The Theory of Membership Degree of Γ-Conclusion in Several n-valued Logic Systems *

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1 erca Joural of Operao eearch hp://ddoorg/046/ajor007 Publhed Ole Jue 0 (hp://wwwscporg/joural/ajor) The Theory of Meberhp Degree of Γ-Cocluo Several -Valued Logc Sye Jacheg Zhag Depare of Maheac Quazhou Noral Uvery Quazhou Cha Eal: zjcqz@6co eceved prl 4 0 reved May 8 0 acceped May 0 0 BSTCT Baed o he aaly of he propere of Γ-cocluo by ea of deduco heore copleee heore ad he heory of ruh degree of forula he pree paper roduce he cocep of he eberhp degree of forula a coequece of Γ (or Γ-cocluo) Łukaewcz -valued propooal logc ye Godel -valued propooal logc ye ad he 0 -valued propooal logc ye The codo ad relaed calculao of forula beg Γ-cocluo were dcued by ee ehod he ae e oe propere of eberhp degree of forula a Γ-cocluo were gve We provde algorh of he eberhp degree of forula a Γ-cocluo by he coruco of heory roo Keyword: N-Valued Propooal Logc Γ-Cocluo Theory oo Meberhp Degree Iroduco Fuzzy logc he heorecal foudao of fuzzy corol Spurred by he ucce applcao epecally fuzzy corol fuzzy logc ha aroued he ere of ay faou cholar a ere of pora reul have bee creaed docue [-5] For he ake of reaog we have o chooe a ube of well-fored forula whch ca reflec coe eeal propere a he ao of he logcal ye ad we he deduce he o-called -cocluo hrough oe reaoable ferece rule [6-9] So a aural queo he are: how o judge wheher or o a geeral forula a cocluo of a gve heory or o wha eed he forula a cocluo of? I bac proble o judge oe hg belog o oe kd arfcal ellgece well kow hua reaog approae raher ha prece aure we bac arg po o eablh graded vero of bac logcal oo I order o eablh a old foudao for fuzzy reaog profeor G J Wag propoed he cocep of roo of heory [] J C Zhag propoed he cocep of geeralzed roo of heory [0] propooal logc ye The graded decrpo ad propere of forula beg -cocluo were dcued d provde algorh of eberhp degree of forula a -cocluo by he coruco of heory roo he above-eoed logc ye The work wa uppored by he Scece ad Techology Ie of he Educao Depare of Fuja Provce of Cha (No 00J05) Prelare I well kow ha dffere plcao operaor ad valuao lace L (e he e of ruh degree for logc) deere dffere logc ye (ee []) Here valuao lace L 0 ad hree popularly ued plcao operaor ad he correpod g -or defed a follow: y L y y > y y a 0 y yl L y G ( y) y > y y ( y) yl G y 0 ( y) ( ) y > y y y > y yl 0 y Thee hree plcao operaor ad plcao operaor 0 G 0 L are called Łukaewcz plcao operaor G Ł Gödel plcao operaor ad he -plcao oper- aor repecvely The -or whch correpod o called alo Nlpoe Copyrgh 0 Sce JO

2 48 J C ZHNG Muor [6] If we f a -or above we he f a propooal calculu (whoe e of ruh value L ): ake for he ruh fuco of he rog cojuco he reduu of becoe he ruh fuco of he plcao operaor ad (0) he ruh fuco of he egao I ore deal we have he followg defo Defo [78] The propooal calculu PC( ) gve by a -or ha he e S of propooal varable p p ad coecve The e F( S ) of well-fored forula PC( ) defed ducvely a follow: each propooal varable a forula f B are forula he B ad B are all forula Defo [89] The foral deducve ye of PC( ) gve by correpodg o L G ad are called Łukaewcz -valued logc ye Ł 0 Gode l -valued logc ye G ad he 0 - valued logc ye L repecvely Defe he above-eoed logc ye : ad he correpodg algebra a ( ) L : a a a L where he -or defed o eark I eay o verfy ha he followg aero are rue: ( ) () G a a for every N ( a ) () () L a for every N ad ( ) () Ł a a ( ) 0 for every N Defo [78] () hooorph v: F ( S ) L of ype ( ) fro F( S) o he valuao lace Ł e v( ) v( ) v ( B) v ( ) v( B) v ( B) v ( ) vb ( ) called a -valuao of F( S ) The e of all -valuao wll be deoed by () forula FS ( ) called a auology wr f v v( ) hold eark [8] I o dffcul o verfy he above-eoed hree logc ye ha B a( ) ( B) ad B a( ) ( B) for every valuao Moreover oe ca check Ł ad L ha B ad ( B) are logcally equvale Defo 4 [8] ue ha p p p a forula geeraed by propooal varable p p p hrough coecve ad Subue for p ad keep he logc coecve uchaged bu epla he a he correpodg operaor defed o he valuao lace L The we ge a fuco : L L ad call he ruh degree fuco of () () Defo 5 [78] () ube of F( S ) called a heory () Le be a heory FS deduco of fro ybol a fe equece of forula uch ha for each a ao of L or or here are jk uch ha follow fro j ad k by MP Equvalely we ay ha a cocluo of (or -cocluo) The e of all cocluo of deoed by D( ) By a proof of we hall heceforh ea a deduco of fro he epy e We hall alo wre place of ad call a heore I eay for he reader o check he followg Propoo Propoo Le be a heory ad FS If he here e a fe ube of ay uch ha Theore (Geeralzed deduco heore) [7 8] Suppoe ha a heory B FS he () Ł B ff N () G B ff B () L B ff B B Defo 6 [8] Suppoe ha p p p a forula of F( S) coag aoc forula p p p ad be he ruh degree fuco of The 0 called he ruh degree of where B he cardal of e B Theore Suppoe ha p p p F ad he Ł ad G T( ) ff a auology e Proof ue ha T Sce he T( ) ) 0 By defe hu e ( ) he a Copyrgh 0 Sce JO

