Abstract:Based on evaluation state theory of residuated lattice and by defining probability measure in set of all evaluation states of residuated lattice and set of all formulate respectively,the probability truth degree of formula in residuated lattices semantics is introduced using the integral method,then the probability logic metric space is established in residuated lattices semantics,the approximate reasoning of quantitative logic methods have been entended to residuated lattices semantics,the feasible graded method of residuated lattices semantics is provided.
[1] PAVELKA J.On fuzzy logic I:Many-valued rules of inference;Ⅱ:Enriched residuated lattice and semanties of propositional calculi;Ⅲ:Semantical completeness of some many-valued propositional calculi[J].Zeitschrf Math Logik und Grundlagen der Math,1979,25:45-52;119-134;447-464.
[2] ADAMS E W.A Primer of Probability Logic[M].Stanford:CSLI Publication,1998.
[3] YING M S.A logic for approximate reasoning[J].Journal of Symbolic Logic,1994,59(3):830-837.
[4] DUBOIS D,PRADE H.Possibility theory and multiple-valued logics[J].Ann Math Artif Intell,2001,32(1):35-66.
[5] WANG G J,FU L,SONG J S.Theory of truth degrees of propositions in two valued logic[J].Science in China Series A Mathematics,2002,45(9):1106-1116.
[6] WANG G J,LEUNG Y.Integrated semantics and logic metric spaces[J].Fuzzy Sets & Systems,2003,136(1):71-91.
[7] ZHOU H J,WANG G J,ZHOU W.Consistency degrees of theories and methods of graded reasoning in n-valued R0 logic (NM-logic)[J].International Journal of Approximate Reasoning,2006,43(2):117-132.
[8] WANG G J,ZHOU H J.Quantitative logic[J].Information Sciences,2009,179(3):226-247.
[9] ZHOU H J.Introduction to Mathematical Logic and Resolution Principle[M].Beijing:Science Press,Oxford:Alpha Science International Limited,2009.
[10] 张东晓,李立峰.二值命题逻辑公式的语构程度化方法[J].电子学报,2008,36(2):320-325. ZHANG Dong-xiao,LI Li-feng.Syntactic graded method of two-valued propositional logic formulas[J].Acta Electronica Sinica,2008,36(2):325-330.(in Chinese)
[11] 罗敏霞,姚宁.L*系统中公式的语构程度化方法[J].电子学报,2011,39(2):424-428. LUO Min-xia,YAO Ning.Syntactic graded method of formulas in the system L*[J].Acta Electronica Sinica,2011,39(2):424-428.(in Chinese)
[12] ZHOU H J,WANG G J.Borel probabilistic and quantitative logic[J].Science China Information Sciences,2011,54(9):1843-1854.
[13] 王国俊.一类一阶逻辑公式中的公理化真度理论及其应用[J].中国科学:信息科学,2012,42(5):648-662. WANG Guo-jun.Axiomatic theory of truth degree for a class of first-order formulas and its application[J].Science China:Information Sciences,2012,42(5):648-662.(in Chinese)
[14] WANG G J.A unified integrated method for evalu-ating goodness of propositions in several propositional logic systems and its applications[J].Chinese Journal of Electronics,2012,21(2):195-201.
[15] 时慧娴,王国俊.多值模态逻辑的计量化方法[J].软件学报,2012,23(12):3074-3087. SHI Hui-xian,WANG Guo-jun.Quantitative method for multi-value modal logics[J].Journal of Software,2012,23(12):3074-3087.(in Chinese)
[16] 周红军,等.Lukasiewicz命题逻辑中命题的Choquet积分真度理论[J].电子学报,2013,23(3):557-563. ZHOU Hong-jun,et al.Theory of Choquet integral truth degrees of propositions in Lukasiewicz propositional logic[J].Acta Electronica Sinica,2013,23(3):557-563.(in Chinese)
[17] 折延宏,贺晓丽.粗糙逻辑中公式的Borel型概率粗糙真度[J].软件学报,2014,25(5):970-983. SHE Yan-hong,HE Xiao-li.Borel probabilistic rough truth degree of formulae in rough logic[J].Journal of Software,2014,25(5):970-983.(in Chinese)
[18] 吴洪博.Lukasiewicz命题逻辑中公式的Γ-真度理论和极限定理[J].中国科学:信息科学,2014,44(12):1542-1559. WU Hong-bo.The theory of Γ-truth degrees of formulas and limit theorem in ?ukasiewicz proposi-tional logic[J].Science China:Information Sciences,2014,44(12):1542-1559.(in Chinese)
[19] XU Y,RUAN D,QIN K Y,et al.Lattice-Valued Logic[M].Berlin Heidelberg:Springer-verlag,2003.
[20] 裴道武.强正则剩余格值逻辑系统LN及其完备性[J].数学学报,2002,45(4):745-752. PEI Dao-wu.A logic system based on strong regular residuated lattices and its completeness[J].Acta Mathematica Sinica,2002,45(4):745-752.(in Chinese)
[21] 傅丽,宋建社.经典命题逻辑的Boole语义理论[J].模糊系统与数学,2007,21(2):46-52. FU Li,SONG Jian-she.Theory of Boolean semantics of classical propositional logic[J].Fuzzy Systems and Mathematics,2007,21(2):46-52.(in Chinese)
[22] 左卫兵.Boole语义的程度化方法[J].电子学报,2012,40(3):441-447. ZUO Wei-bing.Graded method of Boolean semantics[J].Acta Electronica Sinica,2012,40(3):441-447.(in Chinese)
[23] 左卫兵.基于MV代数语义的格值逻辑的程度化方法[J].电子学报,2013,41(10):2035-2040. ZUO Wei-bing.Graded method of lattice-valued logic based on MV-algebra semantics[J].Acta Electronica Sinica,2013,41(10):2035-2040.(in Chinese)
[24] 左卫兵.MTL代数语义上逻辑公式的概率真度[J].电子学报,2015,43(2):293-298. ZUO Wei-bing.Probability truth degrees of formulas in MTL-algebras semantics[J].Acta Electronica Sinica,2015,43(2):293-298.(in Chinese)
[25] WARD M,et al.Residuated lattices[J].Transactions of the American Mathematical Society,1939,45(3):335-354.
[26] GALATOS N,JEPSEN P,et al.Residuated Lattices:An Algebraic Glimpse at Substructural Logics[M].Amsterdam:Elsevier,2007.
[27] 周红军.概率计量逻辑及其应用[M].北京:科学出版社,2015.
[28] 左卫兵.多值逻辑系统中公式的μ-真度理论[J].系统科学与数学,2011,31(7):879-892. ZUO Wei-bing.μ-truth degree of formula in many-valued propositional logic[J].Journal of Systems Science and Mathematical Sciences,2011,31(7):879-892.(in Chinese)