7 2007 7 ACTA ELECTRONICA SINICA Vol. 35 No. 7 July 2007 1,2, 1,3 (1., 710062 ; 21, 710049 ; 31, 716000) :,,., gp, 2. : ; ; ; ; ; gp2 : O142 : A : 037222112 (2007) 0721333208 Geeralizatio of Fuda metal Theore m of Pro bability Lo gic WANG Guo2ju 1,2,HUI Xiao2jig 1,3 (1. Istitute of Mathematics, Shaaxi Normal Uiversity, Xi a, Shaaxi 710062, Chia ; 2. Research Ceter for Sciece, Xi a Jiaotog Uiversity, Xi a, Shaaxi 710049, Chia ; 3. College of Mathematics ad Computer Sciece, Ya Uiversity, Ya, Shaaxi 716000, Chia) Abstract : By itroducig the cocepts of geeratig states set ad geeratig probability a simple ad direct proof of the fudametal theorem of probability logic is proposed. Next, by itroducig the cocept of aturally merged probability the fuda2 metal theorem has bee geeralized. Moreover,the preset paper exteds the cocept of 2cosistecy degree of fiite logic theo2 ries to be the cocept of gp2cosistecy degree ad therefore certai relatioship betwee probability logic ad quatitative logic has bee obtaied. Key words : ucertaity ; geeratig states set ; geeratig probability ; aturally merged probability ; quatitative logic ; gp2co2 sistecy degree 1 { A 1,, A } A 3,, A 1 ( A 2 ( A A 3 ) ),,v A 1,, A, v A 3., { A 1,, A } A 3.,,,., 20 70,(,, [1 5 ])., (), (Ucertaity) (Degree of Essetialess)., 1 :2006206215 ; :2006212220 : (No. 10331010),.. 1 [1] { A 1,, A } A 3 A 3, U( A 3 ) e( A 1 ) U( A 1 ) + + e( A ) U( A ) (1) U( ) e( )., [1 ]1, [6 ], [7 ][8 ]. 1, 1 ( 5), A 3.,, ( [9,10 ]).,
1334 2007,, ( ) ( [ 11 16]),[17][ 18 ], (,, [1924]).,.,. 2 Kol2 mogorov ( [25 ]), ( ), [1 ],. 4 : K1. 0 P( A) 1. K2. A,P( A) = 1. K3. A B,P( A) P( B). K4. A B,. P( A B) = P( A) + P( B). A, B,, A B A B. F( S) ( [16 ]), F P : F [0,1 ]. K3,. 1 [1] A F, P : F [0,1 ]F, U( A) = 1 - P( A) (2) U( A) A P,. 2 [1] { A 1,, A } Α F, A 3 F, { A 1,, A } A 3, { A 1,, A } A 3, A 1,, A,A 3. { A 1,, A } A 3,,, A 3 { A 1,, A } { A i1,, A ik } ( k < )., [1 ] : 3 [1] { A 1,, A } A 3, F { A 1,, A },E F, A 3 F - E, E. A i F, ( A i ) F A i ( ), e ( A i ) = 1/ ( A i ),e( A i ) A i. A i,e( A i ) = 0. 4 [9] S = { q 1, q 2, }, F( S) S (, ), F( S). {0,1} 0 = 1, 1 = 0, a b = 0 a = 1 b = 0, { 0,1} (, ), (, ) v : F( S) { 0,1} F( S).,Π v v( A) = v( A), v( A B) = v( A) v( B), A, B F( S)., A B = A B, A B = ( A B),,{ 0,1} a b = max{ a, b}, a b = mi{ a, b},π v v( A B) = v( A) v( B), v( A B) = v( A) v( B). F( S) F, v : F( S) { 0,1} F v F.,[9 ], Π v, v S v S, F S F,v F v S F, v ( F). Π q S F, v ( q) = 1 v ( q) = 0,ΠA F, v( A) = 1 v( A) = 0., A = q q,π v,v ( A) = 1, v ( A) = 0., B = q q, v ( B) = 0,v ( B) = 1. 5 F = { A 1,, A }, v ( F), T( v) = ( v( A 1 ),, v ( A ) ) { 0,1} F. 1 (i) T( v) 0-1. (ii) v, T( v) { 0, 1} 0-1., A 1,, A,v T( v) = (1,,1). (iii) A 1,, A, v, T( v) { 0,1} 0-1. : 1 F = { A 1,, A }, F 2. 2 F 2. 1 { q, q r, r s, q r} q ( r s), q, r, s, A 1 = q, A 2 = q r, A 3 = r s, A 4 = q r, A 3 = q ( r s).
