若干经典命题逻辑问题的拓扑刻画
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摘要
数理逻辑的特点在于形式化和符号化，它和计算数学有着截然不同的风格，前者注重形式推理而后者注重数值计算；前者强调严格论证而后者允许近似求解，如果说数理逻辑具有刻板的一丝不苟的形象，那么计算数学具有灵活的张弛有度的特征．一个自然的问题是：能不能把数值计算的思想融入到数理逻辑当中以使其具有某种灵活性从而扩大其可能的应用范围呢?回答是肯定的．王国俊教授从基本概念的程度化入手，建立了一种计量逻辑学，从而对上述问题给出了肯定的回答．
计量逻辑学所涉及的逻辑系统包括经典的二值命题逻辑系统L，Lukasiewicz多值命题逻辑系统L_n与Luk以及命题演算系统L~*和L_n~*等．文献[4]在二值命题逻辑中，将重言式概念进行了程度化，引入了公式的真度概念，在此基础之上将逻辑等价概念程度化，引入了公式之间的相似度概念；并从而在L的全体公式集F(S)上引入了伪距离，得到了逻辑度量空间(F(S)，ρ)，并证明了逻辑连接词→，→和∨等关于ρ的连续性．另一方面，二值命题逻辑中理论的发散性与相容性等逻辑性质与它们在空间(F(S)，ρ)中的拓扑性质之间的联系如何?逻辑度量空间(F(S)，ρ)自身的细致结构如何?关于这些深层次的问题尚未及讨论，本文就上述问题进行研究．得到了如下结果：
(1)证明了F(S)中各理论的发散度充满了单位区间[0，1]．
(2)证明了F(S)中一个逻辑闭理论Γ是相容的当且仅当它在逻辑度量空间(F(S)，ρ)中不包含任一半径小于1的圆，从而我们容易得到F(S)中一个逻辑闭理论Γ是相容的当且仅当它在逻辑度量空间(F(S)，ρ)中不含内点．
(3)证明了一个逻辑理论Γ是全发散的当且仅当全体Γ结论之集D(Γ)在逻辑度量空间(F(S)，ρ)中稠密．
(4)证明了任一有限理论Γ的全体结论之集在逻辑度量空间(F(S)，ρ)中是闭集，从而推出了任一有根逻辑闭理论在逻辑度量空间(F(S)，ρ)中也为闭集．
(5)证明了逻辑度量空间(F(S)，ρ)是零维空间，证明了(F(S)，ρ)具有一种类似“樊畿”性质的“有限等球连通性”．即，对任一ε＞0，(F(S)，ρ)中任两点可用有限多个具有相同半径的ε-开球去连接．此外，本文还给出了逻辑度量空间(F(S)，ρ)中任一球面公式真度值的分布以及任一逻辑闭理论的拓扑性质刻画．
The characteristic of mathematical logic lies in symbolism, which is quite different from computational mathematics. The former pays attention to formal deduction and strict proof while the latter stresses numerical computation and approximate solution. If mathematical logic is said to be rigid, then computational mathematics is flexible. One may ask naturally: Can we extend the possible application of mathematical logic by implanting the thoughts of numerical computing into it so as to make it flexible? Professor Wang establishes a quantitative logic by grading the basic concept, and thus gives a positive solution to the questions above. The quantitative logic comprises two-valued propositional logic, the Lukasiewicz many valued logics L_n Luk and many valued logics L~* and L_n~*. The concept of truth degree of formulas in two-valued propositional logic system is proposed by grading the concept of tautologies, and the concepts of similarity degree and pseudometric among formulas are introduced therefrom. Then a logic metric space can be obtained. Furthermore, the continuity of the logic connectives with respect to the pseudo-metricρis proved. On the other hand, for any logic theory, what is the connection between its logic property(such as divergency, consistency) and topological property(such as density, interior point, completeness and the like)? The present paper mainly deals with the questions above, as far as I know, the relevant research has not been considered before. we list the main results as follows:
(1) The divergent degree of all the logic theories can fill the unit interval [0, 1];
(2) Any L-closed theoryΓis consistent if and only ifΓcontains no non-empty circle with positive radius less than 1, and thus the following conclusion is clear: An L-closed theory is consistent if and only ifΓcontains no interior point in logic metric space (F(S),ρ).
(3) A theoryΓis fully divergent if and only if D(Γ) is dense in (F(S),ρ).
(4) The logic conclusion of any finite theoryΓin F(S) is a closed set in (F(S),ρ), the same is true for any L-closed theory with root.
