rocva
Transkript
rocva
Amre6pa H norHKa, 8, N> 5 ( 1970), 530-538 MflK 518.5 O E-TEOPHHX Jl./I.MAKCHMOBA B naaHoS craxbe aayiaioTca TeopHH, ocHosaHHHe aa nc«mc;ieHnax crporoft HMiuimcaiiHH /7 [.1] , £ H /£ [4]. 3-ru HCHHcnemia, B orjiHiHe or 6ojIbIIIHHCTBa H3B6CTHbIX npOnO3HIIHOHajlbHbIX HCHHCJieHHfi, OC , OL -*• npaBHn BbiBona, Kpoxie npaBHjia —' '^ B CB33H C 3THM ~ HMGIOT CpeflH fi , Tanxte paCCMBTpHBaeTCa BOnpOC O peiUGTKe TBOpHft, HC- OCj ft — '"~ B KOTOpblX nocrynnpyioTCH yicaaaHHbie npasMJia BbiBona. HaflneH Knacc Teopafi, ana KOToporo ara pemeTKa flHCTpH6yxnBHa. B n. 2 ycTaHaBnasaeTca cooTHouieHHe Me>Kay KriaccHiecKoft jiorHKOft H HenpOTHBOpeHHBblMH paCUIHpeHHHMH aOTHK I. B COOTB6TCTBHH C o6b!HHWM (JiopMynfai ct H3 Fl , E H OnpeaejieHHeM, R. B bl . B O a O M M H O j K e c T B a /7 HaaoseM nocneao oe. ..., <X.K raKyro, HTO Ana Ka^Knoro c. /\b (fiopMyn BbinonneHO OAHO HS ycnosHft: (a) oc . e /7 (^ 1) «^/ =oe.// —•• Oc£- (#3) oc^. =ct^&o:<i// 4>opMyny c T Ba /7 HJIH oc. naaoseM (coKpameHHO ana , rae ^, i." < L. B b i a o n H M o f t M H O K G i conepJKamee Bee aKCHOMw HcincneHHa MHowecTBo ^opwy^1 E. H xaKOe, HTO us 7"(~ <X cneayer cc € 7" • £ -reopiia, B Koropofi flonycTHMO npaanno noacraHOBKH, HaabiBaerca E - J I O T H K O f t . OieBHflHo, nio6aa H C 7" , H3 KOTOpblX 6bITb /7 Ana nioSbix == - /7 I— oc )» eonn cymecrsyeT Bbiaon ct «3 /?• Cneaya C6U , HaaoaeM f - T e o p n e f i 7" HS £" -xeopua 7" onpeaenaerca MnowecTBOM aKCHOM BWBOflHMfal BC6 (JopMy/Ibl H3 7" (B HaCTHOCTH, 7" ) . ^ -reopuft T H 7"2 onpeaenHM xoiHyio HHJKHKIIO O E - TSOpHHX 531 K3K ,i-«. „ ra H Toinyio BepxHioro rpaHb KBK Torna coBOKynnocTb Bcex pauHit H £"-Teopnii oSpaayex pemexicy OTHocHTenbuo one- + . TEOPEMA. P e u i e T K a B B n a e T C a B c e x ^ - T e o p a f t 7" (E ) A H C T p H 6 y T H B H O f t . MH AOJiwHbi AOKaaaTb, HTO B paccMaxpHBaeMoft peuieTKe BbinojineHO floKa>KeM, rne HTO BbinonHeHO 7^ , 71 , 7~ - npoHSBOJibHbie £ -reopHH, roBopHTfa, ITO (j)opMyna c T b A C3.D . ecnH C V Vcc) HJIH oSosHaiaxb nepea <x ecTb oc H H S I I K I H K T H B ecrfa £> H/IH anta HeKOropofi (Vac Y \? ) ecrbflH3T.K>HKTHBHaanacxb f~[_°Q - & .By- ^opMyny, B Koropoft ecrb aH3t,K)HKTHBHas <X . • ^ - B b l B O f l O M , HTO nnn Kamnoro ) H3 i /7 HaSOBBM HOCJieflOBaTenbHOCTb Ct.,...,^ BbinonneHo OAHO HS ycnoBHft: or. ecTb F[|oc] nna cx^ F[£>~] H nna HeKOTopwx i^ 6 / / < : i, ecTb [V —"""/O ] cxe/7 H ^d'*' H CX^ / HJIH ecTb ^""CVl 6CTb X1" HJIH QC 6CTb * <L . H I — <X oaHanaeT, HTO cyuiecTByeT /1EMMA. flna