7 : 1335 F = { A 1, A 2, A 3, A 4 },F ( 1). 1 q q r r s q r q ( r s) T( v 1 ) 1 1 1 1 1 T( v 2 ) 0 1 1 0 0 T( v 3 ) 1 1 0 1 1 T( v 4 ) 0 1 0 0 0 T( v 5 ) 1 0 1 0? A 3 = q ( r s),, F 4, F 2 4 = 16. (0,0,1,1) F, 0 v( q) = 0, v ( q r) = 0, 4 1. F 1 5, 11.,F, A 3,, F T( v 5 ) = (1, 0,1,0). v( q) = 1 v( q r) = 0 v ( r) = 0. v( r s) = 1. v ( s) 0 1 v( r s) = 1. v( s) T ( v 5 ). v( s) = 1, v ( A 3 ) = v ( q ( r s) ) = 1, v( s) = 0, v ( r) = 0v( A 3 ) = 0. A 3,?., { q, r, s} (1,0,1) (1,0,0) F (1,0,1,0). 3 1,. 6 { A 1,, A } A 3, F = { A 1,, A },,F Auto( F) = { q 1,, q t },Auto( F) 2 t t t 0-1 u 1,, u 2. F T( v) Auto ( F) u i1,, u ik, { u i1,, u ik } T( v), f ( T( v) ) = { u i1,, u ik } (3) 7 { A 1,, A } A 3, F = { A 1,, A }, F m T ( v 1 ),, T( v m ). P T = { T( v 1 ),, T( v m ) } : P( T( v i ) ) = a i, m a i = 1 0 a i 1, i = 1,, m (4) i = 1 (i) F T ( v i ) v i ( A k ) = 1 a i A k T( v i ) P2, A k i2, P i ( A k ). v i ( A k ) = 0, a i A k T( v i ) P2, A k i2, U i ( A k )., v i ( A 3 ) = 1, a i A 3 T( v i ) P2,A 3 i2, P i ( A 3 ). v i ( A 3 ) = 0 v i ( A 3 ) =?, a i A 3 T( v i ) P2,A 3 i2, U i ( A 3 ). (ii) P T ( A k ) = { P i ( A k ) v i ( A k ) = v i ( A 3 ) = 1} (5) U T ( A k ) = { U i ( A k ) v i ( A k ) = 0 v i ( A 3 ) = 0 v i ( A 3 ) =?} (6) P T ( A 3 ) = { a i v i ( A 3 ) = 1} (7) U T ( A 3 ) = { a i v i ( A 3 ) = 0 v i ( A 3 ) =?} (8) P T ( A k ) U T ( A k ) A k P T2 T2, P T ( A 3 ) U T ( A 3 ) A 3 P T2T2. 2 2 F = { q, q r, r s, q r} P. 2 q q r r s q r q ( r s) P T( v 1 ) 1 1 1 1 1 0. 30 T( v 2 ) 0 1 1 0 0 0. 10 T( v 3 ) 1 1 0 1 1 0. 20 T( v 4 ) 0 1 0 0 0 0. 25 T( v 5 ) 1 0 1 0? 0. 15 2 : a 1 = 0130, a 2 = 0110, a 3 = 0120, a 4 = 0125, a 5 = 0115, P T ( A 1 ) = P T ( q) = a 1 + a 3 = 0150, U T ( A 1 ) = a 2 + a 4 = 0135 ; P T ( A 2 ) = P T ( q r) = a 1 + a 3 = 0150, U T ( A 2 ) = a 5 = 0115 ; P T ( A 3 ) = P T ( r s) = a 1 = 0130, U T ( A 3 ) = a 4 = 0125 ; P T ( A 4 ) = P T ( q r) = a 1 + a 3 = 0. 50, U T ( A 4 ) = a 2 + a 4 + a 5 = 0. 50. P T ( A 3 ) = P T ( q ( r s) ) = a 1 + a 3 = 0. 50, U T ( A 3 ) = a 2 + a 4 + a 5 = 0. 50. 3 7 2,, (i) A k (v( A k ) = 1) A k,, P( r s) = P 1 ( r s) + P 2 ( r s) + P 5 ( r s) = a 1 + a 2 + a 5 = 0155
