Approximate Reasoning Model Based on Probability Valuation of Propositional Logic
ZHANG Jia-Lu1, CHEN Xue-Gang2, WU Xia1
1.School of Mathematics and Finance, Xiangnan University, Chenzhou 423000 2.School of Software and Communication Engineering, Xiangnan University, Chenzhou 423000
Abstract：The value domain of proposition logic is extended from two values {0,1} to a probability space, and hence the concept of probability valuation of propositional formulas is introduced. Probability valuation is a generalization of classical propositional valuation and various truth degrees. Based on probability valuation, the concepts of probability truth degree, uncertainty degree, probability truth degree based on the set of all probability valuation of formulas on independent events are introduced. Grounded on the discussion of the properties of probability truth degree, probability truth degree satisfies Kolmogorov axioms on the entire set of propositional formulas. It is proved that the set of probability truth degrees of all formulas based on the set of all probability valuation on independent events has no isolated points in [0,1]. In the form of deduction in propositional logic, the uncertainty degree of conclusion is less than or equal to the sum of the product of uncertainty degree of each premise and its essentialness degree in a formal inference. Based on probability valuation, some concepts of a.e.conclusion, conclusion in probability and conclusion in probability truth of a formula set are introduced, and the relations between these concepts are discussed. Moreover, two different approximate reasoning models based on probability valuation are proposed.
[1] Hjek M. Metamathematics of Fuzzy Logic. Dordrecht, The Netherlands: Kluwer Academic Publishers, 1998 [2] Wang G J. Nonstandard Mathematical Logic and Approximate Reasoning. 2nd Edition. Beijing, China: Science Press, 2008 (in Chinese) (王国俊.非经典数理逻辑与近似推理.第2版.北京:科学出版社, 2008) [3] Wang P Z, Li H X. Fuzzy System Theory and Fuzzy Computer. Beijing, China: Science Press, 1996 (in Chinese) (汪培庄,李洪兴.模糊系统理论与模糊计算机.北京:科学出版社, 1996) [4] Artemov S N, Davoren J M, Nerode A. Modal Logics and Topological Semantics for Hybrid Systems. Technical Report, MSI97-05. Ithaca, USA: Cornell University, 1997 [5] Aiello M, Van Benthem J, Bezhanishvili G. Reasoning about Space: The Modal Way. Journal of Logic and Computation, 2003, 13(6): 889-920 [6] Pavelka J. On Fuzzy Logic I: Many-Valued Rules of Inference. Mathematical Logic Quarterly, 1979, 25: 45-52 [7] Pavelka J. On Fuzzy Logic II: Enriched Residuated Lattices and Semantics of Propositional Calculi. Mathematical Logic Quarterly, 1979, 25: 119-134 [8] Pavelka J. On Fuzzy Logic III: Semantical Completeness of Some Many-Valued Propositional Calculi. Mathematical Logic Quarterly, 1979, 25: 447-464 [9] Ying M S. A Logic for Approximate Reasoning. The Journal of Symbolic Logic, 1994, 59(3): 830-837 [10] Adams E W. A Primer of Probability Logic. Stanford, USA: Center for the Study of Language and Information, 1996 [11] Nillson N J. Probability Logic. Artificial Intelligence, 1986, 28(1): 71-87 [12] Wang G J, Hui X J. Generalization of Fundamental Theorem of Probability Logic. Acta Electronica Sinica, 2007, 35(7): 1333-1340 (in Chinese) (王国俊,惠小静.概率逻辑学基本定理的推广.电子学报, 2007, 35(7): 1333-1340) [13] Liu H L, Gao Q S, Yang B R. Two Levels of Equal Relations in Probabilistic Propositional Logic and Propositional Calculus. Philosophical Researches, 2009, 44(10): 113-120 (in Chinese) (刘宏岚,高庆狮,杨炳儒.概率命题逻辑中命题相等关系的两个层面与命题演算.哲学研究, 2009, 44(10): 113-120) [14] Liu H L, Gao Q S, Yang B R. Probabilistic Propositional Logic is the Event Semantic for Classical Formal System of Propositional Calculus. Journal of Chinese Computer Systems, 2011, 32(5): 978-982 (in Chinese) (刘宏岚,高庆狮,杨炳儒.概率命题逻辑是经典命题演算形式系统的随机事件语义.小型微型计算机系统, 2011, 32(5): 978-982) [15] Wang G J, Zhou H J. Quantitative Logic. Information Sciences, 2009, 179(3): 226-247 [16] Li B J, Wang G J. Theory of Truth Degrees of Formulas in Lukasiewicz-Valued Propositional Logic and a Limit Theorem. Science China: Information Sciences, 2005, 48(6): 727-736 [17] Wang G J, Fu L, Song J S. Theory of Truth Degrees of Propositions in Two-Valued Propositional Logic. Science in China: Series A, 2002, 45(9): 1106-1116 [18] Li J, Wang G J. Theory of α-Truth Degrees in n-Valued Gdel Propositional Logic. Journal of Software, 2007, 18(1): 33-39 (in Chinese) (李 骏,王国俊.Gdel n值命题逻辑中命题的α-真度理论.软件学报, 2007, 18(1): 33-39) [19] Wang G J, Hui X J. Randomization of Classical Inference Patterns and Its Application. Science China: Information Sciences, 2007, 50(6): 867-877 [20] Fu L, Song J S. Theory of Boolean Semantics of Classical Propositional Logic. Fuzzy Systems and Mathematics, 2007, 21(2): 46-52 (in Chinese) (傅 丽,宋建社.经典命题逻辑的Boole语义理论.模糊系统与数学, 2007, 21(2): 46-52) [21] Liu X L, Zhang J L. Random Truth Theory of Proposition Logic and Its Application. Pattern Recognition and Artificial Intelligence, 2013, 26(3): 231-241 (in Chinese) (刘晓玲,张家录.命题逻辑的随机真度理论及其应用.模式识别与人工智能, 2013, 26(3): 231-241) [22] Zhang J L, Chen X G, Zhou X D. Theory of Probability Semantics of Classical Propositional Logic and Its Application. Chinese Journal of Computers, 2014, 37(8): 1775-1785 (in Chinese) (张家录,陈雪刚,赵晓东.经典命题逻辑的概率语义及其应用.计算机学报, 2014, 37(8): 1775-1785) [23] Kolmogorov A N. The Foundation of the Theory of Probability. New York, USA: Chelsea Publishing Company, 1950 [24] Bogachev V I. Measure Theory. New York, USA: Springer-Verlag, 2006
2015, Vol. 28 