(5) The logic metric space (F(S),ρ) is of zero dimension, furthermore, it has the property of so called finite equal-ball connectedness. that is, any two points can be connected by finite equal balls with radius equal or less than any given positive numberε. Furthermore, the topological characterization of any L-closed theory is given.
计量逻辑学所涉及的逻辑系统包括经典的二值命题逻辑系统L，Lukasiewicz多值命题逻辑系统L_n与Luk以及命题演算系统L~*和L_n~*等．文献[4]在二值命题逻辑中，将重言式概念进行了程度化，引入了公式的真度概念，在此基础之上将逻辑等价概念程度化，引入了公式之间的相似度概念；并从而在L的全体公式集F(S)上引入了伪距离，得到了逻辑度量空间(F(S)，ρ)，并证明了逻辑连接词→，→和∨等关于ρ的连续性．另一方面，二值命题逻辑中理论的发散性与相容性等逻辑性质与它们在空间(F(S)，ρ)中的拓扑性质之间的联系如何?逻辑度量空间(F(S)，ρ)自身的细致结构如何?关于这些深层次的问题尚未及讨论，本文就上述问题进行研究．得到了如下结果：
(1)证明了F(S)中各理论的发散度充满了单位区间[0，1]．
(2)证明了F(S)中一个逻辑闭理论Γ是相容的当且仅当它在逻辑度量空间(F(S)，ρ)中不包含任一半径小于1的圆，从而我们容易得到F(S)中一个逻辑闭理论Γ是相容的当且仅当它在逻辑度量空间(F(S)，ρ)中不含内点．
(3)证明了一个逻辑理论Γ是全发散的当且仅当全体Γ结论之集D(Γ)在逻辑度量空间(F(S)，ρ)中稠密．
(4)证明了任一有限理论Γ的全体结论之集在逻辑度量空间(F(S)，ρ)中是闭集，从而推出了任一有根逻辑闭理论在逻辑度量空间(F(S)，ρ)中也为闭集．
(5)证明了逻辑度量空间(F(S)，ρ)是零维空间，证明了(F(S)，ρ)具有一种类似“樊畿”性质的“有限等球连通性”．即，对任一ε＞0，(F(S)，ρ)中任两点可用有限多个具有相同半径的ε-开球去连接．此外，本文还给出了逻辑度量空间(F(S)，ρ)中任一球面公式真度值的分布以及任一逻辑闭理论的拓扑性质刻画．
The characteristic of mathematical logic lies in symbolism, which is quite different from computational mathematics. The former pays attention to formal deduction and strict proof while the latter stresses numerical computation and approximate solution. If mathematical logic is said to be rigid, then computational mathematics is flexible. One may ask naturally: Can we extend the possible application of mathematical logic by implanting the thoughts of numerical computing into it so as to make it flexible? Professor Wang establishes a quantitative logic by grading the basic concept, and thus gives a positive solution to the questions above. The quantitative logic comprises two-valued propositional logic, the Lukasiewicz many valued logics L_n Luk and many valued logics L~* and L_n~*. The concept of truth degree of formulas in two-valued propositional logic system is proposed by grading the concept of tautologies, and the concepts of similarity degree and pseudometric among formulas are introduced therefrom. Then a logic metric space can be obtained. Furthermore, the continuity of the logic connectives with respect to the pseudo-metricρis proved. On the other hand, for any logic theory, what is the connection between its logic property(such as divergency, consistency) and topological property(such as density, interior point, completeness and the like)? The present paper mainly deals with the questions above, as far as I know, the relevant research has not been considered before. we list the main results as follows:
(1) The divergent degree of all the logic theories can fill the unit interval [0, 1];
(2) Any L-closed theoryΓis consistent if and only ifΓcontains no non-empty circle with positive radius less than 1, and thus the following conclusion is clear: An L-closed theory is consistent if and only ifΓcontains no interior point in logic metric space (F(S),ρ).
(3) A theoryΓis fully divergent if and only if D(Γ) is dense in (F(S),ρ).
(4) The logic conclusion of any finite theoryΓin F(S) is a closed set in (F(S),ρ), the same is true for any L-closed theory with root.
(5) The logic metric space (F(S),ρ) is of zero dimension, furthermore, it has the property of so called finite equal-ball connectedness. that is, any two points can be connected by finite equal balls with radius equal or less than any given positive numberε. Furthermore, the topological characterization of any L-closed theory is given.