M y n /? —* £ n w 6 o r o : /7 h- oc. F -BUBOA Ct H3 M H O J K e c T B a s K B H B a n e H T l i o OneBHAHO, nroSoft BHBOA as /7 aBnaeTcs K8K npaaana R.'f H ^5 ecTb nacTHbie cnynaa OSparao, nycTb ex ecTfa /• \_ft~\a & & <j>op /7h- /^ -BMBOAOM HS R. / . Tax H - Ot. /7 , TBK j^S npa 532 & bi nonyiaeM TO /7 /— /-" [" fll . I~IoKa>KeM, HTO npaBHna /3l- F £oq /£/ H d*-F[f>~} H R.8 SBjiaioTCfl npoHSBOflHbiMH. flycxfa . Toraa /?/- FfceJ &, /r^ ] . flanee, ner- KO aoKaaarb HHflyKunei4 no rny6HHe oc a f~ Qx] , HTO => /= « & - F bc-~- =i>/7h F bc St. Coc flOKasaaa. BosBpainaeMca K flOKasarentcTBy reopeMfai. FlycTfa Torna 7",7 l~ ex H 71+77 /—ex , r.e. * O mecTByex /-"-BbiBoa <x • ot ecTfa HJIH 6 e 7~ H, T8K KaK ; Ot e F [AJ /"" -Bbisofla us HJIH 7T , HM66M QC^ 7" ' 71 f 7~<2f. U 7^, o HTO (* 77* * A e 7~A U 7~ . Torna Ae AG 71 . OTorana '— OC , HOSTOMy C 73 ' 71 )"*-f 7^ Z * ecTb HenocpencTBeHHoe cneflcyBHe HOMy K3 npaBHJi Btmona C/2 / cy 7~H U T.. O IIOJIOJKHM, ana onpeneneHHOCTH, * AonycTHM, HTO 7~«a U 7~,ton . CneflOBarenfaHo, O .., (X t = oe us no a/inne ex. e- 7", • C/^ "^ ), C/?S A Torna Q: y<x ___ , ot voc .^ cyxb /-" -Btmoflbi HS 7^ l-cxvoc^ , ^r<- <x v PC ^. /x . Z 7 ot . /,* v/ no OA o^VQc^,..., OCVOC./ 7~ U 7^. KpoMe roro, Flo npeanonoKeHHio HHayKHHH <t>opMyna « v c x ^ ecxfa MenocpeacrBeHHoe OosTOMy CT^, • 7^)f("7^- T^Jh- Of VOc ( 7 ~' , - 7• 1 ) -t K 7' " - 7O ")i—ac Ha £>oc vac -*oc H, acneacrBHe neMMH, ( T."T nonynaeM ~T~ „) TeopeMa aoKasana. C/IEflCTBME 1 . P e u i e r K a £ " - n o r H K B H C T P H - 6 y T H B H a. ECJIH 7", H 7" - c -nomKH, r.e. aaMKHyTH OTHOCHTentno noncTanoBKH, TO Tf • Tg HOBKH. IIosTOMy peuieTKa npaBana H 7^+- 7^ TaKMte saMKHyrw OTHOCHTenfaHO noncTa £" -norHK HBjiaeTca nonpemexKofi T(£) H, cue O , FI y CT b M H O W e c T B a M H B e H H o. 7 ^ , 7 ^ - , 5 - T e o p H n 8 K C H O M T o r fl a flJIS K C H O M ^"*"^Z , /-? ^ C O O T B S T C T - M H o JK e c T T. + 7~ H a K C H O M a n a / ^ • 7\ yTBepwaeHHe Aanee, nio6aa <J>opMyna H3 /7 BbmoflHMa B 7"•f • 71 . S. *« 0 e -^-7- cnenyer, HTO i » £"(— oc v Oc-^Oc 7"(— oc7y VocS 7 H3 /? <X Ha H, KpOMe TOFO, f us /^ a ,6^, ..... 