude alavu maathram (i.iii. settukal) anuvadaneeyamaanu. mukalil vivarichathupole bandhangalumayulla thulyatha kaaranam functionukalude alavum anuvadaneeyamalla. eee niyanthranangalillaatha randaam krama yukthiye chilappol monadic pathippil ninnu verthirichariyaan "phul second oordar lojik" ennu vilikkunnu. graph sidhaanthathile oru algoritam mettaa-sidhaanthamaaya coursellinte sidhaanthathinte pashchaathalatthil monadic second-oordar lojik prathyekichum upayogikkunnu. samboornna ananthamaaya bainari treaude (ezz 2 ezz) emm. ezz. oo sidhaantham nirnnaayakamaan. ithinu vipareethamaayi, eathenkilum ananthamaaya ganathinmelulla poornna second oordar lojikkinu (allengil udaaharanathinu emm. ezz. oo lojik ovar (formula _ 2, +)) yadhaartha second oordar ganithathe vyaakhyaanikkaan kazhiyum.
phastt-oordar lojikkilennapole, second-oordar lojikkilum oru prathyeka second-oordar bhashayil noonn-logical chinnangal ulppettekkaam. ennirunnaalum, iva niyanthrikkappettirkinnul, avaa roopappedunna ellah padangalum onnukil phastt-oordar padangal (phastt-oordar variablinu pakaram vaykkaavunnava) allengil second-oordar padangal (anuyojyamaaya tharatthile second-oordar variablinu pakaram vaykkaavunnava) aayirikkanam.
randaam kramatthilulla yukthiyile oru soothravaakyam onnaam kramathilullathaanennu parayappedunnu (chilappol formula 3 allengil formula 4 ennu soochippikkunnu) athinte quandifierukal (saarvathrikamo asthithvaparamo aayirikkam) onnaam kramatthile veriyabilukalil maathram vyathyaasappedukayaanenkil, athinu randaam kramatthile swatantra variableukal undayirikkaam. oru formula _ 5 (eksisttanshyal second-oordar) formula second oordar veriyabilukalil chila eksisttanshyal quandifierukal koodi ulla onnaanu, i.iii. formula _ 6, ivide formula _ 7 oru phastt oordar formulayanu. asthithvaparamaaya randaam krama soothravaakyangal maathram ulkkollunna randaam krama yukthiyude shakalathe asthithvaparamaaya randaam krama yukthi ennu vilikkukayum eeso ennu churukkipparayukayum cheiyunnu, formula _ 5 allengil He angane polum. formula _ 9 soothravaakyangalude sakalam erattayaayi nirvachikkappedunnu, ithine yoonivezhsal second oordar lojik ennu vilikkunnu. paraspara aavarthanathiloode eathu "k"> 0-num kooduthal expressive sakalangal nirvachikkappedunnu formula _ 10-nuu formula _ 6 undu, ivide formula _ 7 oru formula _ 13 formulayanu, samanamaaya formula _ 14-nuu formula _ 15 undu, ivide formula _ 7 oru formula _ 17 formulayanu. (randaam kramatthile ganithatthinte samana nirmmaanathinaayi vishakalana shreni kaanuka.)
ardhashaasthram.
second oordar lojikkinte ardhashaasthram oro vaakyathinteyum ardham sthaapikkunnu. oru staanderd semantics mathramulla phastt-oordar lojikkil ninnu vyathyasthamayi, second-oordar lojikkinaayi saadhaaranayaayi upayogikkunna rund vyathyastha semantics undu staanderd semantics, henkin semantics. eee oro ardhathilum, phastt-oordar kwandifayarukaludeyum logical connectevukaludeyum vyaakhyaanangal phastt-oordar lojikkil ullathupole thanneyaanu. second oordar veriyabilukalil kwaandifayarukalude shrenikal maathrame rund tharam ardhashaasthrathil vyathyaasappedunnulloo (2001 ll).
phul semantics ennum vilikkappedunna staanderd semaantiksil, quandifierukal "ellah" settukaliloo uchithamaaya tharatthilulla pravartthanangalilo vyathyaasappettirikkunnu. athinaal phastt-oordar veriyabilukalude domaine sthaapichukazhinjaal, sheshikkunna kwaandifayarukalude ardham nischayikkappedum. eee ardhashaasthramaanu randaam nira yukthiykk athinte avishkaara shakti nalkunnathu, avaa eee lekhanathinte backie bhaagathekku anumaanikkappedum.