1336 2007,, A k A k U( r s) = U 3 ( r s) + U 4 ( r s) = a 3 + a 4 = 0145. (ii) A 3, A 3,2, F T ( v 5 ) v( A 3 ), P( A 3 ), A 3. 2 P( A 3 ) P T ( A 3 ) = 0150. U( A 3 ) U T ( A 3 ) = 0150 (9) 2 { A 1,, A } A 3, F = { A 1,, A }, F T = { T ( v 1 ),, T( v m ) }, P T, (i) A k, P( A k ) = { P i ( A k ) v i ( A k ) = 1}, U( A k ) = { U i ( A k ) v i ( A k ) = 0}, P( A k) P T ( A k ), U( A k ) U T ( A k ), (ii) A 3, (10) k = 1,, (11) P( A 3 ) P T ( A 3 ), U( A 3 ) U T ( A 3 ) (12) P T ( A 3 ) P T ( A k ), k = 1,,. (13) (i), A k, A k,(10). (11) (5) (6). (ii) ( 12) (7) ( 8)., T( v j ) F. (5),v j ( A k ) = 1 v j ( A 3 ) = 1 a j P T ( A k ), v j ( A k ) = 0 v j ( A 3 ) = 1 a j P T ( A k ),(7) a j P T ( A 3 ), (13). F = { A 1,, A } P, (7) (8) P T ( A 3 ) U T ( A 3 ), (12) P( A 3 ) U ( A 3 ), 6 8. 8 { A 1,, A } A 3, F = { A 1,, A }, F T = { T ( v 1 ),, T( v m ) }. F Auto ( F) = { q 1,, q t },{ w 1,, w 2 t }. P F, gp Auto( F), f ( T( v) ) = { u i1,, u ik } k P( T( v) ) = gp( u ij ), T( v) T. (14) j = 1 gp P, P = G ( gp). 4 T P gp. (14), P T( v) gp. A 3 gp (, ). 3 2 2, 2 3 : 3 gp q r s q q r r s q r q ( r s) P 0. 30 1 1 1 T( v 1 ) 1 1 1 1 1 0. 30 b 1 0 0 0 b 2 0 0 1 b 3 0 1 1 T( v 2 ) 0 1 1 0 0 0. 10 0. 20 1 1 0 T( v 3 ) 1 1 0 1 1 0. 20 0. 25 0 1 0 T( v 4 ) 0 1 0 0 0 0. 25 b 4 1 0 1 b 5 1 0 0 T( v 5) 1 0 1 0? 1 0. 08 0 0. 07 0. 15 3, ( q, r, s) = (1,1,1) T ( v 1 ), P gp, gp(1,1,1) = P(1,1,1,1) = 0. 30. gp(1,1,0) = P(1,1,0,1) = 0. 20, gp(0,1,0) = P(0,1,0,0) = 0. 25. f ( T( v 2 ) ) = { (0,0,0), (0,0,1), (0,1,1) } gp (0,0,0), (0,0,1) (0,1,1), b 1 + b 2 + b 3 0110. gp (1,0, 1) (1,0,0) b 4 b 5 0115. gp P A 3 = q ( r s) P., b 4 b 5,, b 4 = 0. 08, b 5 = 0. 07, A 3 P( A 3 ) = P T ( A 3 ) + 0108 = 0158. A 3. : 1 { A 1,, A } A 3, F = { A 1,, A }, E = { A k1,, A ks } Α F. T( v), (i) v( A ki ) = 0, i = 1,, s. (ii) v ( A j ) = 1, A j F - E (iii) v( A 3 ) = 0 v( A 3 ) =?. e( A ki ) 1 s, i = 1,, s (15) (ii) (iii) 3 A 3 F - E, E, A ki