6^(~<x.') .. ecxb Bbiaoa /*? . B cany neMMbi, nonynaeM, HTO <X BWBO - /7 , HTO H TpeSosajiocb. Flo auanoruK c noHHTHeM Li V O c i. ociv e/7 x' . . Torna Oc HS H Til— oc « ? T -K- K3 . C npyroft croponbi, nycxb 7~ I— Ot H 71t~ Ot ; ot ..., oc^. (=oc) ecTb BbiBon ex c ^ M H O w e c T B O fljia /? /3 (J fla a s j i a e x c a ' a 533 AUCTpH6yTHBHa. C/IEflCTBHE 2. BOM E - reopHax onpenenaTfa noHame mero ace aKCHOMbi L £ -reopnn MW MOKOM ana nK)6ofl JIOFHKH L, -TeopHH KaK MHOJKecTBa (j>opMyn, conepxa - H aaMKHyxoro OTHOCHxenfaHo npaBiui (./LS) H C^2). SBMOTHM, HTO B flOKasaTejifaCTBe TeopeMbi MH nonfasoBaJiHCfa nnuifa cne— r\- F [oc: , TV F c«-i=> n(B) 7*1-0: V ( X = cnpaBennHBO yTBep>KneHHe: K o e a , H T O c a o f l c T B a M H p H '! jiK>6an ^ ( a ) — (r), e c n H - T e o p a a To n o r H x a j T a o S n a a a e r p e m e T K a // - x e o - 0 H C T p H 6 y T H B H a. B nacTHOcTii, OTC;o,n.a cneayer aaBecTHbifi pesynfaTax, HTO pemeTKa /-T<?opr<H, rue ^ - uriTyHmioHHCTCKaa noniKa, n pemeTKa cynepHHTymiHO - HHCTCKHX nOPHK nHCTpH6yTHBHbI. Mbt paccMOTpHM Tenept. HCHHcneHHe >K6CTBO (flOpMyjl, /7 . Kan noKaaaHO B Q6] , MHO- BblBOflHMblX B 3TOM HCHHCneHHH, COBHaAaeT C MHOJK6CTBOM 534 /I./I.MaKCHMOsa <jx>pMyji, BBIBOAHMUX a E . OflHaica /7 HMOST Sojifaiue HCXOAHWX npaBHn a, nosTOMy npencraejiaeT HHTe'pec Ta*;xce paccMOTpeane HaaoMM /7-TeopHeft f/23; TEOPEMA.. ate 7", £" - xaopHio /7 - 7" , KOTOpaa -T <xvf P e in e T K a /77 - T e o p a ft a H c T p H- 6 y T H B H a. flna noKaaarejifaCTBa nocraTOiHo pacumpHTb HOHHTHB Bbisoaa H F - BHBOfla npHMeHHrenfaHO K fj'n nposepHTi., ITO B n.K>6o& /7-reopHH npaBHJio c-ra.1 F Ipj ana npoH3Boni>Hoft Torna y^ftocfr^ oc VAjQ /7-TeopHH H 7" : 7~/-^°£<xJ H 7" J- ^[rocva]. r H X7 [JC^ &• 1 ^ ) V^] , T.K. E / - C t f e C T « V6^-*-C<X f t - 7 < X ) V 3 nerico npoBepnTfa, 7"/- -i-r(<x.ln<X)vf:'[p~l, 7"/- F{ft1 HaaoBeM OC : . £ h~ F ^ V.5] ~ v> KPOMS TOFO, 7"^- 7 (a & . / « ) . Oo C/Z.S.) nony - , HTO H rpe6oBanoct. £ — ( /7 ) -Teopaio 7~ n o n H o ft, ecnH BJIH jiK)6ofi 7" H OC<=>T IN 7 <X . o6paaoMflOKasbiBaexca,«rro jno6aa coBMecxHaa £ -re - opna MOaceT 6wTfa pacuiHpeHa no nojmoft. /lerno BUneTfa, HTO nojinaa £-teopHa flBJiaexcfl nonnofl no/maa /y'-reopHeft H o6parHO. B CBMOM flene, nycxfa 7~- £" -Teopna. Torna HS 7" A- « H 7"f- "7 ot V^fi cneayer, HTO 7~ P\ & : HHane Suno 6bi 7* H 7 ( T OC V/8J , BonpeKH HenpornBOpeHH BOOTH 7" . FIosToMy 7~f~S Aanee, nro6aa nonHtix . /7/ -xeopHH /7 -TeopHft, coaep^ainHx T npsflCTaBHMa KaK nepeceneuHe scex 7" . B CBMOM nene, ecna 7~ (\<x. , TO 7"f fi<x}-.HenpOTHBopeHHBoe MHOHCSCTBO, T.K. HS 7*"C/{7<xJ f— A&. 