引文
[1] ROSSER J B,TURQUETTE A R.Many-valued Logics[M].Amsterdam:North-Holland, 1952.
[2] PAVELKA J. On fuzzy logic Ⅰ,Ⅱ,Ⅲ[J].Zeitschr f Math Logik u Grundlagen d Math, (9)9, 25:45-52, 119-134, 447-464.
[3] YING MINGSHENG. A logic for approximate reasoning[J].Symbolic Logic, 1994, 59:830-837.
[4] 王国俊．计量逻辑学(Ⅰ)[J]．工程数学学报，2006，23(2)：191-215．
[5] 王伟，王国俊．论Godel蕴涵算子不宜用于建立模糊逻辑系统[J]．模糊系统与数学，2005，19(2)：14-18．
[6] 王国俊，傅丽，宋建社．二值命题逻辑中命题的真度理论[J]．中国科学，A辑，2001，31(11)，998-1008．
[7] 王国俊，数理逻辑引论与归结原理(第二版)[M]．北京：科学出版社，2006．
[8] 邱玉辉，张为群．自动推理导论[M]．成都：电子科技大学出版社，1992．
[9] P HAJEK.Metamathematics of Fuzzy Logic[M].Dordrecht:Kluwer,1998.
[10] 王国俊．非经典数理逻辑与近似推理[M]．北京：科学出版社，2003．
[11] ENGELKING R. General Topology[M]. Berlin: Heldermann Verlag, 1989.
[12] WANG GUOJUN, WANG WEI. Theory of logic metric spaces [J].Acta Mathematica Sinica,2001,44(1):159-168(in Chinese).
[13] 王国俊，紧Hausdorff测度空间上的Riemann积分理论[J]．数学学报，2004，6：1041-1052．
[14] WANG L X. A Course in Fuzzy Systems and Control[M].Upper Saddle River:Prentice Hall PTR,1997.
[15] WANG GUOJUN,LEUNG YI.Intergrated semantics and logic spaces[J].Fuzzy Sets and Systems,2003,136(1):71-91.
[16] ELKENC.The paradoxical controvercy over fuzzy logic[J].IEEE Expert,1994,9(4):47-49.
[17] WATKINS F A. False controvercy:fuzzy and non-fuzzy faus pas[J].IEEE Expert,1995,10(4):4-5.
[18] 吴望名．关于模糊逻辑的一场争论[J]．模糊系统与数学，1995，9(2)：1-9．
[19] 李洪兴．从模糊控制的数学本质看模糊逻辑的成功[J]．模糊系统与数学，1995，9(4)：1-14．
[20] 王国俊．模糊推理与模糊逻辑[J]．系统工程学报，1998，13(2)：1-16．
[21] 王国俊．Fuzzy推理的全蕴涵三Ⅰ算法[J]．中国科学，E辑，1999，29(1)：43-53．
[22] WANG G J. On the logic foundation of fuzzy reasoning[J].Information Sciences,1999,177(1):47-88.
[23] 宋士吉，冯纯伯，吴从忻．关于模糊推理的全蕴涵三Ⅰ算法的约束度理论[J]．自然科学进展，10(2000)，10：884-889．
[24] WANG SHIQIANG, MENG XIAOQING. Applications of Mathematical Logic and Category Theory[M]. Beijing:Beijing Normal University Press, 1999(in Chinese).
[25] WANG GUOJUN,QIN XIAOYAN,ZHOU XIANGNAN.An intrinsic fuzzy set on the universe of discourse of predicate formulas[J]. Fuzzy Sets and Systems, 2006,157(2):3145-3158.
[26] LI BIJING, WANG GUOJUN. Theory of truth degrees of formulas in Lukasiewicz n-valued propositional logic and a limit theorem [J].Science in China, set.F,2005,48(6):727-736.
[27] 熊金成，点集拓扑讲义[M]．北京：人民教育出版社，1982．
[28] 王国俊，宋庆燕．一种新型的三Ⅰ算法及其逻辑基础 [J]．自然科学进展，2003，13(6)：575-581．
[29] PEI DAOWU. Subalgebras and generalized tautologies in multi-valued logic systems[J]. Journal of Shaanxi Normal University, 2000,28(2):18-22.
[30] WANG GUOJUN, FU LI. Unified forms of Triple I method[J].Computer & Mathematics with Applications, 2005,49(6):923-932.
[31] WANG GUOJUN. Formalized theory of general fuzzy reasoning[J]. Information Sciences, 2004,160(3):251-266.
