形式演绎系统L~*中封闭理论的性质及其应用

本文在模糊命题演算的形式演绎系统L~*中引入了封闭理论的概念,讨论了封闭理论的基本性质,并利用封闭理论给出了形式演绎系统L~*的基于公式集的完备性的证明.首先,在形式演绎系统L~*中引入了封闭理论的概念,给出了理论封闭化扩张的方法;其次,在形式演绎系统L~*中引入了完全封闭理论的概念,证明了满足相关条件的完全封闭理论的存在性;第三,对形式演绎系统L~*中的封闭理论确定的同余关系性质进行了讨论,在公式集中引入了强同余关系的概念,给出了封闭理论和强同余关系相互决定的方法;第四,在形式演绎系统L~*中证明了封闭理论型L~*-Lindenbaum代数是R_0代数,并且封闭理论型L~*-Lindenbaum代数是全序的当且仅当封闭理论是完全的;最后,利用完全封闭理论型L~*-Lindenbaum代数完成了形式系统L~*完备性的证明,并改进了原有的结果.

The Properties of Closed Theories with Its Application in the Formal Deductive System L*

In the Formal Deductive System L^* of Fuzzy Propositional Calculus, the notion of closed theory is introduced and its properties are investigated. Furthermore, the completeness of Formal Deductive System L^* is proved through closed theory based on formula set F(S). At first, in the formal deductive system L^*, a concept of closed theory is introduced, and a method for extending theories to closed theories is given; at second, in the formal deductive system L^*, a concept of total closed theory is introduced, and the existence of a total closed theory satisfying relevant conditions is proved; at third, in the formal deductive system L^*, the properties of congruence relations determined by closed theories are investigated, a concept of strong congruence relations is introduced to formulas set F(S), and methods of changing each other between strong congruence relations and closed theories are revealed; at fourth, in the formal deductive system L^*, it is proved that closed theory style L^*-Lindenbaum algebras determined by closed theories are R₀-algebras, and a closed theory-L^*-Lindenbaum algebra is linear if and only if a closed theory is total; at last, the completeness of the formal deductive system L^* is accomplished by making use of total closed theory style L^*-Lindenbaum algebras, and the results obtained before have been improved.