7^8 . cnenyeT (CM. fSJ ) T<- 77<X V71(y68-^fl)H, no npasuny (#3), STOMy 7" ^/ f 7 « x } MO>KHO paCUIHpHTfa flO HOJlHOJi OTcMfla cnenyST, HTO f -TeopHa 2. IloHaTHO, HTO cpeflH /7'-TeOpHH 7^, /fl N.QC. T asnaeTca nepeceneHHeM nonHbix TeopHfl Toraa H xontKO rorna, Koraa ona aanaeTca yxaaano B ^ 7 ] , cymecTByBST 7~h- Of. Ho - /7 -Teopneft. E -xeopaH, He aanaraniHeca £- KaK /7 x ~reopnaMH. A7 -norHK HaxoflHTCa H KnaccHnecKaa norffica. BosHHKaeT sonpoc, conepwHT siti KnaccHHecxaa nortnta see O E - xeopaax 535 npHMep, nro6aa norioca, KOropaa ecTt pacuucpeHHe HHTyniiHOHHCTCKOH, co — AepMurrca BHyxpH KjiaccHiecKOH. Sflecfa o6croHT Aeno HHaieu ECHH K aiccK OM8M n' MM AO6aBHK4 BC6 4>OpMyJIH BHfla 7 (7 (« ~*~fi) ~*~^ V"*~*^ KOTO - pwe He aanaioTca KrtaccuiecKH BUBOAHMBIMH , TO nonyiHM By» /7-jTormcy (H, cneAOBaxenbHO, HenpoTHBOpeiHByra HenpoTHBopeiHBOcTb ee nerxo noKasaTi., Hcnonfaaya Marpniiy, npeano— KeHHyro AnxepMaKOM £l] . Ha MHcrmecTBe KDTCH onepauHH oc 0 7ac 5 1 4 2 3 3 2 4 I 5 0 T r —"^ &• , V {0, 1, 2, 3, 4, 5/ onpenena— cnenyroiKHM o6pa3OM: ^ 0 0 3 3 3 3 3 3 1 0 3 3 0 3 3 2 0 0 3 0 0 3 0 0 0 3 3 3 4 0 0 0 0 3 3 5 0 0 0 0 0 1 2. 3 4 5 3 3 = max. Torna Bee tjiopMynw BHaa ( ^-+-a) npn Jiro6ofi HHTepnpeTauHH npHHHMarar sHaneHHa H3 MHOMcecTBa {O, 5} aHaieHHfl H3 {$, 8} ; , ^opMy™ BHUB i (<x-+-& ) -* T( ^ (ot-^-&)-*-( ^-*~Sl) nony«taeT ana^eHHe 5 . Qce ocTajtbHbie aKCHOMbi xaK>Ke npHKHMaror suaneHiia nnuifa K3 MHO«ecTaa (3, 4, 5} , n npaBHJia BHBOfla (.£,1), (Q.S.") , (fi3) coxpaHaroT STO csoft- CTBO. FIo3TOMy ace BbiBOAHMtie i})opMynbi nonyiaioT aHa'jeHHH HS MHoxcecTBa {3, 4, 5} , H, cneaoBaxenbHO, nocTpoeHHaa norHxa HenpoTHBopeiHBa. PaceMOTpHM , HaKOHeu, norHKy BM K £ cxeMbi axcHOM & , xoTopaa nonyiaeTca Ao6aBneBH- ac-*-C(«k-^-cx)»*-Oc^. 3ra jiormca y*e ne HMSOT HB- npoTHBOpeiHBwx pacuiHpeHHfi, tie coAepxamnxca B miaccH^ecKoft normce. TEOPEMA.. Jl K) 6 a a c o B M e c T H a a floKa>KeM cHanana ABC neMMti. OSosnaiHM nepes / 2 - j i o r H K a Ot,, ijiopwyny -^-a.) v 7 c 7 (o.-*-o. ;-*-ca.-*-o. ); . /I EMMA 1. C n e n y i o i U H e ij'opMyjiw 2) 77 Ct, A Q K a 536 3) OC , & ex0 r\ ~, 7) CX 9) ' -• 8) - CV 0 ; cx o &- ^ «.0 ~ ^ CX0 -, _ V «o - OCC, : 10) (OC.oa —^OC o *0 ) ~- CX * -, AOKASATEJlbCT! O r 4) i \ 7 0 0 V C\ 11) ~ •. . 12) CX0 V 7 CX 0 ~ CXO ; 7 OCo V 7 <X0 - 1 CKo , (Oo c„—"^Co X 0 )^~ ( Xe i• O6-.' HB'-HIM fl'ia Kp-vr KOCTH t— ~*tx-a~'-rp8).(-rp-*-a.). KpoMe roro, (Q.-*-O-) nepea ^a E\— £L&-(ci,-*~&)—*-£ vC. z f— ~*pSi-C1 p~*~p )—*-pv ~i (.1 p-*-p), T.e. E. f— 7<Xj-*- <Xo H, cne R.. ex— /a. a -- y ^ oc- ^ * y , KO T'\ny.n rb /loitaayeMOCTb B £" / T -7 OCp fla.neo, (a) /2- A- . . x -y , ^ y ^ a. v^— y , tjj° ; ' M yn 3) -10), KpOM" rorn, '- /- — CX o . /2 -1™ CXo-^-((Tcxc)—^cvo )-^-OLI). Ha /d /-CcXj-*~*(, /-~-'X,. Flo .rieMMo 1 (6) Hcno.nfa3yH /2 /- Coc-»-( A — 0 i <X.