容纳矛盾逻辑系统与悖论

分析了目前各种容纳矛盾逻辑系统的不足,提出了正域、反域、不动域的概念,进而发现悖论是逻辑思维领域的不动点,建立了一个容纳矛盾的逻辑系统S,并给出了系统S的语义模型,证明了系统S的元定理.在系统S中,命题演算被分成3个独立的域,正域、反域中所有经典逻辑的定理与演算模式都是有效的;不动域是一个包含矛盾的域,在不动域中,可以证明悖论是一个定理.系统S与Da Costa的次协调逻辑系统Cn相比较,它不但可以容纳矛盾,并且可以把矛盾解释清晰.以此逻辑系统为基础,可以建立一个容纳矛盾的数学基础.

A logic system which accommodates contradictions and paradoxes

The insufficiencies of various currently used logic systems which accommodate contradictions were analyzed in this paper.The concepts of a positive field,inverse field,and fixed field were put forward,and the paradox was found to be the fixed point in the field of logical thinking.A new logic system S accommodating the contradictions and a semantic model of the system S were built,and then the metatheorem of system S was proven.In the system S,the propositional calculus was divided into three separate fields.All the schemes of calculus and theorems of classical logic were valid in the positive or inverse fields.The paradox,including the contradiction nature,could be proven as a theorem in the fixed field.Compared with the Da Costa paraconsistent logic system Cn,the system S can not only accommodate the contradictions,but also interpret them clearly.Based on this logic system,a foundation of mathematics which accommodates contradictions can be established.