liyon henkin (1950) second-oordar, hier-oordar sidhaanthangalkkaayi oru badal tharam ardhashaasthram nirvachichu, athil uyarnna oordar domainukalude ardham bhaagikamaayi nirnnayikkunnath type sidhaanthathil varaykkunna, settukaludeyo functionukaludeyo gunangalude vyakthamaaya svayamsidheekaranamaanu. henkin semantics ennathu orutharam palatharam phastt-oordar semanticsaanu, avide semantics staanderd semantixilennapole staanderd modalilekku maathram nischayikkunnathinuparam, svayamsidhathakalude oru klaas modalukal undu. henkin semantixile oru model uyarnna kramatthilulla domainukalude vyaakhyaanamaayi oru koottam settukaloo oru koottam functionukaloo nalkum, athu atharam ellah settukaludeyo functionukaludeyo sariyaaya upavibhagamaayirikkaam. phastt-oordar lojikkinu vendi nilanilkkunna godelinte poornnatha sidhaanthavum compactness sidhaanthavum henkin semantixinoppam randaam-oordar lojikkilekku kondupokunnuvennu henkin thante svayamsidheekaranathinaayi theliyichu. henkin semantixine scholim-lovenheim sidhaanthangal ulkkollunnathinaal, henkin modalukal verum "veshammariya phastt oordar modalukal" aanennu lindstrom sidhaantham soochippikkunnu.
second-oordar ganithashaasthra sidhaanthangalkku, uyarnna-oordar domainukalude staanderd allatha vyaakhyaanangalude nilanilpp henkin upayogicha type sidhaanthathil ninnu uruthirinja prathyeka svayamsidhaanthathinte kuravu mathramalla, godelinte apoornnatha sidhaanthathinte anivaryamaaya anantharafalamaanu staanderd vyaakhyaanamaanu saadhyamaaya eaka maathruka ennu urappakkaan henkinte svayamsidhaanthangalkku kooduthal anubandham nalkaan kazhiyilla. randaam nira ganithashaasthra padanatthil henkin semantics saadhaaranayaayi upayogikkunnu.
randaam krama lojikkinaayi henkin modalukalum poornna modalukalum thammilulla thiranjeduppu settu sidhaanthathinte adisthaanamaayi zfc, "v" enniva thammilulla thiranjeduppinu samaanamaanenn juco vaanan (2001) vaadichu "randaam krama lojikkineppole," v "allengil zfc upayogichu ganithashaasthrathe svayamsidhamaakkano ennu namukk sarikkum thiranjedukkaan kazhiyilla. rund saahacharyangalilum phalam onnuthanneyaanu, kaaranam ganithashaasthrathinte svayamsidhamaakkalaayi "vii" upayogikkunnathinulla ettavum mikacha sramamaanu "isadfsi"."
aavishkaarashakthi.
phastt oordar lojikkinekkaal kooduthal expressive aanu second oordar lojik. udaaharanathinu, domaine ellah yadhaartha samkhyakaludeyum koottamaanengil, oro yadhaartha samkhyayudeyum additteev invezhsinte nilanilpp aadya-krama lojikkil τ "x" H "y" ("x" + "y" = 0) ennu ezhuthi urappikkaam, ennaal oro parimithavum shoonyavumaaya yadhaartha samkhyakalude koottatthinum oru paramaadhikaaramundenn parayunna yadhaartha samkhyakalude koottangalkku ettavum kuranja-mukalilulla-boundu property urappikkaan oraalkk randaam krama lojik aavasyamaanu. domaine ellah yadhaartha samkhyakaludeyum koottamaanengil, inipparayunna randaam krama vaachakam (rund varikalilayi vibhajikkappedunnu) ettavum kuranja upper boundu property prakadippikkunnu
eee soothravaakyam "oronnum, oru settu cheyyuka" ennathinte nerittulla aupachaarikavalkkaranamaanu."eee property thripthippeduthunna eathoru oordar feeld riyal nambar feeldinu isomorphic aanennu kaanikkan kazhiyum. maruvashathu, compactness sidhaantham kaaranam realsil saadhuthayulla phastt-oordar vaakyangalude koottatthinu ekapaksheeyamaayi valia modalukalundu. athinaal ettavum kuranja-mukalilulla-boundu property phastt-oordar lojikkile oru koottam vaakyangalal prakadippikkaan kazhiyilla. (vaasthavathil, oro riyal-clossed feeld signature formula _ 18 leeii athe phastt-oordar vaakyangale riyal nambarukalaayi thripthippeduthunnu.)