-*-/0>^ \}-*-Oh~ (0~*~ y '~*~ f a — ff-h- —< R. h- (.1 p^-p)-*-(L~ <2h ( -ip-~p) -~-(-7 /2 / — i ; o •->- p\ R. t- C 7yO —yD ' - — , ft '••• t ~o-*~/3> - * - < / : ' , • -( - -* ' ' ^ - ( C O E - TeopHax 537 [5] /2 f- fit(r) X2J- Ha (6) H (r) nojiynaeM /2 f— 0£-— ( C *-~*~ CX <,.' H H3 (a) (11). flanee, HS /2 f~ (oc^->-oco) ~ Or Oc.j«- "^ ) cneayer , T.e. ( 1 4 ) . C7oc -*-«,). KpoMe roro, f / «„) ~ a o . flanee, /2. /— CaQ-^ 7cxo ) — 7oco . KpoMe roro, H3 (14) A I— 7cxj*- (Ofj Oco ) , cJieflOBarenbHo, flOKaayeivia (jjopMyna (12). OycTfa renepfa <3"- npOHSBonbHaa HHTepnpeTauHH (jiopMyn B HOII 6yneBOM anre6pe. O6o3HannM nepea ot^ ({lOpMyny, KOTOpaa nonyiaerca us OL B pesynbTare noncTanoBKH 4)oPMyibl[ *„ BMOCTO nepeMeHaux.saa- neHHe KOTOpux ecTfa 1 npH yKasaHHofi HHTepnperauHH, u (jiopMynu BM6CTO nepeM6HHbIX, /1EMMA 2. B e a n n B O 3HaH6HMe KOTOpblX eCTfa flnfl n p e x a u H H - O ' ' <x T O /2 /— oc^. eciiH n e H n e £7 , T O . n w '6 o ft ( J j O p M y n b i o c c n p a cjienyicimee: ~- oco ; O i QC0 e c n n n p H H H M a e r oc B - H H T e p - 3 H a H e H H e 1, n p H H H M a e r a n a - Q. I- <x.0 ~ "oco , AeMMa aoKaabiBaerca HHnyKUHefi no anane ijiopMynbi ex. H ecri. He - nocpencTBeHHoe cneflCTBHe neMMbi 1. BoaspauiaeMca K floxaa arenfacTBy xe— opeMbi. Rycxfa (X - ijrapMyna, ne BfaiBOAiiMaa B KnaccHHecKOM HciHcnenHH. Torna cymecxByex TaKaH HHxepnpeTaUHa pe, HTO oc npHHitMaer aHaneHHe B CHny neMMbi 2, nonynaSM ecrt, /2 -norHKa u Ha /2 /- cco SoBajiocfa O ty B flByanaiHoft SyneBOft anre6— npH STOfi HHXepnpeTamm. /2 /— OL^. • ' oco . IIoaTOMy, eoia oceT , TO 'rf"c*tj-, cJieaoBaTejibHO, H 7" 3/2 nonyiaeM, ITO ~7~ .7" /— 7cxo 7" npoTHBOpeiHsa, ITO H rpe - 538 / i H r e . p a x y p a 1. W. ACKERMANN, Begrundung einer strengen Implikation, J. Symb. Logic , 21 (1 956 ) , 1 1 3-1 28. 2. A.R.AKDERSOH. N. D.BELNAP, The pure calculus of entailment. J. Symb.Logic,27(1962),19-52. 3. K. D.BELRAP, EQ. and tbe first order functional calculus. Zeitschr.f.math.Logik Grundl.Matb. ,6(1960), 217-218. 4. H. D.BELNAP, Intensional models for first degree formulas. J. 3ymb. Logic, 32(1%7), 1-22. 5. /I./I.MaKCHMosa, <t>opMajibHbie Bbisoabi B »c*wcnenmi cxporoS Anre6pa a. norHKa . CeManap, 5, M? 6 (1966), 33-39. b, fi.K. MEYEE, J. M. DUNN. Entailment logics and material implication. Notices A.M. S.15(1968),N7, 660-46. 7. R.K. MEYER, J.M.DUim, £, & and y . J. Symb. Logic 34(1969) ,460474. riocTynn.-io 24 fleKaSpa 1069 r.