7 : 1337 ( A ki ) s, e ( A ki ) = 1/ ( A ki ) (15). 1 : 3 { A 1,, A } A 3, F = { A 1,, A }, T = { T ( v 1 ),, T ( v m ) } F, P T, A 3.,T( v i ) F, v i ( A 3 ) = 0 v i ( A 3 ) =?, A 1,, A v i, v i ( A k1 ) = = v i ( A ks ) = 0, v i ( A kj ) = 1, k j {1,, } - { k 1,, k s } 1 (15)., P ( T ( v i ) ) = a i, 7 U i ( A k1 ) = = U i ( A ks ) = a i, U i ( a 3 ) = a i (16) (15) (16) U i ( A 3 ) = 1 s U i ( A k1 ) + + 1 s U i ( A ks ) e ( A k1 ) U i ( A k1 ) + + e( A ks ) U i ( A ks ). (17) T ( v j ) F v j ( A 3 ) = 0 v j ( A 3 ) =?, (17), { A k1,, A ks }, (17) T( v i )., : v i ( A k ) = 1, U i ( A k ) = 0 ( P i ( A k ) = a i ), v i ( A k ) = 0, P i ( A k ) = 0 ( U i ( A k ) = a i ). (17) : U i ( A 3 ) e( A k ) U i ( A k ) (18) k = 1 I = { i 1 i m, v i ( A 3 ) = 0 v i ( A 3 ) =?} (19) (18) (19) U i ( A 3 ) i I k =1 (8) (6) U i ( A 3 ) = U T ( A 3 ), i I (20) (21) e( A k ) i I U i ( A k ) (20) U i ( A k ) = U T ( A k ) (21) i I U T ( A 3 ) e( A k ) U T ( A k ) (22) k = 1 (11) (12) U( A 3 ) e( A k ) U( A k ) (23) k = 1 (23). 3. 4 (23) 3, ( 23) U( A 3 ).. 9 { A 1,, A } A 3, F = { A 1,, A }. E = { A i1,, A ik } Α F. { A i1,, A ik } A 3,, E., A ij E2, e E ( A ij ) ( j = 1,, k). 10 { A 1,, A } A 3, F = { A 1,, A }, T = { T ( v 1 ),, T ( v m ) } F, P T. { A i1,, A ik } A 3, E{ A i1,, A ik }, T E = { T ( u 1 ),, T( u l ) } E. P E T E, : k 0-1 T( u ) = ( x i1,, x ik ) T E,T m 0-1 i j x ij ( j = 1,, k) T( v 1 ),, T( v t ), P E ( T( u ) ) = P( T( v 1 ) ) + + P( T( v t ) ) (24) P E P. 4 { q, r, s} q r. { q, r} q r. { q, r, s} T 2 3 = 8 3 0-1, P. 4 q r s q r P T( v 1 ) 0 0 0 0 0. 12 T( v 2 ) 0 1 0 0 0. 13 T( v 3 ) 1 0 0 0 0. 15 T( v 4 ) 1 1 0 1 0. 10 T( v 5 ) 0 0 1 0 0. 11 T( v 6 ) 0 1 1 0 0. 14 T( v 7 ) 1 0 1 0 0. 20 T( v 8 ) 1 1 1 1 0. 05 E = { q, r} 4, T E = { (0,0), (0, 1), (1,0), (1,1) } T (0,0,0) (0,0, 1) (0,0), P E (0,0) = P(0,0,0) + P(0,0,1) = 0. 12 + 0. 11 = 0. 23,, P E (0,1) = P(0,1,0) + P(0,1,1) = 0. 13 + 0. 14 = 0. 27, P E (1,0) = 0115 + 0120 = 035, P E (1,1) = 0. 10 + 0. 05 = 0. 15, P E P. 5.