randaam kramatthil, "domaine parimithamaanu" allengil "domaine kanakkakkavunna cardinality ullathaanu" ennu parayunna aupachaarika vaakyangal ezhuthaan kazhiyum."domaine parimithamaanennu parayaan, domainil ninnu thannilekku varunna oro surjective function kuthivaykkalaanennu parayunna vaachakam upayogikkuka. domaine kanakkakkavunna cardinality undennu parayaan, domaininte oro rund ananthamaaya upavibhagangalkkum idayil oru bijection undennu parayunna vaachakam upayogikkuka. compactness sidhaanthathil ninnum mukalilekkulla lovenheim-scom sidhaanthathil ninnum ithu pinthudarunnu, phastt-oordar lojikkil yathaakramam parimithiyo ennalo savisheshamaakkaan kazhiyilla.
randaam krama lojikkinte chila sakalangal poornna randaam krama lojikkinekkaal karsanamaayi kuranja expressionaanu aanenkilum, esso polulla randaam krama lojikkinte chila sakalangal onnaam krama lojikkinekkaal kooduthal expressionaanu. henkin quandifierukal, hintica, sandu ennivayude independence frondly lojik, vaananente dependence lojik enniva upayogichu vipuleekaricha phastt-oordar lojik polulla quantifier dependensikalude noonn-leanier oordaring anuvadikkunna phastt-oordar lojikkinte chila vipuleekaranangalumaayulla vivarthana thulyathayum esso aasvadikkunnu.
deductive sisttangal.
oru yukthiyude deductive sisttam ennathu soothravaakyangalude eathu sreniyaanu saadhuvaaya thelivukal ennu nirnnayikkunna anumana niyamangaludeyum logical aksiyangaludeyum oru koottamaanu. staanderd semanticsinu onnum poortthiyaakkaan kazhiyillengilum second oordar lojikkinaayi niravadhi deductive sisttangal upayogikkaam (chuvade kaanuka). eee samvidaanangal oronnum shabdamullathaanu, athaayath avaa theliyikkaan upayogikkaavunna eathu vaakyavum uchithamaaya ardhashaasthrathil yukthisahamaayi saadhuthayullathaanu.
randaam krama nibandhanakalkkulla pakara niyamangalumayi vardhippicha aadya krama lojikkinulla (swaabhaavika kizhivu polulla) oru staanderd deductive sisttam ulkkollunnathaan upayogikkaan kazhiyunna ettavum durbalamaaya deductive sisttam. eee deductive sisttam saadhaaranayaayi second oordar ganithashaasthra padanatthil upayogikkunnu.
shapiro (1991), henkin (1950) enniva pariganikkunna deductive sisttangal augmented phastt-oordar deductive schemilake comprehension axiums choys axiums cherkkunnu. eee sidhaanthangal staanderd second oordar semanticsinu anuyojyamaanu. avaa henkin modalukalil parimithappedutthiyirikku henkin semanticsinu anuyojyamaanu, avaa grahyavum choys aaksiyamukalum thripthippeduthunnu.
phastt oordar lojikkilekkulla noonn-reducibility.
poornna second oordar semantics upayogichu yadhaartha samkhyakalude second oordar sidhaanthathe inipparayunna reethiyil phastt oordar sidhaanthathilekku kuraykkaan sramikkaam. aadyam domaine ellah yadhaartha samkhyakaludeyum koottatthil ninnu rund tharam domainilekku vikasippikkuka, randaamathe tharatthil ellah "settu" yadhaartha samkhyakalum adangiyirikkunnu. bhaashayilekku oru puthiya bainari pravachanam cherkkuka angathwa bandham. pinneedu second oordaraayirunna vaakyangal phastt oordaraayi maarunnu, pakaram muun second oordar quandifierukal randaam tharathilaayirikkum. oru ghadakam oru samkhyayaano atho oru koottamaano ennu parayunna eaka pravachanangal cherkkunnathiloodeyum domaine oru koottam yadhaartha samkhyakaludeyum yadhaartha samkhyakalude power settinteyum samyojanamaayi edukkunnathiloodeyum eee kuravu otappetta sidhaanthathil sramikkaam.