3 J C ZHNG 49 auology Coverely aue h a a auolog y e he 0 0 o T( ) Th coplee he proof Theore [8] Suppoe h a FS ( ) he Ł ( ) ff a auology e Theore 4 Suppoe ha B F( S ) If for every B he B Proof Suppoe ha p p p ad B Bp p p are all a forula of F( S ) coag aoc forula p p p follow fro B ha ad hece ( ) ( p ) ( p ) ( p ) ( B) B ( p ) ( p ) ( p ) B B B B () B () B B B 0 I eay o verfy ha () B () B B B he ( ) ( B) Propere of he oo of Theore Defo 7 [] Suppoe ha a heory D( ) If for every B D( ) we have B he called he roo of Theore 5 Suppoe ha a fe heory ay he () Ł () a roo of L a roo of () G a roo of Proof () I followg for referece [ 4] ha D( ) for every B D( ) here e N uch ha B by Theore I eay o ~ check ha by eark followg fro B B ha l l l where l l l hu B by Hypohecal h how ha a roo of () I followg for referece [4] ha D( ) for every BD( ) followg fro Theore ha B ce B ad B are provably equvale ad o B Th how a roo of () I fo llowg fro referece [4] ha D( ) for ever y BD( ) we ge B by Theore eay o verfy ha B C ad B C are provably equvale hece B ad B are provably equ- vale ad o B Th how ha a roo of 4 Meberhp Degree of Forula I Γ-Cocluo I followg le u fr ake a aaly o he co- d o of forula a -cocluo Ł Suppoe ha a heory ad a -cocluo follow fro Propoo ad Theore ha here e a fe rg of forula l ad l N uch ha l l l hold e he forula l a heore of Ł le B l l hece B a auology follow fro Theore ha B Coverely f here e a -cocluo B uch ha B he followg fro Copyrgh 0 Sce JO

4 50 J C ZHNG Theore ha B a auology hu B a heore of Ł e B hold ad B we have ha by MP e a -cocluo Moreover he l arger he eberhp degree of uch forula are he ore cloer o be -cocluo Hece aural ad reaoable for u ug he upreu of ruh degree of all forula wh he for B where BD( ) o eaure a -cocluo Defo 8 Suppoe ha a heory FS ( ) Defe B he T T up BD( ) called he eberhp degree of forula a -cocluo I eay o verfy ha 0 T ad followg Propoo by Defo 8 Propoo I Ł G ad L If a -cocluo he T Theore 6 I Ł G ad L f a fe heory ay h e a -cocluo ff T Proof The ecey par by propoo oly eceary o pr ove he uffcecy Le T For every uber >0 here e a forula B D( ) uch ha B > by Defo 8 () I L follow fro Theore 5 ha a roo of ad B hold Hece for every we have ( B ) follow fro propere of plcao operaor ha B > ce arbrary we have hu a auology ad a heore ogeher wh he reul he by MP e D( ) () I Ł o ce ha a roo of by Theore 5 hece he proof of () lar o ha he proof of () ad o oed () I G oce ha a roo of by Theore 5 hece he proof of () lar o ha he proof of () I fac B a heore by Defo 7 hece we have B ad B > hu hold he a heore ogeher wh he reul we have by MP The proof copleed Theore 7 Suppoe ha he () L T () Ł T () G T Proof () Sce a roo of by Theore 5 hece for every BD( ) we have B Thu for every ( B) ad B hold he B by Theore 4 I follow fro D( ) ha up B B D( ) e T () Noce ha Ł a roo of by T heore 5 he proof of () lar o ha he proof of () ad o oed () Noce ha G a roo of by Theore 5 he proof of () lar o ha he proof of ( ) ad o oed Theore 8 Suppoe ha a fe heory The () L T () Ł T up N N up () G T up N Proof () For every BD( ) followg fro Propoo ha here e a fe rg of forula uch ha B I fo- llow fro Theore ha B a heore hece B a auology by copleee heore ad for every ( ) B we have B by Theore 4 I followg for referece [4] ha D( ) he T up N () Noce ha Ł ~ by eark he Proof of () lar o ha he Proof of () Copyrgh 0 Sce JO

