基于BL系统的演绎系统集代数的剩余格属性
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摘要
在命题逻辑系统BL中提出了演绎系统的概念,并且给出了由F(S)的子集生成演绎系统的方法,证明了命题逻辑系统BL中演绎系统和结论之集的同一性;其次,在命题逻辑系统BL中证明了在包含偏序关系下所有演绎系统构成的集族D(F)中上,下确界的存在性;最后,在D(F)中定义了二元运算∧,∨,*,→,证明了集代数(D(F),∧,∨,*,→,0,1)是满足可除性的完备剩余格。
Firstly,the concept of deduction system of the basic propositional logic system BL is defined,and the method generating deduction system from a subset of formula set F( S) is obtained,and the identity of deduction system and conclusions set is proved. Secondly,in propositional logic system BL the existence of supremum and infimum is proved in family D( F) consisting of all deductive systems with partial order relation of =; Moreover,binary operations *,→ are defined in family D( F),and it is proved that set algebra( D( F),∧,∨,*,→,⊥,F( S)) is a complete residuated lattice satisfying divisibility.
Firstly,the concept of deduction system of the basic propositional logic system BL is defined,and the method generating deduction system from a subset of formula set F( S) is obtained,and the identity of deduction system and conclusions set is proved. Secondly,in propositional logic system BL the existence of supremum and infimum is proved in family D( F) consisting of all deductive systems with partial order relation of =; Moreover,binary operations *,→ are defined in family D( F),and it is proved that set algebra( D( F),∧,∨,*,→,⊥,F( S)) is a complete residuated lattice satisfying divisibility.
引文
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[13]吴洪博,寇海燕.余Frame范畴中余积的构造[J].数学学报(中文版),2016,59(1):75-90.WU Hongbo,KOU Haiyan.The construction of Co-Product in the category of Co-Frame[J].Acta M athematica Sinica(Chinese Series),2016,59(1):75-90.
[14]TURUNEN E.Mathematics behind fuzzy logic[M].Heidelberg:Physica-Verlag,1999.
[15]刘春辉.有界Heyting代数的模糊LI-理想[J].工程数学学报,2016,33(4):391-401.LIU Chunhui.Fuzzy LI-ideals in bounded Heyting algebras[J].Chinese Journal of Engineering M athematics,2016,33(4):391-401.
[16]王国俊,折延宏.二值命题逻辑中理论的发散性,相容性及其拓扑刻画[J].数学学报,2007,50(4):841-850.WANG Guojun,SHE Yanhong.Topological description of divergency and consistency of tw o-valued propositional theories[J].Acta Mathematica Sinica,2007,50(4):841-850.
[17]折延宏,王国俊.三值命题逻辑系统L*3中逻辑理论性态的拓扑刻画[J].数学学报,2009,52(6):1225-1234.SHE Yanhong,WANG Guojun.Topological characterizations of properties of logic theories in three-valued propositional logic system L*3[J].Acta Mathematica Sinica,2009,52(6):1225-1234.
[18]吴洪博.推理闭包算子及其诱导的空间[J].西北大学学报(自然科学版),2008,38(1):9-13.WU Hongbo.The reasoning closure operator w ith its induced space[J].Journal of Northw est University(Natural Science Edition),2008,38(1):9-13.
[2]PAVELKA J.On fuzzy logic I:many-valued rules of inference,II:enriched residuated lattice and semantics of propositional calculi,III:semantical completeness of some many-valued propositional calculi[J].Zeitschrf M ath Logik und Grundlagender M ath,1979,25:45-52;119-134;447-464.
[3]徐扬.格蕴涵代数[J].西南交通大学学报,1993,89(1):20-27.XU Yang.Lattice implication algebras[J].Journal of Southw est Jiaotong University,1993,89(1):20-27.
[4]XU Yang,RUAN Da,QIN Keyun,et al.Lattice-valued logic[M].Heidelberg:Springer-Verlag,2003.
[5]王国俊.非经典数理逻辑与近似推理[M].2版.北京:科学出版社,2008.WANG Guojun.Nonclassical mathematical logic and approximate reasoning[M].2nd.Beijing:Science Press,2008.
[6]张小红.模糊逻辑及其代数分析[M].北京:科学出版社,2008.ZHANG Xiaohong.Fuzzy logic and its algebra analysis[M].Beijing:Science Press,2008.
[7]Hjek P.Metamathematics of fuzzy logic[M].Dordrecht:Kluwer Academic Publishers,1998.
[8]吴洪博,石慧君.Heyting代数及其H-空间化表示形式[J].电子学报,2012,40(5):995-999.WU Hongbo,SHI Huijun.Heyting system and its representation by H-spatilization[J].Acta Electronica Sinica,2012,40(5):995-999.
[9]吴洪博.基础R0代数与基础L*系统[J].数学进展,2003,32(5):565-576.WU Hongbo.Basis R0-algebra and basis L*system[J].Advance in M athematics,2003,32(5):565-576.
[10]吴望名.Fuzzy蕴涵代数[J].模糊系统与数学,1990,4(1):56-64.WU Wangming.Fuzzy implication algebras[J].Fuzzy Systems and M athematics,1990,4(1):56-64.
[11]刘春辉.Heyting代数的模糊滤子格[J].山东大学学报(理学版),2013,48(12):57-60.LIU Chunhui.Lattice of fuzzy filter in a Heyting algebra[J].Journal of Shandong University(Natural Science),2013,48(12):57-60.
[12]吴洪博,汪宁.基于正则FI代数的MT理想及其应用[J].电子学报,2013,41(7):1389-1394.WU Hongbo,WANG Ning.M T ideals of regular FI algebra w ith their applications[J].Acta Electronica Sinica,2013,41(7):1389-1394.
[13]吴洪博,寇海燕.余Frame范畴中余积的构造[J].数学学报(中文版),2016,59(1):75-90.WU Hongbo,KOU Haiyan.The construction of Co-Product in the category of Co-Frame[J].Acta M athematica Sinica(Chinese Series),2016,59(1):75-90.
[14]TURUNEN E.Mathematics behind fuzzy logic[M].Heidelberg:Physica-Verlag,1999.
[15]刘春辉.有界Heyting代数的模糊LI-理想[J].工程数学学报,2016,33(4):391-401.LIU Chunhui.Fuzzy LI-ideals in bounded Heyting algebras[J].Chinese Journal of Engineering M athematics,2016,33(4):391-401.
[16]王国俊,折延宏.二值命题逻辑中理论的发散性,相容性及其拓扑刻画[J].数学学报,2007,50(4):841-850.WANG Guojun,SHE Yanhong.Topological description of divergency and consistency of tw o-valued propositional theories[J].Acta Mathematica Sinica,2007,50(4):841-850.
[17]折延宏,王国俊.三值命题逻辑系统L*3中逻辑理论性态的拓扑刻画[J].数学学报,2009,52(6):1225-1234.SHE Yanhong,WANG Guojun.Topological characterizations of properties of logic theories in three-valued propositional logic system L*3[J].Acta Mathematica Sinica,2009,52(6):1225-1234.
[18]吴洪博.推理闭包算子及其诱导的空间[J].西北大学学报(自然科学版),2008,38(1):9-13.WU Hongbo.The reasoning closure operator w ith its induced space[J].Journal of Northw est University(Natural Science Edition),2008,38(1):9-13.