ennaal domainil "ellah" settu yadhaartha samkhyakalum ulppeduthiyathaayi avakashappettathaayi shradhikkuka. lovenheim-scholim sidhaantham kaanikkunnathupole aaa aavasyakatha oru phastt-oordar vaakyamaayi kuraykkaan kazhiyilla. aaa sidhaantham soochippikkunnathu, yadhaartha samkhyakalude chila ennathil ananthamaaya upavibhaagangalund, avayude amgangale nammal "aantharika samkhyakal" ennu vilikkum, koodaathe aantharika samkhyakalude chila ennathil ananthamaaya shekharavum undu, avayude amgangale nammal "aantharika settukal" ennu vilikkum, athaayath aantharika samkhyakalum aantharika settukalum adangiya domaine yadhaartha samkhyakaludeyum yadhaartha samkhyakaludeyum domainukal thripthippeduthunna athe aadya krama vaakyangale thripthippeduthunnu. prathyekichum, phalatthil, ithu parayunna orutharam ettavum kuranja-mukalilulla-badhamaaya sidhaanthathe thripthippeduthunnu
ellah aantharika samkhyakaludeyum gananakshamatha (avaa oru saandramaaya kramatthilulla ganam roopappeduthunnu enna vasthuthayumayi samyojichu) soochippikkunnathu aaa ganam poornnamaaya ettavum kuranja-mukalilulla-badhamaaya soothravaakyathe thripthippeduthunnilla ennaanu. ellah "aantharika" settukaludeyum gananakshamatha soochippikkunnathu ithu ellah "aantharika" samkhyakalude koottatthinteyum "ellah" upavibhaagangaludeyum koottamalla ennaanu (kaaranam kaantorinte sidhaantham soochippikkunnathu ennaavunna ananthamaaya oru koottatthinte ellah upavibhaagangaludeyum koottam ennappedaatha ananthamaaya oru koottamaanu). eee nirmmaanam scholaminte vairudhyavumaayi adutha bandhamullathaan.
athinaal yadhaartha samkhyakaludeyum yadhaartha samkhyakalude koottangaludeyum aadya krama sidhaanthathinu niravadhi modalukalundu, avayil chilath kanakkakkaavunnavayaanu. ennirunnaalum, yadhaartha samkhyakalude randaam krama sidhaanthathinu oru maathruka maathrameyulloo.
oru archemedian poornna kramatthilulla feeld maathrame ulloo enna shaasthreeya sidhaanthathil ninnu ithu pinthudarunnu, koodaathe oru archemedian poornna kramatthilulla feeldinte ellah soothrangalum randaam kramatthilulla yukthiyil prakadippikkaavunnavayaana. ithu kaanikkunnath yadhaartha samkhyakalude randaam krama sidhaanthathe oru onnaam krama sidhaanthamaayi churukkaan kazhiyillennu, athaayath yadhaartha samkhyakalude randaam krama sidhaanthathinu oru maathruka maathrameyulloo enna ardhathil, ennaal anubandha onnaam krama sidhaanthathinu niravadhi maathrkakalundu.
staanderd semanticsulla second-oordar lojik phastt-oordar lojikkinekkaal kooduthal expressive aanennu kaanikkunna kooduthal theevramaaya udaaharanangalund. oru parimithamaaya randaam krama sidhaanthamundu, athinte eaka maathruka thudarcha sidhaantham nilanilkkunnundenkil yadhaartha samkhyakal maathramaan, thudarcha sidhaantham nilanilkkunnillenkil athinu maathrkayilla (cf. shapiro 2000, pii. 105). eee sidhaanthathil oru parimitha sidhaantham ulppedunnu, ithu yadhaartha samkhyakale oru samboornna archemedian oordar feeld aayi chithreekarikkunnu, koodaathe domaine aadyathe ennaanaavaatha cardinalityanennu parayunna oru svayamsidhaantham. randaam kramatthilulla yukthiyile oru vaachakam sthirathayullathaano enna chodyam valare sookshumamaanennu eee udaaharanam vyakthamaakkunnu.