[32] 李骏，黎锁平，夏亚峰．n值Lukasiewicz逻辑中命题的真度理论[J]．数学学报，2004，47(4)：769-780．
[33] 李骏，王国俊．逻辑系统L_n~*中命题的真度理论[J]．中国科学，2006，36(6)：631-643．
[34] 裴道武．模糊全蕴涵算法及其还原性[J]．数学研究与评论，2004，24(2)：359-368．
[35] 王国俊．三Ⅰ方法与区间值模糊推理[J]．中国科学(E辑)，2000，30(4)：331-340．
[36] HALMOS P R.Measure Theory[M].Springer-Verlag,1974.
[37] PEI DAOWU, WANG GUOJUN. Extension L_n~* of the formal system L~* and its completeness[J]. Science in China, set.F, 2003,46(2):81-98.
[38] 彭家寅，侯健，李洪兴．基于某些常见蕴涵算子的反向三Ⅰ算法[J]．自然科学进展，2005，15(4)：404-410．
[39] 苏忍锁，王国俊．RL型蕴涵与Fuzzy推理的三Ⅰ算法[J]．陕西师范大学学报，2004，32(3)：7-11．
[40] 傅丽，王国俊．三Ⅰ算法的统一形式[J]．陕西师范大学学报，2004，32(3)：12-17．
[41] SONG S,FENG C,LEE E S.Triple I method of fuzzy reasoning[J]. Comput. Math. Appl. 2002,44(12): 1567-1579.
[42] 吴洪博．修正的Kleen系统中的广义重言式理论[J]．中国科学，E辑，2002，32(2)：224-229．
[43] WANG G J, WANG H. Non-fuzzy versions of fuzzy reasoning in classical logics[J].Information Sciences,2001,138(3):211-236.
[44] ZHOU HONGJUN,WANG GUOJUN.A new theory consistency index based on deduction theories in several logic systems[J].Fuzzy Sets and Systems, 2006,157(3):427-443.
[45] 王国俊．广义MP规则[J]．陕西师范大学学报，2000，28(3)：1-9．
[46] ZHOU XIANGNAN, WANG GUOJUN. Consistency degrees of theories in some systems of propositional fuzzy logic[J]. Fuzzy Sets and Systems, 2005,152(3):321-331.
[47] NALAD,LETTIERI A.On normal forms in Lukasiewicz logic[J].Arch.Math. logic.2004,43(6):795-823.
[48] AGUZZOLISS.The complexity of McNaughton foundations of one variable[J].Adv.Appl.Math.1998,21(1):58-77.
[49] CIGNOLI R,D'OTTAVIANO I,MUNDICI D. Algebraic Foundation of Many-valued Reasoning[M].Dordrecht:Kluwer Academic Publishers,2000.
[50] 杨晓斌，张文修．Lukasiewicz系统中的广义重言式理论[J]．陕西师范大学学报，1998，26(4)：6-9．
[2] PAVELKA J. On fuzzy logic Ⅰ,Ⅱ,Ⅲ[J].Zeitschr f Math Logik u Grundlagen d Math, (9)9, 25:45-52, 119-134, 447-464.
[3] YING MINGSHENG. A logic for approximate reasoning[J].Symbolic Logic, 1994, 59:830-837.
[4] 王国俊．计量逻辑学(Ⅰ)[J]．工程数学学报，2006，23(2)：191-215．
[5] 王伟，王国俊．论Godel蕴涵算子不宜用于建立模糊逻辑系统[J]．模糊系统与数学，2005，19(2)：14-18．
[6] 王国俊，傅丽，宋建社．二值命题逻辑中命题的真度理论[J]．中国科学，A辑，2001，31(11)，998-1008．
[7] 王国俊，数理逻辑引论与归结原理(第二版)[M]．北京：科学出版社，2006．
[8] 邱玉辉，张为群．自动推理导论[M]．成都：电子科技大学出版社，1992．
[9] P HAJEK.Metamathematics of Fuzzy Logic[M].Dordrecht:Kluwer,1998.
[10] 王国俊．非经典数理逻辑与近似推理[M]．北京：科学出版社，2003．
[11] ENGELKING R. General Topology[M]. Berlin: Heldermann Verlag, 1989.
[12] WANG GUOJUN, WANG WEI. Theory of logic metric spaces [J].Acta Mathematica Sinica,2001,44(1):159-168(in Chinese).
[13] 王国俊，紧Hausdorff测度空间上的Riemann积分理论[J]．数学学报，2004，6：1041-1052．
[14] WANG L X. A Course in Fuzzy Systems and Control[M].Upper Saddle River:Prentice Hall PTR,1997.
