Abstract：The probabilistic truth degree functions of propositions in n-valued ukasiewicz propositional logic are abstracted as a modality,and their three basic identities are abstracted as axioms of the modality. Then,a modal logic system for reasoning about probabilistic truth degrees of propositions is proposed. The syntax and semantics are constructed. The completeness theorem with respect to probabilistic truth degree functions is finally obtained. A logic foundation is set for probabilistically quantitative logic.
[1] Fagin R,Halpern J Y,Megiddo N. A Logic for Reasoning about Probabilities. Information and Computation,1990,87(1/2): 78-128 [2] Adam E W. A Primer of Probability Logic. Stanford,USA: CSLI Publications,1998 [3] Mundici D. Averaging the Truth-Value in ukasiewicz Logic. Studia Logica,1995,55(1): 113-127 [4] Flaminio T,Godo L. A Logic for Reasoning about the Probability of Fuzzy Events. Fuzzy Sets and Systems,2007,158(6): 625-638 [5] Dubois D,Prade H. Possibilistic Logic: A Retrospective and Prospective View. Fuzzy Sets and Systems,2004,144(1): 3-23 [6] Kroupa T. Extension of Belief Functions to Infinite-Valued Events. Soft Computing,2012,16(11): 1851-1861 [7] Li Xiang,Liu Baoding. Foundation of Credibilistic Logic. Fuzzy Optimization and Decision Making,2009,8(1): 91-102 [8] Zhou Chunlai. Probability Logic of Finitely Additive Beliefs. Journal of Logic,Language and Information,2010,19(3): 247-282 [9] Wang Guojun,Fu Li,Song Jianshe. Theory of Truth Degrees of Propositions in Two-Valued Logic. Science in China: Series A,2002,45(9): 1106-1116 [10] Wang Guojun,Wang Wei. Logic Metric Spaces. Acta Mathematica Sinica,2001,44(1): 159-168 (in Chinese) (王国俊,王 伟.逻辑度量空间.数学学报,2001,44(1): 159-168) [11] Li Bijing,Wang Guojun. Theory of Truth Degrees of Formulas in ukasiewicz n-Valued Propositional Logic and a Limit Theorem. Science in China: Series F,2005,48(6): 727-736 [12] Wang Guojun,Leung Y. Integrated Semantics and Logic Metric Spaces. Fuzzy Sets and Systems,2003,136(1): 71-91 [13] Wang Guojun. Quantitative Logic (I). Chinese Journal of Engineering Mathematics,2006,23(2): 191-215 (in Chinese) (王国俊.计量逻辑学(I).工程数学学报,2006,23(2): 191-215) [14] Wang Guojun,Zhou Hongjun. Quantitative Logic. Information Sciences,2009,179 (3): 226-247 [15] Zhou Hongjun,Wang Guojun. Borel Probabilistic and Quantitative Logic. Science China: Series F,2011,54(9): 1843-1854 [16] Zhou Hongjun. Theory of Borel Probability Truth Degrees of Propositions in ukasiewicz Propositional Logic and a Limit Theorem. Journal of Software,2012,23(9): 2235-2247 (in Chinese) (周红军.ukasiewicz命题逻辑中命题的Borel概率真度理论和极限定理.软件学报,2012,23(9): 2235-2247) [17] Zhou Hongjun. Probabilistically Quantitative Logic.Xian,China: Shaanxi Normal University General Publishing House Co.Ltd,2009 (in Chinese) (周红军.概率计量逻辑.西安:陕西师范大学出版总社有限公司,2009) [18] Wang Guojun,Song Jianshe. Graded Method in Propositional Logic. Acta Electronica Sinica,2006,34(2): 252-257 (in Chinese) (王国俊,宋建社.命题逻辑中的程度化方法.电子学报,2006,34(2): 252-257) [19] Li Jun,Wang Guojun. Theory of Truth Degrees of Propositions in the Logic System L*n. Science in China: Series F,2006,49(4): 471-483 [20] Han Banghe,Wang Guojun. The Conditional Truth Degree of Formulas in Classical Propositional Logic.Fuzzy Systems and Mathematics,2007,21(4): 9-15 (in Chinese) (韩邦合,王国俊.二值逻辑中命题的条件真度理论.