randaam krama yukthiyude adhika parimithikal adutha vibhaagathil vivarichittundu.
lohashasthraparamaaya phalangal.
oresamayam eee moonnu aavasyamulla aatribyoottukale thripthippeduthunna randaam-oordar soothravaakyangalkku deductive sisttam (athaayath "provability" enna ashayam illa) illa ennathu godelinte apoornnatha sidhaanthathinte oru anantharafalamaanu
randaam krama yukthi oru samboornna thelivu sidhaantham amgeekarikkillennu paranjaanu eee anantharafalam chilappol prakadippikkunnathu. eee kaaryatthil staanderd semanticsulla second-oordar lojik phastt-oordar lojikkil ninnu vyathyasthamaan; kwin (1970, pipi. 90-91) randaam nira yukthiye "lojik" allennu chinthikkunnathinulla oru kaaranamaayi oru samboornna proof sisttathinte abhavam choondikkaanichu.
mukalil soochippichathupole, henkin semantixinoppam second-oordar lojikkinu phastt-oordar lojikkinulla staanderd deductive sisttam mikachathum poornnavum phalapradavumaanennu henkin theliyichu, koodaathe grahyavum choys thathwangalumulla deductive sisttam eee thathwangal thripthippeduthunna modalukal maathram upayogichu henkin semanticsinu mikachathum poornnavum phalapradavumaanu.
compactness sidhaanthavum lovenheim-scholim sidhaanthavum randaam krama yukthiyude poornna modalukalkk yojikkunnilla. ennirunnaalum avaa henkin modalukalkk baadhakamaan.
charithravum tarkka moolyavum.
sii ganithashaasthra samoohathinu pravachana yukthi avatharippichu. ezz. second-oordar lojik" enna padam srishtichathum aadhunika roopavumaayi ettavum saamyamullathumaya peezhs (putnaam 1982). ennirunnaalum, innu yukthiyude mikka vidyaarthikalum peersinu niravadhi varshangalkk munbu thante kruthi prasidheekaricha freejinte kruthikalekkurichu kooduthal parichitharaanu, ennaal bertrand russel alfred north whitehead avare prashastharaakkunnathuvare avarude kruthikal ariyapettirunnilla. vasthukkalude alavine gunangaludeyum settukaludeyum alavukalil ninnu verthirichariyaan freej vyathyastha variableukal upayogichu; ennaal rund vyathyastha tharam lojikkukal cheiyunnathaayi addeham svayam kandilla. rassalinte vairudhyam kandethiyathinushesham addehathinte samvidaanathil entho kuzhappamundennu manassilakki. oduvil, freejinte yukthiye vividha reethikalil niyanthrikkunnath-ippol phastt-oordar lojik ennu vilikkappedunnathilekka-eee prashnam illaathaakkiyathaayi yukthivaadikal kandethithi phastt-oordar lojikkil maathram settukalum propertykalum alakkaan kazhiyilla. lojiksinte kramangalude ippol nilavaaramulla shreni eee samayam muthalaan.
phastt-oordar lojikkinte upakaranathinullil (palatharam poornnathayude vilaykk, ennaal rassalinte virodhabhaasam pole moshamaayi onnumilla) oru svayamsidha samvidaanamaayi settu sidhaantham roopappeduthaan kazhiyumennu kandethi, ithu ganithashaasthrathinu settukal pradhaanamaayathinaal (sermelo-frenkel settu sidhaantham kaanuka) cheythu. phastt oordar kwaandifikkeshanekkaal kooduthal logical upakaranangalonnum aakarshikkathe ganithashaasthram, maathrashaasthram, mattu sakthamaaya logical sidhaanthangal enniva svayamsidhamaayi roopappeduthaan kazhiyum, ithu, godel skolamum phastt oordar lojikkinodu chernnu, randaam oordar lojikkile (allengil eathenkilum uyarnna) joliyil pothuvaaya thakarchayilekku nayichu.