1338 2007 5 q r q r P E T( u 1 ) 0 0 0 0. 23 T( u 2 ) 0 1 0 0. 27 T( u 3 ) 1 0 0 0. 35 T( u 4 ) 1 1 1 0. 15 4 () { A 1,, A } A 3, F = { A 1,, A }, T F, P T., { A i1,, A ik } A 3, E = { A i1,, A ik }, T E E, P E P., A ij, A ij P A ij P E ( j = 1,, k). A ij P E T E i j 0 k P E. (24) T i j 0 m 0-1 P. 1 (i) 4. 5, P E P,, U( A k ) A k, A k.,, F = { A 1,, A },E = { A i1,, A ik },,{ A i1,, A ik } A 3, U E ( A 3 ) A 3, 3 4 U E ( A 3 ) e E ( A i1 ) U( A i1 ) + + e E ( A ik ) U( A ik ) (25) U( A ij ) A ij. ( E) (25), E, (25) U E ( A 3 ) ( E) (26) F T = { T ( v 1 ),, T ( v m ) }, P T. P E P (10), gp P (8), P E gp P E., 4 A 3 gp P P E,(26) U( A 3 ) ( E) (27) F E 1,, E 3 : 5 { A 1,, A } A 3, F = { A 1,, A }, E 1,, E F, U( A 3 ) mi{ ( E 1 ),, ( E ) } (28), 5 1. 5, [9,10 ]. [9,10 ][16],: 11 A 3 = A 3 ( q 1,, q t ) t, A 3 ( x 1,, x t ) A 3 Boole A 3 :{0,1} t { 0,1}. ( A 3 ) A 3. ( A 3 ) = 1 2 t A 3-1 (1) (29), A 3 2 t, A 3-1 (1) A 3,, ( A 3 ) A 3., { A 1,, A } A 3, F = { A 1,, A }, T = { T( v 1 ),, T( v m ) } F, P T. Auto( F) = { q 1,, q t }, 6 T Auto( F) }. Auto ( F) } 2 t,,1/ 2 t. 8,P gp, (29) A 3. : 6 { A 1,, A } A 3, F = { A 1,, A }, T = { T ( v 1 ),, T ( v m ) } F, P T. P gp,a 3 : U( A 3 ) = 1 - ( A 3 ) = 1 2 t A 3-1 (0), ( A 3 ) = 1 - U( A 3 ). (30)., (30), A 3 U( A 3 ) A 1,, A,, { A 1,, A } A 3, gp A 1,, A. 3 (23)., 3,.
7 : 1339 7 { A 1,, A } A 3, F = { A 1,, A }, T = { T ( v 1 ),, T ( v m ) } F, T P gp.,,, e( A k ) 1 (1 - U( A k ) ( A 3 ) ), k = 1,, (31) P gp, 6 U( A 3 ) = 1 - ( A 3 ),3 (23) e( A k ) U( A k ) 1 - ( A 3 ) (32) k = 1 e( A 1 ) = = e( A ) U ( A 1 ) = = U ( A ) 0 (32) (31). gp gp2. 12 F = { A 1,, A }, A 3 = A 1 A, { A 1,, A } A 3. T = { T( v 1 ),, T( v m ) } F, P T, gp P, 1 - U( A 3 ) F gp2, gp ( F). gp2 [10 ] 2 : 8 F = { A 1,, A }, gp Auto( F),F gp2 F 2: gp ( F) = ( F) (33) 6 (30) gp ( F) = 1 - U( A 3 ) = ( A 3 ) (34),[10] ( F) = 1 - div ( F) = 1 - (1 - ( A 1 A ) ) = ( A 3 ). (35) (34) (35) (33). 8, 2 gp2 gp, gp2 2.,. 6.,,.,, gp, 2 gp gp2... : [1 ] Adams E W. A Primer of Probability Logic [ M ]. Staford : CSLI Publicatios,1998. [ 2 ] Hailperi T. Setetial Probability Logic [ M ]. Lodo : Associ2 ated Uiversity Presses,1996. [3] Coletti G,Scozzafava R. Probabilistic Logic i A Coheret Set2 tig[ M ]. Lodo : Kluwer Academic Publishers,2002. [ 4 ] Dubois D, Prade H. Possibility theory, Probability theory ad multiple2valued logics [J ]. Aals of Mathematics ad Artificial Itelligece,2001,32 (1-4) :35-66. [5 ] Baioletti M, Capotopti A, et al. Simplificatio rules for the co2 heret probability assessmet problem[j ]. Aals of Mathemat2 ics ad Artificial Itelligece,2002,35 (1-4) :11-28. [6 ] Adams E W, Levie H. O the ucertaities trasmitted from premises to coclusios i deductive ifereces [J ]. 