5 J C ZHNG 5 ad o oed () Noce ha G B ad B Provably equvale he Proof of () lar o ha he Proof of () ad o oed Theore 9 Suppoe ha a heory T ad T C he T C 0 Proof () If we ge 00 he T C 0 () If > we ge >0 ad >0 for ay gve pove uber uch ha >0 ad >0 here e forula B D( ) uch ha B > ad B C> I follow fro propere of egular plcao operaor ha B B B ad B B C B C I eay o verfy ha B B C ad B B C are provably equvale (e logcally equvale) hece B B C B B C I follow fro he heory of ruh degree of forula ad B B > B B C> ha B B B B C 0 Buca B B B B C ad C B B B B are provably equvale (e logcally equvale) hece B 0 B C B B D he eay o verfy ha ( ) T C 0 by he defo of he eberhp degree of forula Eaple Suppoe ha p p S p S I Ł L ad G copue T p repecvely Soluo ( ) I L aue ha p p p Sce ad hu ad We have hece 0 0 oherwe 0 oherwe 0 0 oherwe ad () 5 p p p 7 () () I G aue ha p p p Sce ad he Tp p p p hu he or or 0 oherwe oherwe ( ) p p p 5 () T p 8 () I L aue ha p p p Sce ad he hu 0 0 or 0 oherwe 0 0 oherwe p p p () Copyrgh 0 Sce JO

6 5 J C ZHNG 7 he Tp 8 Suppoe ha p p p p p S Ł copue T p Soluo () ue ha p p p p Sce ad Eaple 0 0 oherwe 0 0 or 0 oherwe 0 0 or or 0 oherwe 0 0 or or 0 oherwe ( ) p p p p a -cocluo () hu Tp p p p p EFEENCES he p [] H W Lu ad G J Wag Ufed For of Fully Iplcaoal erco Mehod for Fuzzy eaog Iforao Scece Vol 77 No 007 pp do:006/j [] J Pavelka O Fuzzy Logc II-Erched eduaed La- ce ad Seac of Propooal Calcul Zechrf für aheache Logk ud Grudlage der Maheak Vol 5 0 pp 9-4 [] G J Wag ad H Wag No-Fuzzy Vero of Fuzzy eaog Clacal Logc Iforao Scece Vol 8 No -4 0 pp -6 do:006/s (0)00- [4] G J Wag O he Logc Foudao of Fuzzy eaog Iforao Scece Vol 7 No pp do:006/s (98)00- [5] M S Yg Copace he Löwehe-Skole Propery ad he Drec Produc of Lace of Truh Value Zechrf für aheache Logk ud Grudlage der Maheak Vol 8 99 pp 5-54 [6] F Eeva ad L Godo Moodal -Nor Baed Logc: Toward a Logc for Lef-Couou -Nor Fuzzy Se ad Sye Vol 4 No 00 pp 7-88 do:006/s065-04(0) [7] P Hájek Meaaheac of Fuzzy Logc Kluwer cadec Publher Dordrech 998 [8] G J Wag ad H J Zhou Iroduco o Maheacal Logc ad eoluo Prcple d Edo Scece Cha Pre Bejg 006 ( Chee) [9] G J Wag Foral Deducve Sye for Fuzzy Propooal Calculu Chee Scece Bulle Vol 4 No pp 5-55 [0] J C Zhag Soe Propere of he oo of Theore Propooal Logc Sye Copuer ad Maheac wh pplcao Vol 55 No pp do:006/jcawa [] J C Zhag ad X Y Yag Soe Propere of Fuzzy eaog Propooal Fuzzy Logc Sye Iforao Scece Vol 80 No 00 pp do:006/j [] S Gowald Treae o May-Valued Logc Sude Logc ad Copuao eearch Sude Pre Baldock 00 [] G J Wag Theory of No-Clacal Maheacal Logc ad pproae eaog Scece Cha Pre Bejg 000 [4] D Dubo J Lag ad H Prade Fuzzy Se pproae eaog Fuzzy Se ad Sye Vol 40 No 99 pp 4-44 do:006/065-04(9)90050-z Copyrgh 0 Sce JO