eee nirasikkal chila yukthivaadikal sajeevamaayi munnott kondupoyi, prathyekichu dablyu. vii. kwina. fx" polulla pravachana bhasha vaakyangalil "x" oru vasthuvine soochippikkunna oru variablaayo peraayo kanakkakkanamennum athinaal "ellah karyangalkkum" ennathupole alakkaan kazhiyumennum kwain kaazhchappaad munnott kondupoyi. . ."ennaal" "eph" "ennathu oru apoornna vaakyathinte" "churukkapperaayi" "kanakkaakkanam, oru vasthuvinte peralla (oru vasthu poleyulla amoortha vasthuvinte polum allah)". udaaharanathinu, "ennaanu ardham. . . oru naayayaan."ennaal ithupolulla onninekkurichu namukk alakkaan kazhiyumennu karuthunnathil ardhamilla. (atharamoru nilapad ashayam-vasthu vyathyasathekkurichulla freejinte swantham vaadangalumaayi thikachum poruthappedunnu). athinaal oru variable oru pravachanam upayogikkunnathu athu oru perinte sthaanam kaivashappedutthuka ennathaanu, athu vyakthigatha variableukal maathram kaivashappedutthanam. eee nyaayavaadam jorjju boolos nirasichu.
adutha kaalathaayi second oordar lojik oru veendedukkal undaakki, phastt oordar quandification (boolos 1984) pole vasthukkalude athe domainil bahuvachana kwaandifikkeshanaayi second oordar kwandifikkeshanekkurach boolosinte vyaakhyaanam prachodippichu. chila vimarshakar parasparam maathram aaradhikkunnu", "" prathisrutha varante chila aalukal mattarkkum oppam illaathe warehouseilekku pooyi "thudangiya vaakyangalude noonn phastt oordaraisebility avakaashappedunnathilekku boolos viral choondunnu, ithu second oordar kwaandifikkeshante poornna shakthiyal maathrame prakadippikkaan kazhiyoo ennu addeham vaadikkunnu. ennirunnaalum, saamaanyavalkkaricha alavum bhaagikamaayi krameekaricha (allengil branching) alavum oru prathyeka klaas noonn phastt oordar cheyyaavunna vaakyangal prakadippikkaan mathiyakum, iva second oordar alaveduppinodu yojikkunnilla.
computational sankeernathayumaayulla bandham.
parimithamaaya gadanakalile randaam krama yukthiyude vividha roopangalude avishkaara shakti computational sankeernnatha sidhaanthavumaayi adutha bandhamullathaan. vivaranathmaka sankeernnatha padanangalude mekhala, avayil bhashakal (parimithamaaya stringukalude koottangal) prakadippikkunnathin aavashyamaaya yukthiyude shakti upayogichu computational sankeernnatha klaasukale savisheshamaakkaan kazhiyum. oru parimitha aksharamalayile "a" enna string "w" = "w" 1···· "" wn "" d "= {1 domaine ulla oru parimitha gadanayaal prathinidheekarikkam.., n", "oro" a "" ω "a" num "" ω "a" ennu yoonari pravachikkunnu, "" i "enna soochikakalaal thripthippettathinaal" "vi" = "" a ", koodaathe eathu soochikayaanennu advitheeyamaayi thirichariyaan sahayikkunna adhika pravachanangal (saadhaaranayaayi, oraal pingamiyude functionte graph" d "allengil oordar relations <edukkunnu, orupakshe mattu ganitha pravachanangalumayi). neremarichu, eathenkilum parimithamaaya gadanayude (oru parimithamaaya oppinu mukalilulla) kayley tableukal oru parimithamaaya string upayogichu encode cheyyaan kazhiyum.
eee thirichariyal parimithamaaya gadanakalil randaam krama yukthiyude vakabhedangalude inipparayunna swabhaavasavisheshathakalilekku nayikkunnu
eee klaasukal thammilulla bandhangal parimithamaaya gadanakalile yukthikalude aapekshika prakadanatthe nerittu badhikkunnu; udaaharanathinu, ph = pspace aanenkil, randaam krama lojikkilekku oru traansitteevu closher oopparettare cherkkunnath parimithamaaya gadanakalil kooduthal aavishkaarasheshi undakkilla.