模糊系统与数学,2007,21(4): 9-15) [21] Wang Guojun,Hui Xiaojing. Randomization of Classical Inference Patterns and Its Application. Science in China: Series F,2007,50(6): 867-877 [22] Wu Hongbo. The Generalized Truth Degree of Quantitative Logic in the Logic System L*n (n-Valued NM-Logic System). Computers Mathematics with Applications,2010,59(8): 2587-2596 [23] Zhang Dongxiao,Li Lifeng. Syntactic Graded Method of Two-Valued Propositional Logic Formulas. Acta Electronica Sinica,2008,36(2): 325-330 (in Chinese) (张东晓,李立峰.二值命题逻辑公式的语构程度化方法.电子学报,2008,36(2): 325-330) [24] Liu Huawen,Wang Guojun,Zhang Chengyi. Approximate Reasoning in Several Logic Systems. Journal of Shandong University: Natural Science,2007,42(7): 1-6 (in Chinese) (刘华文,王国俊,张诚一.几种逻辑系统中的近似推理理论. 山东大学学报:理学版,2007,42(7): 1-6) [25] Zhou Xiangnan,Wang Guojun. Consistency Degrees of Theories in Some Systems of Propositional Fuzzy Logic. Fuzzy Sets and Systems,2005,152(2): 321-331 [26] Ma Lina,Wang Guojun. Formalized Theory of α-Triple I Solutions for Problems of Generalized Modus Ponens. Journal of Shaanxi Normal University: Natural Science Edition,2011,39(2): 13-16,22 (in Chinese) (马丽娜,王国俊.广义MP问题的α-三I解的形式化理论.陕西师范大学学报:自然科学版,2011,39(2): 13-16,22) [27] Zhou Hongjun,Wang Guojun. Generalized Consistency Degrees of Theories in Several Standard Complete Logic Systems. Fuzzy Sets and Systems,2006,157(15): 2058-2073 [28] Zhou Hongjun,Wang Guojun,Zhou Wei. Consistency Degrees of Theories and Methods of Graded Reasoning in n-Valued R0-Logic. International Journal of Approximate Reasoning,2006,43(2): 117-132 [29] Wang Guojun. A Unified Integrated Method for Evaluating Goodness of Propositions in Several Propositional Logic Systems and Its Applications. Chinese Journal of Electronics,2012,21(2): 195-201 [30] Li Bijing,Wang Guojun. Logic Pseudo-Metric Spaces of Regular Implication Operators. Acta Electronica Sinica,2010,38 (3): 497-502 (in Chinese) (李璧镜,王国俊.正则蕴涵算子所对应的逻辑伪度量空间.电子学报,2010,38 (3): 497-502) [31] Hu Mingdi,Wang Guojun. Distribution of the Symmetrical Logic Formulas in the Classical Logic Metric Space. Acta Electronica Sinica,2011,39(2): 419-423 (in Chinese) (胡明娣,王国俊.对称逻辑公式在经典逻辑度量空间中的分布.电子学报,2011,39(2): 419-423) [32] Zhou Hongjun,Wang Guojun. Characterizations of Maximal Consistent Theories in the Formal Deductive System L*(NM-Logic) and Cantor Space. Fuzzy Sets and Systems,2007,158(23): 2591-2604 [33] Zhou Hongjun. Topological Characterizations of Maximally Consistent Theories in the Formal Deductive System L*. Acta Electronica Sinica,2011,39(12): 2895-2899 (in Chinese) (周红军. 形式系统L*中极大相容逻辑理论的拓扑刻画. 电子学报,2011,39(12): 2895-2899) [34] Liu Lianzhen. States on Finite Monoidal t-norm Based Algebras. Information Sciences,2011,181(7): 1369-1383 [35] Zhou Hongjun,Zhao Bin. Generalized Bosbach and Riean States Based on Relative Negations in Residuated Lattices. Fuzzy Sets and Systems,2012,187(1): 33-57 [36] Hájek P,Godo L,Esteva F. Fuzzy Logic and Probability // Proc of the 11th Conference on Uncertainty in Artificial Intelligence.Montreal,Canada,1995: 237-244 [37] Godo L,Esteva F,Hájek P. Reasoning about Probability Using Fuzzy Logic. Neural Network World,2005,10(5): 811-824 [38] Hájek P. Complexity of Fuzzy Probability Logics II. Fuzzy Sets and Systems,2007,158(23): 2605-2611 [39] Wang Guojun,Zhou Hongjun. Introduction to Mathematical Logic and Resolution Principle. Beijing,China: Science Press,2009 [40] Gottwald S. A Treatise on Many-Valued Logics. Baldock,France: Research Studies Press,2001
2013, Vol. 26 