[15] WANG GUOJUN,LEUNG YI.Intergrated semantics and logic spaces[J].Fuzzy Sets and Systems,2003,136(1):71-91.
[16] ELKENC.The paradoxical controvercy over fuzzy logic[J].IEEE Expert,1994,9(4):47-49.
[17] WATKINS F A. False controvercy:fuzzy and non-fuzzy faus pas[J].IEEE Expert,1995,10(4):4-5.
[18] 吴望名．关于模糊逻辑的一场争论[J]．模糊系统与数学，1995，9(2)：1-9．
[19] 李洪兴．从模糊控制的数学本质看模糊逻辑的成功[J]．模糊系统与数学，1995，9(4)：1-14．
[20] 王国俊．模糊推理与模糊逻辑[J]．系统工程学报，1998，13(2)：1-16．
[21] 王国俊．Fuzzy推理的全蕴涵三Ⅰ算法[J]．中国科学，E辑，1999，29(1)：43-53．
[22] WANG G J. On the logic foundation of fuzzy reasoning[J].Information Sciences,1999,177(1):47-88.
[23] 宋士吉，冯纯伯，吴从忻．关于模糊推理的全蕴涵三Ⅰ算法的约束度理论[J]．自然科学进展，10(2000)，10：884-889．
[24] WANG SHIQIANG, MENG XIAOQING. Applications of Mathematical Logic and Category Theory[M]. Beijing:Beijing Normal University Press, 1999(in Chinese).
[25] WANG GUOJUN,QIN XIAOYAN,ZHOU XIANGNAN.An intrinsic fuzzy set on the universe of discourse of predicate formulas[J]. Fuzzy Sets and Systems, 2006,157(2):3145-3158.
[26] LI BIJING, WANG GUOJUN. Theory of truth degrees of formulas in Lukasiewicz n-valued propositional logic and a limit theorem [J].Science in China, set.F,2005,48(6):727-736.
[27] 熊金成，点集拓扑讲义[M]．北京：人民教育出版社，1982．
[28] 王国俊，宋庆燕．一种新型的三Ⅰ算法及其逻辑基础 [J]．自然科学进展，2003，13(6)：575-581．
[29] PEI DAOWU. Subalgebras and generalized tautologies in multi-valued logic systems[J]. Journal of Shaanxi Normal University, 2000,28(2):18-22.
[30] WANG GUOJUN, FU LI. Unified forms of Triple I method[J].Computer & Mathematics with Applications, 2005,49(6):923-932.
[31] WANG GUOJUN. Formalized theory of general fuzzy reasoning[J]. Information Sciences, 2004,160(3):251-266.
[32] 李骏，黎锁平，夏亚峰．n值Lukasiewicz逻辑中命题的真度理论[J]．数学学报，2004，47(4)：769-780．
[33] 李骏，王国俊．逻辑系统L_n~*中命题的真度理论[J]．中国科学，2006，36(6)：631-643．
[34] 裴道武．模糊全蕴涵算法及其还原性[J]．数学研究与评论，2004，24(2)：359-368．
[35] 王国俊．三Ⅰ方法与区间值模糊推理[J]．中国科学(E辑)，2000，30(4)：331-340．
[36] HALMOS P R.Measure Theory[M].Springer-Verlag,1974.
[37] PEI DAOWU, WANG GUOJUN. Extension L_n~* of the formal system L~* and its completeness[J]. Science in China, set.F, 2003,46(2):81-98.
[38] 彭家寅，侯健，李洪兴．基于某些常见蕴涵算子的反向三Ⅰ算法[J]．自然科学进展，2005，15(4)：404-410．
[39] 苏忍锁，王国俊．RL型蕴涵与Fuzzy推理的三Ⅰ算法[J]．陕西师范大学学报，2004，32(3)：7-11．
[40] 傅丽，王国俊．三Ⅰ算法的统一形式[J]．陕西师范大学学报，2004，32(3)：12-17．
[41] SONG S,FENG C,LEE E S.Triple I method of fuzzy reasoning[J]. Comput. Math. Appl. 2002,44(12): 1567-1579.
[42] 吴洪博．修正的Kleen系统中的广义重言式理论[J]．中国科学，E辑，2002，32(2)：224-229．
[43] WANG G J, WANG H. Non-fuzzy versions of fuzzy reasoning in classical logics[J].Information Sciences,2001,138(3):211-236.
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