Generalizatio n of Funda mental Theore m of Pro bability Lo gic
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1 ACTA ELECTRONICA SINICA Vol. 35 No. 7 July ,2, 1,3 (1., ; 21, ; 31, ) :,,., gp, 2. : ; ; ; ; ; gp2 : O142 : A : (2007) 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 , Chia ; 2. Research Ceter for Sciece, Xi a Jiaotog Uiversity, Xi a, Shaaxi , Chia ; 3. College of Mathematics ad Computer Sciece, Ya Uiversity, Ya, Shaaxi , 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 : ; : : (No ),.. 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 ]).,
2 ,, ( ) ( [ 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).
3 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 ) T( v 2 ) T( v 3 ) T( v 4 ) T( v 5 ) ? 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 T2T F = { q, q r, r s, q r} P. 2 q q r r s q r q ( r s) P T( v 1 ) T( v 2 ) T( v 3 ) T( v 4 ) T( v 5 ) ? : 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 = 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 = ,, (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
4 ,, A k A k U( r s) = U 3 ( r s) + U 4 ( r s) = a 3 + a 4 = (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 ) = 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 ), { 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 (, ) , 2 3 : 3 gp q r s q q r r s q r q ( r s) P T( v 1 ) b b b T( v 2 ) T( v 3 ) T( v 4 ) b b T( v 5) ? , ( q, r, s) = (1,1,1) T ( v 1 ), P gp, gp(1,1,1) = P(1,1,1,1) = gp(1,1,0) = P(1,1,0,1) = 0. 20, gp(0,1,0) = P(0,1,0,0) = 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 gp (1,0, 1) (1,0,0) b 4 b 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 ) = 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
5 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 ) 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) (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 = , P. 4 q r s q r P T( v 1 ) T( v 2 ) T( v 3 ) T( v 4 ) T( v 5 ) T( v 6 ) T( v 7 ) T( v 8 ) 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. 23,, P E (0,1) = P(0,1,0) + P(0,1,1) = = 0. 27, P E (1,0) = = 035, P E (1,1) = = 0. 15, P E P. 5.
6 q r q r P E T( u 1 ) T( u 2 ) T( u 3 ) T( u 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), , [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 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) : [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) : [6 ] Adams E W, Levie H. O the ucertaities trasmitted from premises to coclusios i deductive ifereces [J ]. 1975,Sy2 these,30 (3) : [7 ] Nillso N J. Probability logic [J ]. Artificial Itelligece, 1986, 28 (1) : [8 ] Pearl J. Probabilistic Reasoig i Itelliget Systems [ M ]. Sa Mateo, Califoria :Morga Kaufma Publishers,1988. [9 ]. (I) [J ]., 2006, 23 (2) : 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) : [12 ],,. [J ].,A,2001,31 (11) : 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) : (i Chiese) [13 ] Wag G J. Compariso of deductio theorems i diverse logic systems [ J ]. New Mathematics ad Natural Computatio, 2005,1 (1) : [14 ],. ukasiewicz [J ]. ( E ),2005,35 (6) : 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) : (i
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