1975,Sy2 these,30 (3) :429-460. [7 ] Nillso N J. Probability logic [J ]. Artificial Itelligece, 1986, 28 (1) :71-87. [8 ] Pearl J. Probabilistic Reasoig i Itelliget Systems [ M ]. Sa Mateo, Califoria :Morga Kaufma Publishers,1988. [9 ]. (I) [J ]., 2006, 23 (2) : 191-215. Wag G J. Quatitative logic ( I) [J ]. J oural of Egieerig Mathematics,2006,23 (2) :1-26. (i Chiese) [10 ]. () [ M ]. :,2006. [ 11 ] Wag G J, Leug Y. Itegrated sematics ad logic metric spaces [J ]. Fuzzy Sets ad Systems,2003,136 (1) :71-91. [12 ],,. [J ].,A,2001,31 (11) :998-1008. Wag G J, Fu L,Sog J S. Theory of truth degrees of proposi2 tios i two2valued logic[j ]. Sciece i Chia (Ser. A),2002, 45 (9) :1106-1116. (i Chiese) [13 ] Wag G J. Compariso of deductio theorems i diverse logic systems [ J ]. New Mathematics ad Natural Computatio, 2005,1 (1) :65-77. [14 ],. ukasiewicz [J ]. ( E ),2005,35 (6) :561-569. Li B J, Wag G J. Theory of truth degrees of formulas i ukasiewicz 2valued propositioal logic ad a limit theorem [J ]. Sciece i Chia (Ser. E), 2005, 48 (6) :727-736. (i
1340 2007 Chiese) [15 ],,. [J ].,2005,33 (1) :1-6. Wag G J, Qig X Y, Zhou X N. Theory of quasi2truth de2 grees of formulas i two2valued predict logic [J ]. J oural of Shaaxi Normal Uiversity,2005,33 (1) :1-6. (i Chiese) [16 ],. [J ].,2006,34 (2) :252-257. Wag G J, Sog J S. Graded method i propositioal logic [J ]. Acta Electroica Siica,2006,34 (2) :252-257. (i Chi2 ese) [17 ] Pavelka J. O fuzzy logic I. II. III[J ]. Z. Math Logic Grudla2 ge Math,1979,25 :45-52;119-134;447-464. [18 ] Yig M S. A logic for approximate reasoig [J ]. J oural of Symbolic Logic,1994,59 (3) :830-837. [19 ]. I [J ]. ( E ),1999,29 (1) :43-53. Wag G J. Fully implicatio Triple I method of fuzzy reaso2 ig[j ]. Sciece i Chia (Ser. E),1999,29 (1) :43-53. (i Chiese) [20 ] Wag G J, Fu L. Uified forms of Triple I method[j ]. Com2 puters & Mathematics with Applicatios,2005,49 :923-932. [21 ],. I [J ]. ( E ),2002,32 (2) :58-66. [22 ] Sog S J, Feg C B,Lee E S. Triple I method of fuzzy rea2 soig[j ]. Computers &Mathematics with Applicatios,2002, 44 (3) :1567-1579. [ 23 ] Pei D W, Wag G J. The extesios L 3 of formal systems L 3 ad their completeess [ J ]. Iformatio Scieces, 2003, 152 (2) :155-166. [24 ],,. I [J ].,2005,15 (4) :404-410. Peg J Y, Hou J,Li H X. Opposite directio Triple I method based o some commoly used implicatio operators [ J ]. Progress i Natural Sciece,2005,15 (4) :404-410. (i Chi2 ese) [ 25 ] Kolmogorov N. The Foudatio of Probability [ M ]. New : York : Chelsea Publishig Compay,1950.,1935 11,.. E2mail :gjwag @su. edu. c,1973 5,.. E2mail :xjhui @stu. su. edu. c